2.1. Quasicontinuum model

We consider the dispersive quasicontinuum model (modified regularized Boussinesq equation)

(2.1)\begin{equation} u_{tt}-\dfrac{1}{12}u_{xxtt}=(f(u_x))_x, \end{equation}

which approximates the FPU problem

(2.2)\begin{equation} \ddot{u}_n=f(u_{n+1}-u_n)-f(u_n-u_{n-1}) \end{equation}

for a chain of masses with displacements $u_n(t)$ and interaction force $f(w)$. This can be shown [Reference Rosenau33] by rewriting (2.2) in terms of strain variables $w_n=u_n-u_{n-1}$, using a Padé approximation of the discrete operator and integrating the resulting PDE to obtain the displacement formulation (2.1). In what follows, we consider the cubic interaction force

(2.3)\begin{equation} f(w)=w+\alpha w^2+\beta w^3, \end{equation}

which corresponds to the $\alpha$- $\beta$-FPU problem (2.2).

Observe that if $\beta \gt \alpha^2/3$ or $\beta \lt 0$, the change of variables

\begin{equation*} u(x,t) = \left ( 1-\frac{\alpha^2}{3\beta} \right )^{1/2} y(x,\tau) - \frac{\alpha}{3\beta} x, \quad \tau = \left ( 1-\frac{\alpha^2}{3\beta} \right )^{1/2}t, \end{equation*}

yields the reduced equation

\begin{equation*} y_{\tau\tau} - y_{xx} - \beta\left (y_x^3 \right )_x - \frac{1}{12} y_{xx\tau\tau} = 0 \end{equation*}

that involves no quadratic term in the interaction force. Because of this, we can, in the aforementioned regimes, set $\alpha = 0$. The additional rescaling $\sqrt{|\beta|} y \to y$ allows one to take $\beta = \pm 1$ without loss of generality. Similarly, if $\beta=0$ but $\alpha$ is nonzero (quadratic interaction), the rescaling $\alpha u \to u$ means that we can set $\alpha = 1$. This change of variables is reminiscent of the transformation of solutions of the Gardner equation to solutions of the modified Korteweg-de Vries or Korteweg-de Vries equations [Reference Kamchatnov, Kuo, Lin, Horng, Gou, Clift, El and Grimshaw26]. In what follows, we will proceed with $\alpha,\beta \in \mathbb{R}$ and either $\beta \gt \alpha^2/3$, $\beta \lt 0$, or $\beta = 0$ and then restrict to $\alpha = 0$, $\beta = \pm 1$ and $\alpha= 1$, $\beta=0$ when convenient.

Linearizing (2.1) about the uniformly deformed state by setting $u(x,t) = x \overline{w} + \epsilon e^{i(kx-\omega_0t)}$ and taking the limit $\epsilon \to 0$, we obtain the linear dispersion relation

(2.4)\begin{align} \omega_0^2(k,\overline{w}) = \frac{k^2 c_s^2(\overline{w})}{1+k^2/12} \lt 12 c_s^2(\overline{w}), \end{align}

where

(2.5)\begin{align} c_s(\overline{w})=(f'(\overline{w}))^{1/2}=(1+2\alpha \overline{w}+3\beta \overline{w}^2)^{1/2} \end{align}

is the sound (long-wave) speed.

2.2. Conservation laws

Equation (2.1) corresponds to the Lagrangian density

(2.6)\begin{align} \mathbb{L}=\dfrac{1}{2}v^2-\phi(w)+\dfrac{1}{24}w_t^2, \end{align}

where $w=u_x$ is the strain, $\phi(w)$ is the elastic energy density determining the force $f(w) = \phi'(w)$, and $v=u_t$ is the particle velocity. The regularized Boussinesq equation (2.1) can be written as the first-order system in conservative form

(2.7)\begin{align} w_t - v_x = 0, \quad v_t-\left (f(w) + \dfrac{1}{12}w_{tt} \right )_x =0. \end{align}

Translational invariance of $\mathbb{L}$ implies two additional conservation laws that are equivalent for smooth solutions [Reference Gavrilyuk, Nkonga, Shyue and Truskinovsky21]. The first one is the energy balance,

(2.8)\begin{align} \dfrac{\partial}{\partial t}\left(\phi(w)+\dfrac{v^2}{2}+\dfrac{1}{24}w_t^2\right)-\dfrac{\partial}{\partial x}\left[\left(f(w)+\dfrac{1}{12}w_{tt}\right)v\right]=0, \end{align}

where $\phi(w)+\frac{v^2}{2}+\frac{1}{24}w_t^2$ is the energy density, and $-(f(w)+\frac{1}{12}w_{tt})v$ is the energy flux. The second (Bernoulli–Eshelby) conservation law is

(2.9)\begin{align} \dfrac{\partial}{\partial t}\left(wv+\dfrac{1}{12}w_x w_t\right)-\dfrac{\partial}{\partial x}\left(w f(w)+\dfrac{1}{12}w w_{tt}+\dfrac{1}{2}v^2-\phi(w)+\dfrac{1}{24}w_t^2\right)=0, \end{align}

where $w v+\frac{1}{12}w_x w_t$ is the momentum density, and $\phi(w)-w f(w)-\frac{1}{24}w_t^2-\frac{1}{2}v^2-\frac{1}{12}w w_{tt}$ is the momentum flux.

2.3. Averaged conservation laws

As shown in Section 3, under certain conditions, there exists a four-parameter family of periodic travelling wave solutions

(2.10)\begin{align} w = w(\theta;\mathbf{r}), \quad v = v(\theta;\mathbf{r}), \end{align}

of (2.7) with modulated phase

\begin{align*} \theta = S(X,T)/\epsilon, \quad \theta_x = S_X = k(X,T), \quad \theta_t = S_T = -\omega(X,T), \end{align*}

and parameters $\mathbf{r} = (r_1,r_2,r_3,r_4)$ that are allowed to vary on slow spatio-temporal scales $X = \epsilon x$, $T = \epsilon t$, $0 \lt \epsilon \ll 1$: $\mathbf{r} = \mathbf{r}(X,T)$. The nonlinear dispersion relation yields $k(X,T) = k(\mathbf{r}(X,T))$, $\omega(X,T) = \omega(\mathbf{r}(X,T))$. We can formally obtain the Whitham modulation equations by averaging three conservation laws (2.7), (2.8) over a $2\pi$ period according to

(2.11a)\begin{align} \overline{w}_T - \overline{v}_X = 0, \end{align}

(2.11b)\begin{align} \overline{v}_T - \overline{f(w)}_X = 0, \end{align}

(2.11c)\begin{align} \left ( \overline{\phi(w)} + \frac{1}{2}\overline{v^2} + \frac{\omega^2}{24} \overline{w_\theta^2} \right )_T - \left ( \overline{v f(w)} - \frac{\omega^2}{12}\overline{w_\theta v_\theta} \right )_X = 0 , \end{align}

accompanied by the conservation of waves

(2.11d)\begin{align} S_{XT} = S_{TX} \quad \iff \quad k_T + \omega_X = 0 . \end{align}

An additional conservation law, derivable from the previous set, is the average of momentum (2.9)

\begin{align*} \left(\overline{wv}-\dfrac{1}{12}k\omega \overline{w_\theta^2} \right)_T-\left(\overline{w f(w)}+ \dfrac{1}{2} \overline{v^2}- \overline{\phi(w)} - \dfrac{1}{24} \omega^2\overline{w_\theta^2} \right)_X =0. \end{align*}

$2\pi$

$F(\theta)$

\begin{align*} \overline{F} \equiv \frac{1}{2\pi} \int_0^{2\pi} F(\theta)\,\mathrm{d}\theta . \end{align*}

3.1. Case I: $\beta \gt 0$, two centres

As illustrated in Fig. 1(a), in this case there are two centres separated by a saddle point and thus two families of periodic orbits described by (3.5), with $w_1 \leq w \leq w_2$ (below the saddle point) and $w_3 \leq w \leq w_4$ (above it), respectively. The first family of orbits is enclosed by the homoclinic trajectory corresponding to a compressive (depression or dark) solitary wave, and the second by a homoclinic orbit corresponding to a tensile (elevation or bright) solitary wave.

To obtain the first family, we need to solve (3.5) for $w(\xi)$ in the interval $(w_1,w_2)$, where $w_1$ and $w_2$ must be real and distinct. Using separation of variables and an appropriate substitution to evaluate the integral involving $w$ [Reference Byrd and Friedman4], we obtain

(3.21)\begin{equation} \begin{split} w(\theta) &= \frac{w_1+w_4n\, \mathrm{sn}^2\left ( \frac{K(m)}{\pi} \theta; m \right )}{1 +n\,\mathrm{sn}^2\left ( \frac{K(m)}{\pi} \theta; m \right )}, \quad m = \frac{(w_4-w_3)(w_2-w_1)}{(w_4-w_2)(w_3-w_1)}, \quad n=\dfrac{w_2-w_1}{w_4-w_2}, \end{split} \end{equation}

where $\text{sn}$ is the Jacobi elliptic sine function and $K(m)$ is the complete elliptic integral of the first kind. The solution (3.21) is $2\pi$-periodic, $w(\theta + 2\pi) = w(\theta)$, in the phase $\theta = k\xi = kx - \omega t$ with wavenumber $k$ and nonlinear frequency $\omega$ given by

(3.22)\begin{equation} k = \frac{\pi}{K(m)} \sqrt{\frac{3\beta}{2c^2}(w_4-w_2)(w_3-w_1)}, \quad \omega=ck, \end{equation}

where we recall (3.8), the wavelength $L = 2\pi/k$, and the temporal period $T = 2\pi/\omega$. The physical parametrization of this solution is $(k,a,\overline{w})$, where the wavenumber $k$ in (3.22), the amplitude $a = w_2 - w_1$ and the mean

(3.23)\begin{equation} \overline{w}=\frac{1}{2\pi}\int_0^{2\pi} w(\theta)\,\mathrm{d}\theta = w_4-\dfrac{\Pi(-n,m)}{K(m)}(w_4-w_1) \end{equation}

are all determined in terms of the roots $w_1$, $w_2$, $w_3$ in view of (3.7). The function $\Pi(-n,m)$ is the complete elliptic integral of the third kind with $n$ given in (3.21).

Note that the solution (3.21) only requires $w_1$ and $w_2$ to be real and also works in the case when $w_3$ and $w_4$ are non-real complex conjugates. In this case, $m$ and $n$ are complex but the solution itself and $k$, $a$, $\overline{w}$ are real. An example of a periodic orbit with complex $w_{3,4}$ is the black trajectory in Fig. 1(a). In the case when $w_3$ and $w_4$ are real, the root ordering $w_1 \lt w_2 \lt w_3 \lt w_4$ implies that

(3.24)\begin{equation} n \gt 0, \quad 0 \lt m \lt 1. \end{equation}

If these roots are complex conjugates, $w_{3,4}=\rho \mp i\eta$, $\eta \gt 0$, (3.7) implies that $\rho=-(4\alpha/(3\beta)+w_1+w_2)/2$, and the ordering $w_2 \lt \rho$ thus means that $4\alpha + 3\beta (w_1 + 3 w_2) \lt 0$, which together with $w_1 \lt w_2$ yields

(3.25)\begin{equation} \begin{split} &\text{Re}(n)=-\dfrac{6\beta (w_2 - w_1) (4\alpha + 3\beta (w_1 + 3 w_2))}{36 \beta^2 \eta^2 + (4 \alpha + 3\beta (w_1 + 3 w_2))^2} \gt 0, \\ &\text{Im}(n)=-\dfrac{36 \beta^2 \eta (w_2 - w_1)}{36\beta^2 \eta^2 + (4\alpha + 3\beta (w_1 + 3 w_2))^2} \lt 0. \end{split} \end{equation}

As we will show below, in this case, we can obtain $m$ in terms of real and imaginary parts of $n$.

The bifurcation from real to complex conjugate roots occurs in the limit $w_3 \to w_4$ where $m \to 0$ so that the solution (3.21) is the trigonometric wave

(3.26)\begin{equation} w(\theta) = \frac{w_1 + w_4 n \sin^2(\theta/2)}{1 + n \sin^2(\theta/2)}, \end{equation}

whose wavenumber, amplitude and mean are

(3.27)\begin{equation} k = 2 \sqrt{\frac{3\beta}{2c^2}(w_4-w_2)(w_4-w_1)}, \quad a = w_2 - w_1, \quad \overline{w} = w_4 - \frac{w_4-w_1}{\sqrt{1+n}} . \end{equation}

Since $2w_4 = -4 \alpha/(3\beta) - w_1 - w_2$ by virtue of (3.7), the trigonometric wave (3.26) is a two-parameter family of solutions. An example orbit given by the trigonometric solution (3.26) is the magenta trajectory in Fig. 1(a).

In the limit $w_2 \to w_1$ in eq. (3.21), $m \to 0$, and the solution exhibits the following asymptotic behaviour:

\begin{equation*} w(\theta) = w_1 - \frac{1}{2}(w_2 -w_1)\cos\theta + \dots , \quad \theta = kx - \omega_0 t, \end{equation*}

where, upon utilizing (3.7), (3.8), (3.22) and (3.23), we obtain the linear dispersion relation (2.4) for small-amplitude (linear) waves propagating on the constant background $\overline{w} = w_1$.

In the limit $w_2 \to w_3$, $m \to 1$, and the solution (3.21) approaches the depression solitary wave

(3.28)\begin{equation} w(\xi) = \frac{w_1 + w_4 n\, \mathrm{tanh}^2(\xi/\ell)}{1 + n\, \mathrm{tanh}^2(\xi/\ell)} , \quad n = \frac{w_3-w_1}{w_4-w_3}, \quad \ell = \left (\frac{3\beta}{2c^2}(w_4-w_3)(w_3-w_1) \right )^{-1/2}, \end{equation}

with amplitude $a = w_3-w_1$ propagating on the background $\overline{w} = w_3$. An example of the corresponding trajectory is shown by the left cyan loop in Fig. 1(a). These depression solitary waves form a two-parameter family of solutions with velocity-amplitude relation

(3.29)\begin{equation} c^2 = c_s^2(\overline{w}) + \frac{1}{2} a^2 \beta-\frac{2}{3} a(\alpha + 3 \beta\overline{w}) . \end{equation}

Strict inequality $w_3 \lt w_4$ and (3.7) imply that these waves exist when

(3.30)\begin{equation} \overline{w} \lt -\dfrac{\alpha}{3\beta},~ a \gt 0 \quad \text{or} \quad a \gt 4\left(\overline{w}+\dfrac{\alpha}{3\beta}\right) \gt 0, \end{equation}

which means that $c^2 \gt c_s^2(\overline{w})$ in (3.29), and thus the waves are supersonic. Note that this implies that for $\overline{w} \gt -\alpha/(3\beta)$ the amplitude has a strictly positive lower bound.

In the limit $w_2 \to w_3 \to w_4$, we have $\ell \to \infty$, so that $\tanh^2(\xi/\ell) \sim (\xi/\ell)^2$ in eq. (3.28), and the algebraically decaying depression solitary wave is obtained,

(3.31)\begin{equation} w(\xi) = \frac{2 w_1 - 3 \beta w_4 \left ( w_1(w_4+w_1) - (w_4-w_1)^2 \xi^2 \right )}{2 - 3 \beta \left ( w_4(w_4+w_1) - (w_4-w_1)^2 \xi^2 \right )}, \end{equation}

with amplitude $a = w_4 - w_1$ and background $\overline{w} = w_4$. By virtue of (3.7), this is a one-parameter family of algebraic solitary waves subject to the amplitude-background constraint $a = 4(\overline{w} + \alpha/(3\beta)) \gt 0$. These waves propagate with sound velocity,

(3.32)\begin{equation} c^2 = 1 + 3\beta\overline{w}(\tfrac{1}{2} a - \overline{w})=c_s^2(\overline{w}), \end{equation}

and thus represent the nontrivial sonic limit of (3.28) [Reference Vainchtein37]. The algebraic solitary wave solution (3.31) can alternatively be obtained by taking the limit $w_2 \to w_4$ in the trigonometric wave (3.26) because in this limit $k \to 0$ while $n \to \infty$, so that $n \sin^2(k\xi) \sim n (k\xi)^2 \to 6\beta(w_4-w_1)^2\xi^2/c^2$.

The second periodic travelling wave family (illustrated by the right set of orbits in Fig. 1(a)) solves (3.5) in $(w_3,w_4)$ with real and distinct $w_3$ and $w_4$. It is readily obtained from the first family by using the symmetry of the problem, which amounts to making the following replacements in (3.21):

(3.33)\begin{equation} \begin{split} &w(\theta) \to -w(\theta)-\dfrac{2\alpha}{3\beta},\\ &w_1 \to -w_4-\dfrac{2\alpha}{3\beta}, \quad w_2 \to -w_3-\dfrac{2\alpha}{3\beta}, \quad w_3 \to -w_2-\dfrac{2\alpha}{3\beta}, \end{split} \end{equation}

where the roots are renumbered to keep their nondecreasing order. The solution has the amplitude $a = w_4 - w_3$ and the same wavenumber (3.22), wavelength, frequency and period as the solutions (3.21). It leads to a two-parameter family of trigonometric solutions in the limit $w_2 \to w_1$, a two-parameter elevation solitary wave family in the limit $w_3 \to w_2$, and a one-parameter family of sonic algebraic elevation waves when $w_3 \to w_2 \to w_1$.

For simplicity, we now consider $(\alpha,\beta)=(0,1)$, recalling that one can always reduce the case $\beta \gt \alpha^2/3$ to these parameter values upon a proper rescaling, as discussed in Section 2.1. The periodic travelling wave solution (3.21) can be parameterized by $\mathbf{W}$ in (3.19), which involves the first three roots and the mean particle velocity. First, we consider the case when all four roots are real. In this case, (3.7) and (3.8) for $\alpha=0$ and $\beta=1$ imply

\begin{equation*} c^2=1+\dfrac{1}{4}\left((w_1+w_2)^2+(w_2+w_3)^2+(w_1+w_3)^2\right) \geq 1, \end{equation*}

so that $c$ is always real (eq. (3.10) holds), and we have $|c|\geq 1$. A second, useful parameterization involves the elliptic function parameters $n$, $m$, and the wave amplitude $a$:

(3.34)\begin{equation} \mathbf{A} = (n,m,a,\overline{v})^T. \end{equation}

The transformation $\mathbf{W} = \mathbf{W}(\mathbf{A})$ and its Jacobian determinant are

(3.35)\begin{equation} \begin{split} w_1 &= a \Xi(n,m)\gamma_1(n,m), \quad w_2 = a \Xi(n,m) \gamma_2(n,m), \quad w_3 = a \Xi(n,m) \gamma_3(n,m), \\ \Xi &=\frac{1}{4n(m+n)} , \quad \det \left (\frac{\partial \mathbf{W}}{\partial \mathbf{A}} \right ) = \frac{a^2(1+n)}{4n^2(m+n)^2} , \end{split} \end{equation}

where

\begin{equation*} \begin{split} \gamma_1 &= -3n^2 - 2(1+m)n - m, \quad \gamma_2 = n^2 - 2n(1-m) - m, \\ \gamma_3 &= n^2 + 2n(1-m) - m . \end{split} \end{equation*}

For $a \gt 0$ and (3.24), the transformation (3.35) is invertible. In these variables, the existence of periodic travelling waves is restricted by (3.24):

(3.36)\begin{equation} \mathbf{A} \in E_a = \left \{ (n,m,a,\overline{v})^T \in \mathbb{R}^4 \,|\, n \gt 0, ~0 \lt m \lt 1,~a \gt 0 \right \}. \end{equation}

Another parameterization results by transforming $a$ in $\mathbf{A}$ to the wave frequency

(3.37)\begin{equation} \omega = \mathrm{sgn}(c)\frac{\pi a}{K(m)} \sqrt{\frac{3(1+n)}{2n(m+n)}} , \end{equation}

which is invertible when (3.36) holds. Similar transformations to $k$ and $\overline{w}$ can also be obtained.

For the case of complex conjugate roots $w_{3,4}=-\frac{1}{2}(w_1+w_2) \mp i\eta$, $\eta \gt 0$, it is convenient to use

(3.38)\begin{equation} \mathbf{\tilde{A}}=(n_r,n_i,a,\overline{v})^T \end{equation}

as the parametrisation, where $n_r=\text{Re}(n) \gt 0$ and $n_i=\text{Im}(n) \lt 0$ by (3.25). This yields the transformation $\mathbf{W}(\mathbf{\tilde{A}})$

(3.39)\begin{equation} \begin{split} w_1=-\dfrac{a}{4}\left(3 + \dfrac{2 n_r}{n_i^2 + n_r^2}\right), \quad &w_2=\dfrac{a}{4}\left(1 - \dfrac{2 n_r}{n_i^2 + n_r^2}\right), \quad w_3=\dfrac{a}{4}\left(1 + \dfrac{2 n_r}{n_i^2 + n_r^2}\right)+i\dfrac{a n_i}{n_i^2 + n_r^2}, \\ &\det \left (\frac{\partial \mathbf{W}}{\partial \mathbf{\tilde{A}}} \right ) = \frac{i a^2}{2(n_i^2 + n_r^2)^2}, \end{split} \end{equation}

that is invertible for $a \gt 0$ with

\begin{equation*} m=\dfrac{2 n_i}{n_i + i(1 + n_r)}. \end{equation*}

Recalling (3.25) and (3.10), we observe that periodic travelling waves exist when

(3.40)\begin{equation} \begin{split} &\mathbf{\tilde{A}} \in \tilde{E}_a = \bigg\{ (n_r,n_i,a,\overline{v})^T \in \mathbb{R}^4\,|\, n_r \gt 0, ~n_i \lt 0,~a \gt 0,\\ &\qquad\qquad\qquad 16 + a^2 \left(3 + \dfrac{4 (n_i^2 (n_r - 2) + n_r^2 (1 + n_r))}{(n_i^2 + n_r^2)^2}\right) \gt 0 \bigg\}. \end{split} \end{equation}

The transition from real roots to a pair of complex conjugate roots occurs when $w_3 \to w_4$, so that $m \to 0$. Passing to this limit in (3.35), (3.37) and (3.27) for $(\alpha,\beta) = (0,1)$, we can eliminate $n$ and identify the crossover dispersion relation

\begin{equation*} \omega^2 = k^2 \frac{48 + 9a^2}{48-2k^2}, \quad 0 \le k \lt 2 \sqrt{6} , \end{equation*}

holding for the trigonometric solution (3.26) with wave speed satisfying $c^2 = 1 + \frac{3}{16} a^2 + \frac{1}{24} \omega^2$.

3.2. Case II, β < 0, single centre

In this case, as shown in Fig. 1(b), there are two saddle points separated by a centre, and hence periodic orbits satisfy (3.5) in $(w_2,w_3)$, where $w_2$ and $w_3$ are real and distinct. One family of periodic travelling waves is given by [Reference Chong, Herrmann and Kevrekidis6]

(3.41)\begin{equation} w(\theta)=\dfrac{w_2-w_1n\,\text{sn}^2\left (\frac{K(m)}{\pi}\theta;m \right )} {1-n\, \text{sn}^2 \left (\frac{K(m)}{\pi} \theta;m \right )}, \quad m=\dfrac{(w_4-w_1)(w_3-w_2)}{(w_4-w_2)(w_3-w_1)} , \quad n = \frac{w_3-w_2}{w_3-w_1}, \end{equation}

where root ordering $w_1 \lt w_2 \lt w_3 \lt w_4$ implies

(3.42)\begin{equation} 0 \lt n \lt m \lt 1 . \end{equation}

The physical parameters $(k,a,\overline{w})$ are the wavenumber

(3.43)\begin{equation} k = \frac{\pi}{K(m)}\sqrt{\frac{3|\beta|}{2 c^2}(w_4-w_2)(w_3-w_1)}, \end{equation}

the non-negative amplitude $a = w_3 - w_2$, and the mean strain

(3.44)\begin{equation} \overline{w} = w_1+\dfrac{\Pi(n,m)}{K(m)}(w_2-w_1). \end{equation}

Examples of orbits computed using (3.41) are shown in Figure 1(b).

In the limit $w_3 \to w_2$, $m \to 0$, and

\begin{equation*} w(\theta) = w_2 - \frac{1}{2}(w_3-w_2)\cos(\theta) + \dots, \quad \theta = kx - \omega_0t, \end{equation*}

subject to the linear dispersion relation (2.4) with $\overline{w} = w_2$.

When $w_3 \to w_4$, $m \to 1$ and we obtain a two-parameter family of depression solitary waves [Reference Vainchtein and Truskinovsky36]

(3.45)\begin{equation} w(\xi) = \frac{w_2 - w_1 n\, \mathrm{tanh}^2(\xi/\ell)}{1 - n\, \mathrm{tanh}^2(\xi/\ell)} , \quad n = \frac{w_4-w_2}{w_4-w_1}, \quad \ell = \left ( \frac{3|\beta|}{2c^2}(w_4-w_2)(w_4-w_1) \right )^{-1/2} , \end{equation}

with amplitude $a = w_4 - w_2$, propagating on the background $\overline{w} = w_4$ with velocity-amplitude relation

(3.46)\begin{equation} c^2 = c_s^2(\overline{w}) + \frac{1}{2} \beta a^2 - \frac{2}{3}a(\alpha+3\beta\overline{w}). \end{equation}

An example of the corresponding homoclinic trajectory is shown by the cyan loop in Fig. 1(b). Using root ordering $w_1 \lt w_2 \lt w_4$, (3.7) and requiring that $c^2 \gt 0$ in (3.46), we obtain the existence conditions

(3.47)\begin{equation} 0 \lt a \lt 2 \left ( \overline{w} + \frac{\alpha}{3 \beta} \right ) , \quad \overline{w} + \frac{\alpha}{3\beta} \gt 0, \quad c_s^2(\overline{w}) + \frac{1}{2} \beta a^2 - \frac{2}{3}a(\alpha+3\beta\overline{w}) \gt 0. \end{equation}

Note that these conditions impose an upper bound on the amplitude of the solitary wave, as well as upper and lower bounds on $\overline{w}$, in contrast to the waves (3.28), whose amplitude can be arbitrarily large for any $\overline{w}$ but may have a strictly positive lower bound (recall (3.30)). We also remark that the first two conditions in (3.47) imply that the solitary waves are supersonic, i.e., $c^2 \gt c_s^2(\overline{w})$ in (3.46), and thus the third inequality is automatically satisfied when $c_s^2(\overline{w}) \geq 0$. However, this inequality must be imposed when $c_s^2(\overline{w}) \lt 0$ to ensure that $c^2 \gt 0$.

Another family of periodic travelling waves is related to the first family via the symmetry transformation (3.33). One can show that the two families differ by a $\pi$ shift in phase $\theta$ and have the same mean, wave number and amplitude. In the limit $w_2 \to w_1$, the second family of periodic orbits approaches the two-parameter family of elevation solitary waves that can be obtained from (3.45) using (3.33).

The case in which $w_2 \to w_1$ and $w_3 \to w_4$ results in heteroclinic orbits described by supersonic kink solutions that are given by [Reference Vainchtein and Truskinovsky36]

(3.48)\begin{equation} w(\xi)=\dfrac{w_1+w_4}{2} \pm \dfrac{w_4-w_1}{2}\tanh(\xi/\ell), \quad \ell = \frac{2|c|}{\sqrt{6|\beta|}(w_4-w_1)}, \end{equation}

where

(3.49)\begin{equation} w_1+w_4=-\dfrac{2\alpha}{3\beta}, \quad c^2=1-\dfrac{\alpha^2}{3\beta}+\beta\left(w_1+\dfrac{\alpha}{3\beta}\right)^2. \end{equation}

This is a one-parameter family of kink and antikink solutions. If the plus sign is chosen in (3.48), $w(\xi) \to w_4$ as $\xi \to \infty$ and $w(\xi) \to w_1$ as $\xi \to -\infty$ (an antikink). With the minus sign, we have $w(\xi) \to w_1$ as $\xi \to \infty$ and $w(\xi) \to w_4$ as $\xi \to -\infty$ (a kink). The amplitude $a=w_4-w_1$ of the kink is the upper bound (3.47) of the solitary wave amplitude for solutions (3.45) and their symmetric counterparts. When $(\alpha,\beta) = (0,-1)$, we have $w_1=-w_4$, $c^2=1-w_1^2=1-w_4^2$ by (3.49), and the solution (3.48) simplifies to

\begin{equation*} w(\xi) = \pm w_4 \,\mathrm{tanh} \left ( w_4 \sqrt{\frac{6}{1-w_4^2}} \ \xi \right ). \end{equation*}

Examples of periodic orbits approaching the heteroclinic ones are shown in Fig. 1(c). The two families of periodic solutions yield the same trajectories. Near the heteroclinic orbits, the solution acquires the flat-top structure of a periodic kink-antikink lattice shown in Fig. 1(d).

In what follows, we consider the case $(\alpha,\beta)=(0,-1)$ without loss of generality. The periodic travelling wave solution (3.41) can be parameterized by $\mathbf{W}$ in (3.19) subject to the existence inequalities (3.9) and (3.10). As before, we can also introduce the alternative parameterization $\mathbf{A}$ in (3.34) involving the amplitude $a$.

The transformation $\mathbf{W} = \mathbf{W}(\mathbf{A})$ and its Jacobian determinant are

(3.50)\begin{equation} \begin{split} w_1 &= a \Xi(n,m)\gamma_1(n,m), \quad w_2 = a \Xi(n,m) \gamma_2(n,m), \quad w_3 = a \Xi(n,m) \gamma_3(n,m), \\ \Xi &=\frac{1}{4n(m-n)} , \quad \det \left (\frac{\partial \mathbf{W}}{\partial \mathbf{A}} \right ) = \frac{a^2(1-n)}{4n^2(m-n)^2} , \end{split} \end{equation}

where

\begin{equation*} \begin{split} \gamma_1 &= -n^2 + 2(1+m)n - 3m, \quad \gamma_2 = 3n^2 - 2n(1+m) + m, \\ \gamma_3 &= -n^2 - 2n(1-m) + m . \end{split} \end{equation*}

When $a \gt 0$ and (3.42) hold, the transformation (3.50) is invertible. Periodic travelling wave existence is subject to (3.42) and a real wave speed requiring (3.10):

\begin{equation*} \begin{split} \mathbf{A} \in E_a &= \bigg \{ (n,m,a,\overline{v})^T \in \mathbb{R}^4 \,|\, 0 \lt n \lt m \lt 1, ~ 0 \lt a \lt a_{\rm max}(n,m)\bigg \}, \\ a_{\rm max} &= \frac{\sqrt{2}}{\Xi(n,m)P(n,m)}, \end{split} \end{equation*}

where

\begin{equation*} \begin{split} P(n,m) &= \sqrt{\gamma_1 \gamma_2 + \gamma_1 \gamma_3 + \gamma_2 \gamma_3-(\gamma_1 + \gamma_2 + \gamma_3)^2} \\ &= \sqrt{6 n^4 - 8(1+m)n^3 + 4(2+m+2m^2)n^2 -8 m(1+m)n + 6m^2}. \end{split} \end{equation*}

The amplitude upper bound $a_{\rm max}(n,m)$ is an increasing function of its arguments with $a_{\rm max}(0,0) = a_{\rm max}(0,m) = \lim_{n\to m}a_{\rm max}(n,m) = 0$ for $0 \lt m \lt 1$ and $\lim_{n \to 1}\lim_{m\to 1} a_{\rm max}(n,m) = 2$ so that all periodic travelling waves are constrained to have amplitudes in the interval $0 \lt a \lt 2$.

We can obtain other parameterizations by, for example, transforming $a$ in $\mathbf{A}$ to the wave frequency

\begin{equation*} \omega = \mathrm{sgn}(c)\frac{\pi a}{K(m)} \sqrt{\frac{3(1-n)}{2n(m-n)}} , \end{equation*}

which is invertible when (3.42) holds. Similar transformations to $k$ and $\overline{w}$ can also be obtained.

3.3. Case III: $\beta=0$, single family of periodic orbits

In this case, eq. (3.20) has a quadratic polynomial in the right-hand side, and we focus on the case of two real roots, corresponding to two equilibrium points $(w_*,0)$ in the phase plane. If $\alpha \gt 0$, there is a saddle point on the left, and a centre on the right, so that the homoclinic trajectory enclosing the periodic orbits corresponds to a tensile (bright) solitary wave (see Figure 2). If $\alpha \lt 0$, the centre is on the left, and the saddle is on the right so that the homoclinic trajectory corresponds to a compressive (dark) solitary wave. Recall that, in this case, the periodic orbits satisfy (3.13). For $\alpha \gt 0$, we have $w_2 \leq w \leq w_3$, and the solution is given by [Reference Kamchatnov25]

(3.51)\begin{equation} w(\theta)=w_3-(w_3-w_2)\text{sn}^2\left(\frac{K(m)}{\pi}\theta\right), \quad m=\dfrac{w_3-w_2}{w_3-w_1}, \end{equation}

Figure 2.

Periodic orbits for different values of $B$ at $\alpha=1$, $\beta=0$, $c=2.5$, $A=-8.5$ corresponding to (3.51) with $m=0.9225$, $a=4.6671$, $\overline{w}=2.8434$ (red), $m=0.66971$, $a=3.6995$, $\overline{w}=3.5976$ (green), $m=0.3994$, $a=2.2333$, $\overline{w}=4.0457$ (purple), $m=0.22765$, $a=1.2224$, $\overline{w}=4.1915$ (magenta). Manifolds associated with the saddle point (including the homoclinic orbit) are shown in cyan.

with amplitude $a=w_3-w_2$, wavenumber

(3.52)\begin{equation} k=\dfrac{\pi}{K(m)}\sqrt{\dfrac{2|\alpha|}{c^2}(w_3-w_1)} \end{equation}

and mean

(3.53)\begin{equation} \overline{w}=w_1+(w_3-w_1)\dfrac{E(m)}{K(m)}, \end{equation}

where $E(m)$ is the complete elliptic integral of the second kind. Example trajectories are shown in Fig. 2.

The harmonic limit when $w_3 \to w_2$, and thus $m \to 0$, is described by

\begin{equation*} w(\theta) = w_2 + \frac{1}{2}(w_3-w_2)\cos(\theta) + \dots, \quad \theta = kx-\omega_0t, \end{equation*}

with $\omega_0$ corresponding to the linear dispersion relation (2.4) on the background $\overline{w} = w_2$. In the limit $w_2 \to w_1$, $m \to 1$, and we obtain the two-parameter family of elevation solitary waves

\begin{equation*} w(\xi)=w_3-(w_3-w_1)\text{tanh}^2(\xi/l), \quad l=\left(\dfrac{2\alpha}{c^2}(w_3-w_1)\right)^{-1/2}, \end{equation*}

with amplitude $a=w_3-w_1$, propagating on the background $\overline{w}=w_1$ with velocity-amplitude relation

\begin{equation*} c^2=c_s^2(\overline{w})+\dfrac{2\alpha}{3}a, \end{equation*}

where we used (3.15). See the cyan loop in Fig. 2 for an example of the corresponding homoclinic orbit.

For $\alpha \lt 0$, we have $w_1 \leq w \leq w_2$, and the solution can be obtained from (3.51) using the symmetry transformation

\begin{equation*} w(\theta) \to -w(\theta), \quad w_3 \to -w_1, \quad w_2 \to -w_2, \quad w_1 \to -w_3, \end{equation*}

where again the roots are renumbered to keep their nondecreasing order. In the limit when $w_2 \to w_3$, it yields a family of depression solitary waves.

In what follows, we assume $\alpha = 1$. In addition to the parameterization $\mathbf{W}$ in (3.19) in terms of the roots, it is advantageous to introduce the parameterization

(3.54)\begin{equation} \overline{\mathbf{A}} = (m,a,\overline{v},\overline{w})^T . \end{equation}

The transformation $\mathbf{W} = \mathbf{W}(\overline{\mathbf{A}})$ and its Jacobian determinant are

\begin{equation*} \begin{split} w_1 &= \frac{m\overline{w} K(m) - aE(m)}{m K(m)}, \quad w_2 = \frac{m K(m)(\overline{w}-a) - a(E(m)-K(m))}{mK(m)}, \\ w_3 &= \frac{m\overline{w} K(m) - a(E(m)-K(m))}{mK(m)} , \quad \det\left ( \frac{\partial \mathbf{W}}{\partial \overline{\mathbf{A}}} \right ) = - \frac{a}{m^2} , \end{split} \end{equation*}

demonstrating invertibility for $a \gt 0$ and $m \gt 0$.

The positivity of the squared wave speed (3.15) implies that the mean strain (3.53) exhibits the lower bound

(3.55)\begin{equation} \overline{w} \gt \overline{w}_{\rm min} = a \frac{3 E(m)-(2-m)K(m)}{3 m K(m)} - \frac{1}{2} . \end{equation}

The minimum mean is a decreasing function of $m$, exhibiting the limits $\lim_{m \to 1} \overline{w}_{\rm min} = -\frac{1}{2} - \frac{a}{3}$, $\lim_{m \to 0} \overline{w}_{\rm min} = -\frac{1}{2} + \frac{\omega_0^2}{24}$. Thus, the minimum mean satisfies

(3.56)\begin{equation} -\frac{a}{3} \le \overline{w}_{\rm min} +\frac{1}{2} \le \frac{\omega_0^2}{24} . \end{equation}

The periodic travelling wave existence criteria are

(3.57)\begin{equation} \overline{\mathbf{A}} \in \overline{E}_a = \bigg \{(m,a,\overline{v},\overline{w})^T \in \mathbb{R}^4 \, | \, 0 \le m \le 1, ~ a \gt 0, ~ \overline{w} \gt \overline{w}_{\rm min} \bigg \} . \end{equation}

It first attains the value $\overline{w}_{\rm min} = -\frac{1}{2}$ when $m = m_* \approx 0.96115$ satisfying $3 E(m_*)-(2-m_*)K(m_*)$ = 0. That is, for $0 \lt m \lt m_*$, the periodic travelling wave existence criteria include $\overline{w} \gt \overline{w}_{\rm min}(m,a) \gt -\frac{1}{2}$. When $m_* \le m \le 1$, $\overline{w} \gt \overline{w}_{\rm min}(m,a) \ge -\frac{1}{2} - \frac{a}{3}$, i.e., the mean can fall below the value $-\frac{1}{2}$. It will be convenient for later analysis to express the admissible set $\overline{E}_a$ as the union of two (not disjoint) subsets

(3.58)\begin{equation} \begin{split} \overline{E}_a &= \overline{E}^{(a)}_a \cup \overline{E}^{(b)}_a , \\ \overline{E}^{(\rm a)}_a &= \bigg \{\overline{\mathbf{A}} \in \mathbb{R}^4 \, | \, 0 \lt m \lt 1, ~ 0 \lt a \lt a_{\rm max}(m,\overline{w}), ~ \overline{w} \gt -\frac{1}{2} \bigg \} , \\ \overline{E}^{(\rm b)}_a &= \bigg \{\overline{\mathbf{A}} \in \mathbb{R}^4 \, | \, m_* \approx 0.96115 \lt m \lt 1, ~ 0 \lt a , ~ \overline{w}_{\rm min}(m,a) \lt \overline{w} \bigg \} . \end{split} \end{equation}

Here, the upper bound on the amplitude of the periodic travelling wave is

(3.59)\begin{equation} a_{\rm max}(m,\overline{w}) = \begin{cases} 3 \left ( \frac{1}{2} + \overline{w} \right ) \frac{m K(m)}{3 E(m) - (2-m)K(m)}, & 0 \lt m \lt m_* , \\ \infty, & m_* \lt m \lt 1 . \end{cases} \end{equation}

Since

(3.60)\begin{equation} \omega = \mathrm{sgn}(c)\frac{\pi\sqrt{2a/m}}{K(m)}, \end{equation}

we can invertibly transform $a \to \omega$ in (3.54) within the admissible region.

When $a \to a_{\rm max}$ for $0 \lt m \lt m_*$, the periodic travelling wave velocity $c \to 0$ but the frequency $\omega$ remains finite so that the wavenumber $k \to \infty$. Consequently, this is the short-wavelength limit.

4.1. Averaged conservation laws revisited

Recall that the averaged conservation laws are given by (2.11). Averaging of a $2\pi$-periodic function $F(\theta)$ can now be expressed as

\begin{equation*} \overline{F} = \frac{1}{2\pi}\int_0^{2\pi}F(\theta)\,\mathrm{d}\theta = \frac{|c|k}{2\pi\sqrt{6|\beta|}} \oint \frac{F(w)}{\sqrt{|G_4(w)|}} \,\mathrm{d}w, \end{equation*}

in the case of the quartic potential $\phi(w)$. For example, taking three roots, say $w_1$, $w_2$, $w_3$ incorporating the constraint (3.7), and the averaged particle velocity $\overline{v}$ in (3.18) as the modulation parameters $\mathbf{W} = (w_1,w_2,w_3,\overline{v})$, eqs. (2.11) are four quasilinear modulation equations for $\mathbf{W}$. The first conservation law (2.11a) is $\overline{w}_T - \overline{v}_X = 0$, where $\overline{w}$ is defined in terms of the roots $w_j$ through (3.8) and one of (3.23) or (3.44) or (3.53), respectively. Using $\omega = ck$, eq. (3.8) and one of (3.22), (3.43), (3.52) for $k$, the conservation of waves (2.11d) gives another modulation equation. The average velocity (2.11b) and energy (2.11c) equations require further calculation of the density and fluxes. Rather than pursue this further, we instead develop the averaged variational principle.

4.2. Average Lagrangian

Writing the Lagrangian density (2.6) in terms of $u(x,t)$, we obtain

(4.1)\begin{equation} \mathbb{L}(u_t,u_x,u_{xt}) = \dfrac{1}{2}u_t^2 - \phi(u_x) + \dfrac{1}{24} u_{xt}^2, \quad \phi'(u_x)=f(u_x) \end{equation}

and note that it depends only on derivatives of $u(x,t)$. Furthermore, the periodic travelling wave solutions are obtained for $v = u_t$ and $w = u_x$. Consequently, the most general periodic travelling wave can be written as

(4.2)\begin{equation} u(x,t) = \overline{w} x + \overline{v} t + U(\theta), \quad U(\theta + 2\pi) = U(\theta), \quad \theta = kx - \omega t . \end{equation}

The term $\overline{w} x + \overline{v} t$ corresponds to the so-called pseudo-phase and enables $U(\theta)$ to be $2\pi$-periodic.

In the general case (2.3), the Lagrangian density (4.1) is

(4.3)\begin{equation} \mathbb{L}=\dfrac{1}{2}u_t^2-\dfrac{1}{2}u_x^2-\dfrac{1}{3}\alpha u_x^3-\dfrac{1}{4}\beta u_x^4+\dfrac{1}{24}u_{xt}^2. \end{equation}

Note that (4.2) implies that $u_t=\overline{v}-c(w(\theta)-\overline{w})$, $u_x=w(\theta)$ and $u_{xt}=-\omega w'(\theta)$, so that (4.3) can be written solely in terms of $w = w(\theta)$

\begin{equation*} \mathbb{L}=\dfrac{1}{2}\left(\overline{v}-c(w-\overline{w})\right)^2-\dfrac{1}{2}w^2-\dfrac{\alpha}{3}w^3-\dfrac{\beta}{4} w^4+\dfrac{1}{24}\omega^2(w')^2. \end{equation*}

Recalling that, in view of (3.4), we have

(4.4)\begin{equation} \dfrac{\omega^2(w')^2}{24}=-\dfrac{\beta}{4}w^4-\dfrac{\alpha}{3}w^3+\dfrac{1}{2}(c^2-1)w^2+\dfrac{1}{2}Aw+\dfrac{1}{2}B, \end{equation}

we arrive at

\begin{equation*} \mathbb{L}=\dfrac{1}{12}\omega^2(w')^2-\left(c^2\overline{w}+\overline{v} c+\dfrac{1}{2}A\right)w+\dfrac{1}{2}\overline{v}^2+\dfrac{1}{2}c^2\overline{w}^2+c\,\overline{v} \,\overline{w}-\dfrac{1}{2}B. \end{equation*}

Thus, the averaged Lagrangian density is given by

\begin{equation*} \mathcal{L}(\overline{w},\overline{v},k,\omega,A,B) = \frac{1}{2\pi} \int_0^{2\pi} \mathbb{L}(\theta)\,\mathrm{d}\theta =\dfrac{\omega^2}{12k}\mathcal{W}\left(\dfrac{\omega}{k},A,B\right)+\dfrac{1}{2}\overline{v}^2-\dfrac{\omega^2}{2k^2}\overline{w}^2-\dfrac{1}{2}A\overline{w}-\dfrac{1}{2}B, \end{equation*}

where

(4.5)\begin{equation} \begin{split} \mathcal{W}(c,A,B)&=\dfrac{k}{2\pi}\int_0^{2\pi}(w'(\theta))^2d\theta=\dfrac{1}{2\pi}\oint kw'(\theta)dw\\ &=\dfrac{1}{2\pi}\oint \left[\dfrac{12}{c^2}\left(-\dfrac{\beta}{2}w^4-\dfrac{2\alpha}{3}w^3+(c^2-1)w^2+Aw+B\right)\right]^{1/2}dw. \end{split} \end{equation}

Here, we used (4.4), and the integration in $w$ is around the periodic orbit.

The variational equations along with the consistency relations are [Reference Whitham39]

(4.6)\begin{equation} \begin{split} &\mathcal{L}_A=0, \quad \mathcal{L}_B=0,\\ &\dfrac{\partial}{\partial T}\mathcal{L}_\omega-\dfrac{\partial}{\partial X}\mathcal{L}_k=0, \quad k_T+\omega_X=0,\\ &\dfrac{\partial}{\partial T}\mathcal{L}_{\overline{v}}+\dfrac{\partial}{\partial X}\mathcal{L}_{\overline{w}}=0, \quad \overline{w}_T-\overline{v}_X=0. \end{split} \end{equation}

The first equation, $\mathcal{L}_A=0$, yields

\begin{equation*} \overline{w}=\dfrac{c\omega\mathcal{W}_A}{6}, \end{equation*}

which is automatically satisfied because, by (3.5) or (3.13) and (4.5), we have

\begin{equation*} \mathcal{W}_A=\begin{cases}\dfrac{6|c|}{2\pi c^2\sqrt{6|\beta|}}\oint\dfrac{w \ dw}{\sqrt{|G_4(w)|}}, & \beta \neq 0 \\ \dfrac{6|c|}{2\pi c^2\sqrt{8|\alpha|}}\oint\dfrac{w \ dw}{\sqrt{|G_3(w)|}}, & \beta=0, \end{cases} \end{equation*}

which in either case yields

\begin{equation*} \mathcal{W}_A=\dfrac{6}{2\pi c^2k}\int_0^{2\pi} wd\theta=\dfrac{6}{c\omega}\overline{w}. \end{equation*}

The second equation in (4.6), $\mathcal{L}_B=0$, implies that

\begin{equation*} k=\dfrac{6}{\mathcal{W}_B c^2}, \end{equation*}

which yields the dispersion relation

\begin{equation*} \omega^2=(ck)^2=\dfrac{6}{k \mathcal{W}_B}. \end{equation*}

We also have the identity

\begin{equation*} \mathcal{W}_c = -\frac{1}{c} \mathcal{W} + \frac{12}{\omega} \overline{w^2} . \end{equation*}

The remaining four equations in (4.6) yield

(4.7a)\begin{equation} \overline{w}_T - \overline{v}_X = 0, \end{equation}

(4.7b)\begin{equation} \overline{v}_T - (c^2\overline{w}+\dfrac{1}{2}A)_X = 0, \end{equation}

(4.7c)\begin{equation} \left(\dfrac{c}{12}\mathcal{W} + c\dfrac{\overline{w^2} - \overline{w}^2}{k} \right)_T +\left(c^2\dfrac{\overline{w^2} - \overline{w}^2}{k}\right )_X = 0 , \end{equation}

(4.7d)\begin{equation} k_T + \omega_X = 0 . \end{equation}

Here, (4.7a) and (4.7d) coincide with (2.11a) and (2.11d), respectively, while (4.7b) is equivalent to (2.11b) because, by (3.2), we have

\begin{equation*} c^2\overline{w}+\dfrac{1}{2}A=\overline{f(w)+\dfrac{\omega^2}{12}w''(\theta)}=\overline{f(w)}. \end{equation*}

Equation (4.7c) is the conservation of wave action, which makes (4.7) consistent with (2.11) but, as we will see, remains distinct from the remaining equations in the limit $k \to 0$, whereas eq. (2.11c) becomes redundant with the remaining modulation equations.

Two special limits are of interest. In the solitary-wave limit, we have $k \to 0$, and hence $\omega \to 0$, so that (4.7d) is trivially satisfied. Thus, in this limit, the modulation system consists of three equations (4.7a), (4.7b) and (4.7c) that will be analysed separately for each case. In the harmonic limit, we have $\overline{w^2} \to \overline{w}^2$ and $\mathcal{W} \to 0$, while $k$ is nonzero, making the third equation (4.7c) trivial. Thus, the modulation system (4.7) in this limit reduces to three equations, (4.7a), (4.7b) and (4.7d). These equations are given by

(4.8)\begin{equation} \begin{split} \overline{w}_T-\overline{v}_X&=0,\\ \overline{v}_T-c_s^2(\overline{w})\overline{w}_X&=0,\\ \omega_T+\mathrm{sgn}(\omega)\dfrac{(c_s^2(\overline{w})-\omega^2/12)^{3/2}}{c_s^2(\overline{w})}\omega_X -\dfrac{(\alpha+3\beta\overline{w})\omega}{c_s^2(\overline{w})}\overline{v}_X&=0, \end{split} \end{equation}

where we recall that $c_s(\overline{w})$ is the sound speed defined in (2.5). We recognize the first two equations as the dispersionless limit of (2.7), as expected. The third equation is found to be equivalent to the modulation equation for linear waves – conservation of waves (4.7d) with $\omega = \omega_0(k,\overline{w})$, the linear dispersion relation (2.4), which can be inverted to yield

\begin{equation*} k(\omega,\overline{w})=\dfrac{|\omega|}{\sqrt{c_s^2(\overline{w})-\omega^2/12}}, \end{equation*}

where we note that $\omega^2 \lt 12c_s^2(\overline{w})$ by (2.4). The coefficient of $\omega_X$ in this equation equals the group velocity $\mathrm{sgn}(\omega) /k_\omega$, while the coefficient of $\overline{v}_X$ is $k_{\overline{w}}/k_\omega$. In both the solitary wave and harmonic limits, the first two mean equations, (4.7a) and (4.7b), decouple from the third.

We will parameterize the Whitham equations (4.7) using a convenient set of modulation variables, i.e., one of $\mathbf{W}$ in (3.19), $\mathbf{A}$ in (3.34), $\mathbf{\tilde{A}}$ in (3.38), or $\overline{\mathbf{A}}$ in (3.54), depending on the situation. Note that (4.7) have the form $\mathbf{P}_T+\mathbf{Q}_X=0$, where $\mathbf{P}$ and $\mathbf{Q}$ are the density and flux vectors, respectively. For each parametrization $\mathbf{r}$, we can find $\mathbf{P}(\mathbf{r})$ and $\mathbf{Q}(\mathbf{r})$ and rewrite the Whitham modulation equations (4.7) in the hydrodynamic form

(4.9)\begin{equation} \mathbf{r}_T + \mathbb{A}(\mathbf{r})\mathbf{r}_X = \mathbf{0} , \end{equation}

where $\mathbb{A}(\mathbf{r}) = (\partial \mathbf{P}/\partial \mathbf{r})^{-1} \partial \mathbf{Q}/\partial \mathbf{r}$ is obtained using symbolic algebra in Mathematica.

The modulation system $\mathbf{r}_T + \mathbb{A}(\mathbf{r})\mathbf{r}_X = \mathbf{0}$ has four eigenvalue-eigenvector pairs $\{(\lambda_j,\mathbf{z}_j)\}_{j=1}^4$ satisfying

\begin{equation*} \mathbb{A}(\mathbf{r}) \mathbf{z}_j(\mathbf{r}) = \lambda_j(\mathbf{r}) \mathbf{z}_j(\mathbf{r}), \quad j = 1, 2,3, 4. \end{equation*}

We order the eigenvalues first by their real parts and, for complex conjugate eigenvalues, then by their imaginary parts. We say that the modulation system is strictly hyperbolic if $\lambda_j \le \lambda_{j+1} \in \mathbb{R}$ and $\lambda_j = \lambda_{j+1}$ if and only if the $j^{\rm th}$ and $(j+1)^{\rm th}$ modulation equation are equivalent. Here we use an extended definition of strict hyperbolicity (see, e.g., [Reference Levermore30]) in which merged characteristic velocities correspond to a reduction in order of the modulation system. For example, this is precisely the case in the solitary wave and harmonic limits of (4.7), where either the conservation of waves (4.7d) or conservation of wave action (4.7c) are identically zero in the respective limits. Both limits thus maintain strict hyperbolicity of the modulation system provided that $c_s^2(\overline{w})=f'(\overline{w}) \gt 0$. The $j^{\rm th}$ characteristic field is said to be genuinely nonlinear if [Reference Lax29]

(4.10)\begin{equation} \mu_j(\mathbf{r}) = \nabla_{\mathbf{r}} \lambda_j(\mathbf{r}) \cdot \mathbf{z}_j(\mathbf{r}) \ne 0 , \quad j = 1,2,3,4. \end{equation}

Together, strict hyperbolicity and genuine nonlinearity characterize the convexity of the system.

In particular, the system of the first two mean equations in (4.8), which decouples from the remaining equation in the harmonic and solitary-wave limits, has eigenvalues $\lambda_{1,2}=\mp c_s(\overline{w})$ and the corresponding eigenvectors $\mathbf{z}_{1,2}=(1,\pm c_s(\overline{w}))^T$, yielding $\mu_{1,2}=\mp f''(\overline{w})/(2\sqrt{f'(\overline{w})})$. This subsystem is strictly hyperbolic as long as $c_s^2(\overline{w})=f'(\overline{w}) \gt 0$ and it is genuinely nonlinear when $f''(\overline{w}) \neq 0$.

Considering now the harmonic-limit system (4.8), we find that the eigenvalues are

\begin{equation*} \lambda_1=-c_s(\overline{w}), \quad \lambda_2= \mathrm{sgn}(\omega)\dfrac{(c_s^2(\overline{w})-\omega^2/12)^{3/2}}{c_s^2(\overline{w})}, \quad \lambda_3=c_s(\overline{w}), \end{equation*}

and the corresponding eigenvectors are

\begin{equation*} \begin{split} &\mathbf{z}_1=\left(1,c_s(\overline{w}),\dfrac{(\alpha+3\beta\overline{w})c_s(\overline{w})\omega}{\mathrm{sgn}(\omega) (c_s^2(\overline{w})-\omega^2/12)^{3/2}+c_s^3(\overline{w})}\right)^T, \quad \mathbf{z}_2=(0,0,1)^T, \\ &\mathbf{z}_3=\left(1,-c_s(\overline{w}),-\dfrac{(\alpha+3\beta\overline{w})c_s(\overline{w})\omega}{\mathrm{sgn}(\omega)(c_s^2(\overline{w})-\omega^2/12)^{3/2}-c_s^3(\overline{w})}\right)^T, \end{split} \end{equation*}

yielding

\begin{equation*} \mu_1=-\dfrac{f''(\overline{w})}{2\sqrt{f'(\overline{w})}}, \quad \mu_2=-\dfrac{(f'(\overline{w})-\omega^2/12)^{1/2}}{4f'(\overline{w})}|\omega|, \quad \mu_3=\dfrac{f''(\overline{w})}{2\sqrt{f'(\overline{w})}} . \end{equation*}

Since the harmonic wave frequency is bounded from above (2.4), the system is strictly hyperbolic when $c_s^2(\overline{w}) = f'(\overline{w}) \gt 0$ and genuinely nonlinear when $f''(\overline{w}) \neq 0$ and $\omega \neq 0$. In particular, in the case $(\alpha,\beta)=(1,0)$ considered below, the harmonic limit is convex (strictly hyperbolic and genuinely nonlinear) when $\overline{w} \gt -1/2$ and $\omega \neq 0$. For cubic nonlinearity with $(\alpha,\beta)=(0,1)$, the harmonic limit system is always strictly hyperbolic, while in the case $(\alpha,\beta)=(0,-1)$ strict hyperbolicity requires $|\overline{w}| \lt 1/\sqrt{3}$. In both cases, we must have $\omega \neq 0$ and $\overline{w} \neq 0$ for genuine nonlinearity.

4.3. Quadratic nonlinearity: $(\alpha,\beta) = (1,0)$

We now consider the special case of $(\alpha,\beta) = (1,0)$. Note that, if $\beta = 0$, we can set $\alpha = 1$ without loss of generality using the transformation $\alpha u \to u$, as noted in Section 2.1. We wish to express the Whitham eqs. (4.7) in terms of the parameter set $\mathbf{W} = (w_1,w_2,w_3,\overline{v})^T$. For this, we will need to calculate

\begin{equation*} \begin{split} \mathcal{W} &= \frac{1}{2\pi} \sqrt{\frac{8}{c^2}} \oint \sqrt{-G_3(w)}\,\mathrm{d} w = \frac{1}{\pi}\sqrt{\frac{8}{c^2}} \int_{w_2}^{w_3} \sqrt{-G_3(w)}\,\mathrm{d} w, \\ \overline{w^2} &= \frac{1}{2\pi} \int_0^{2\pi} w^2(\theta)\,\mathrm{d}\theta = \frac{1}{2\pi} \sqrt{\frac{\omega^2}{8}} \oint \frac{w^2}{\sqrt{-G_3(w)}}\,\mathrm{d}w = \frac{1}{\pi} \sqrt{\frac{\omega^2}{8}} \int_{w_2}^{w_3} \frac{w^2}{\sqrt{-G_3(w)}}\,\mathrm{d}w . \end{split} \end{equation*}

Evaluating the elliptic integrals, we obtain [Reference Byrd and Friedman4]

\begin{equation*} \mathcal{W} = \frac{4\sqrt{2}}{15\pi |c|} \frac{(w_3-w_2)(w_2-w_1)^2}{\sqrt{w_3-w_1}} \frac{2(1-m+m^2) E(m) - (2-m)(1-m)K(m)}{m(1-m)^2} \end{equation*}

and

\begin{align*} \begin{split} \overline{w^2} &= \frac{|\omega|}{3\sqrt{2}\pi(m-1)^2\sqrt{w_3-w_1}} \biggl\{2(w_2-w_1)\biggl ( (1-2m)w_1+(2-m)w_2 \biggr )E(m)\\ & + (m-1)\left((3m-2)w_1^2-2 w_1w_2 + w_2^2 \right)K(m)\biggr\}. \end{split} \end{align*}

Using equations (3.15), (3.16), (3.51), (3.52), (3.53) with $\alpha=1$, we obtain

(4.11)\begin{equation} \begin{split} A &= - \frac{2}{3}(w_1 w_2 + w_1 w_3 + w_2 w_3), \\ c^2 &= 1 + \frac{2}{3}(w_1+w_2+w_3) , \quad \omega = c k, \\ k &= \dfrac{\sqrt{2}\pi}{|c|K(m)}\sqrt{w_3-w_1}, \quad m = \frac{w_3-w_2}{w_3-w_1} , \\ \overline{w} &= w_1+(w_3-w_1)\dfrac{E(m)}{K(m)}. \end{split} \end{equation}

Consequently, we have expressed each density and flux of the Whitham equations (4.7) in terms of the parameters $\mathbf{W}$ in (3.19).

4.3.1. Solitary-wave limit

We now pass to the solitary-wave limit $w_2 \to w_1$ in the Whitham equations (4.7). The terms in the wave-action density eq. (4.7c) exhibit the limits

\begin{equation*} \begin{split} \lim_{w_2 \to w_1} \frac{c}{12} \mathcal{W} &= \frac{\sqrt{2}\mathrm{sgn}(c)}{2\pi} \frac{4}{45} (w_3-w_1)^{5/2}, \\ \lim_{w_2 \to w_1} c \frac{\overline{w^2} - \overline{w}^2}{k} &= \frac{\sqrt{2}\mathrm{sgn}(c)}{2\pi} \frac{2}{3} c^2(w_3 - w_1)^{3/2} , \\ \lim_{w_2 \to w_1} \overline{w} &= w_1 , \\ \lim_{w_2 \to w_1} \overline{w^2} &= w_1^2 , \\ \lim_{w_2 \to w_1} c^2 &= 1 + \frac{2}{3}(2 w_1 + w_3). \end{split} \end{equation*}

Since the solitary wave amplitude is $a = w_3 - w_1$ (cf. Section 3.3), we can express the above quantities in terms of $\overline{\mathbf{A}} =(1,a,\overline{v},\overline{w})^T$ to obtain

(4.12a)\begin{equation} \overline{w}_T - \overline{v}_X = 0 , \end{equation}

(4.12b)\begin{equation} \overline{v}_T - f(\overline{w})_X = 0 , \end{equation}

(4.12c)\begin{equation} \left ((1+2\overline{w})a^{3/2} + \tfrac{4}{5} a^{5/2} \right )_T + \left ( c \left ( (1+2\overline{w})a^{3/2} + \tfrac{2}{3} a^{5/2} \right ) \right )_X = 0 , \end{equation}

where

\begin{equation*} c^2 = 1 + 2\overline{w} + \frac{2}{3} a , \quad f(\overline{w}) = \overline{w} + \overline{w}^2 . \end{equation*}

Note that in eq. (4.12c), we have multiplied through by the inconsequential constant $2\pi/(\sqrt{2}\mathrm{sgn}(c))$ to simplify the expression. We recognize eqs. (4.12a) and (4.12b) as the dispersionless limit of eq. (2.7) and eq. (4.12c) is the equation for the solitary-wave amplitude field $a = a(X,T)$. Expanding eq. (4.12c) and replacing $\overline{w}_T = \overline{v}_X$, we obtain

(4.13)\begin{equation} a_T + c \,a_X + \frac{4 a}{3(1+2\overline{w}+\frac{4}{3}a)} \overline{v}_X + \frac{2 c a}{1+2\overline{w}+\frac{4}{3}a} \overline{w}_X = 0 . \end{equation}

Because it is decoupled from the other equations (4.12a), (4.12b), the characteristic velocity for (4.13) is $\lambda=c$, as it should be in the solitary-wave limit.

Equation (4.12) exhibits the ordered characteristic velocities (for the branch of fast waves with $c \gt 0$)

\begin{equation*} \lambda_1 = -\sqrt{1+2\overline{w}}, \quad \lambda_2 = \sqrt{1+2\overline{w}}, \quad \lambda_3 = c = \sqrt{1 + 2\overline{w} + \tfrac23 a} . \end{equation*}

The corresponding eigenvectors are

\begin{equation*} \mathbf{z}_1 = \begin{bmatrix} 1 \\ \sqrt{1+2\overline{w}} \\ - \frac{1+2\overline{w}+2a+\sqrt{(1+2\overline{w})(1+2\overline{w}+\frac{2}{3}a)}}{1+2 \overline{w} + \frac{4}{3}a} \end{bmatrix}, \quad \mathbf{z}_2 = \begin{bmatrix} 1 \\ \sqrt{1+2\overline{w}} \\ - \frac{1+2\overline{w}+2a-\sqrt{(1+2\overline{w})(1+2\overline{w}+\frac{2}{3}a)}}{1+2 \overline{w} + \frac{4}{3}a} \end{bmatrix}, \quad \mathbf{z}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} . \end{equation*}

4.3.2. Convexity

We first examine the convexity (hyperbolicity, genuine nonlinearity) of the Whitham equations in the solitary-wave limit (4.12). The mean lower bound $\overline{w} \gt -\frac{1}{2}-\frac{a}{3}$ for solitary waves (recall (3.56)) ensures that $\lambda_3 \in \mathbb{R}$ but it does not preclude the possibility of complex characteristic velocities $\lambda_1$ and $\lambda_2$. Thus, to ensure strict hyperbolicity, we further require $\overline{w} \gt -\frac{1}{2}$ and $a \gt 0$. To assess genuine nonlinearity, we compute

\begin{equation*} \begin{split} \mu_j &= \nabla \lambda_j \cdot \mathbf{z}_j = \partial_{\overline{w}} \lambda_j = \frac{1}{\lambda_j} \ne 0, \quad j = 1,2, \\ \mu_3 &= \nabla \lambda_3 \cdot \mathbf{z}_3 = \partial_a \lambda_3(a,\overline{w}) = \frac{1}{3 \lambda_3} \ne 0 . \end{split} \end{equation*}

Thus, the solitary-wave modulation system is genuinely nonlinear, provided it is strictly hyperbolic.

We now examine the convexity of the full Whitham system (4.7) by numerically computing the eigenvalues and eigenvectors of the modulation matrix $\mathbb{A}(\mathbf{r})$ in (4.9) as $\overline{\mathbf{A}}$ is varied. Recalling the admissibility criteria (3.57), (3.58), we will study convexity in two separate regimes:

1. (a) $\overline{E}_a^{(\rm a)}$: $\overline{w} \gt -\frac{1}{2}$, $0 \lt a \lt a_{\rm max}$, and $0 \lt m \lt 1$,

2. (b) $\overline{E}_a^{(\rm b)}$: $\overline{w}_{\rm min}(m,a) \lt \overline{w}$, $a \gt 0$, and $m_* \lt m \lt 1$,

where $\overline{w}_{\rm min}$ is given by (3.55), $a_{\rm max}$ is defined in (3.59) and $m_* \approx 0.96115$ satisfies $3\ E(m_*) - (2-m_*)K(m_*) = 0$. For case (a), we introduce the change of variables

\begin{equation*} \mathrm{(a)} \quad u \to (1+2\overline{w}) u + \overline{v} t + \overline{w} x, \quad t \to \frac{t}{\sqrt{1+2\overline{w}}}, \quad \overline{w} \gt -\frac{1}{2}, \end{equation*}

to eq. (2.1), leaving it unchanged since $\overline{w} \gt -\tfrac12$. Consequently, in this case, we will set $\overline{v} = \overline{w} = 0$ without loss of generality and consider variation of the remaining two modulation parameters $0 \lt m \lt 1$, $0 \lt a \lt a_{\rm max}(m,\overline{w} = 0)$. For case (b), we note that eq. (2.1) is invariant under the alternative change of variables

\begin{equation*} \mathrm{(b)} \quad u \to \frac{u}{a} + \overline{v} t + \left ( \frac{1}{2a} - \frac{1}{2} \right ) x, \quad t \to \sqrt{a} t, \quad a \gt 0, \end{equation*}

Now, we will set $a = 1$ without loss of generality and consider convexity when $m_* \lt m \lt 1$ and $\overline{w} \gt \overline{w}_{\rm min}(m,a=1)$.

Figure 3 depicts the (white) region of strict hyperbolicity when $\overline{\mathbf{A}} \in \overline{E}_a^{(\rm a)}$. We find that $\lambda_3 \lt \lambda_4$ are real and distinct for all computed admissible parameter values. There exists a greyscale region in which the modulations exhibit complex conjugate characteristic velocities $\lambda_1$ and $\lambda_2$. Such regions correspond to modulational instability of the underlying periodic travelling wave [Reference Whitham39]. At the boundaries of these regions, $\lambda_1 = \lambda_2 \in \mathbb{R}$. The coalescence of two characteristics coincides with loss of genuine nonlinearity, so $\mu_1 = \mu_2 = 0$ also. The smallest values of $m$ and $a$ where complex characteristics occur are depicted as black dots.

Figure 3.

Hyperbolicity of the regularized Boussinesq–Whitham equations for $(\alpha,\beta) = (1,0)$ (case III), quadratic interaction force with $\overline{\mathbf{A}} = (m,a,\overline{v}=0,\overline{w}=0) \in \overline{E}_a^{(a)}$: Strict hyperbolicity (white region). Greyscale is the magnitude of the imaginary part of $\lambda_2$ where the system is not hyperbolic. Periodic travelling waves do not exist in the inadmissible pink region. Black dots depict points corresponding to the values $(m,a) \in \{(0.83,2.69),~(0.89,2.27)\}$ in (a) and $(m,\omega) = (0.83,3.41)$ in (b) where complex characteristic velocities are first encountered. The solid black curves correspond to parameters where the second characteristic field loses genuine nonlinearity, $\mu_2 = \nabla \lambda_2 \cdot \mathbf{z}_2 = 0$, and $\lambda_1 = \lambda_2$ at the boundary of the non-hyperbolic region. Panels (a) and (b) are equivalent under the transformation (3.60).

Figure 3 Long description

The image A showing a contour plot of Im lambda 2 as a function of m and a. The x-axis label is m, with no unit specified, ranging from 0 to 1. The y-axis label is a, with no unit specified, ranging from 0 to 5. A vertical scale bar is labeled Im lambda 2, with no unit specified, ranging from 0 to 0.7. A boundary curve separates a large region where Im lambda 2 is shown as 0 from a region above the curve. The boundary passes through approximately (m equals 0, a equals 0), then rises with m. At m equals 0.2 the boundary is near a about 0.3. At m equals 0.4 the boundary is near a about 0.8. At m equals 0.6 the boundary is near a about 1.6. At m equals 0.8 the boundary is near a about 3.0. The boundary reaches the upper edge near m between 0.85 and 0.9. For m between about 0.8 and 1, a shaded lobe indicates Im lambda 2 greater than 0. The lobe spans roughly a from about 2.3 up to 5. Within this lobe, the darkest shading occurs near m close to 1 and a near 4 to 5, approaching the upper scale value 0.7. A closed contour loop is visible inside the lobe. The image B showing a contour plot of Im lambda 2 as a function of m and a. The x-axis label is m, with no unit specified, ranging from 0 to 1. The y-axis label is a, with no unit specified, ranging from 0 to 4. A vertical scale bar is labeled Im lambda 2, with no unit specified, ranging from 0 to 1.5. A boundary curve runs across the plot at a between about 3.4 and 3.6 for m from 0 to about 0.8, then bends upward for m above about 0.8 and approaches a near 4 as m approaches 1. For a below this boundary, most of the plotted area corresponds to Im lambda 2 shown as 0. For m close to 1, a shaded region indicates Im lambda 2 greater than 0 over a range from about 2.5 up to 4. The shading increases toward m equals 1, with the darkest values near m close to 1 and a near 4, approaching the upper scale value 1.5. A contour line within this region curves from about (m about 0.85, a about 3.6) toward (m close to 1, a about 2.7).

We also compute the quantities $\mu_j$ in (4.10) in the strictly hyperbolic region and find that only $\mu_2 = 0$ (i.e., $\mu_1 \ne 0$, so $\lambda_1 \ne \lambda_2$) on a curve bifurcating from $(a,m) = (0,0)$ or $(\omega,m) = (\sqrt{12},0)$. The curve is depicted in Fig. 3. Nonzero $\mu_j$ is a necessary monotonicity condition for the existence of a $j$-integral curve in which $\mathbf{r}'(\alpha) = \mathbf{z}_j(\mathbf{r}(\alpha))$ where $\alpha$ evolves along the characteristic $\frac{\mathrm{d}x}{\mathrm{d}t} = \lambda_j$: $\alpha_t + \lambda_j(\mathbf{r}(\alpha)) \alpha_x = 0$. Simple waves are used in the solution of the Riemann problem consisting of step initial conditions at the origin. The short-wave limit ( $k \to \infty$) occurs when $a$ approaches the pink admissibility boundary ( $a \to a_{\rm max}$). At this boundary, the modulation matrix $\mathbb{A}$ exhibits a double zero eigenvalue and two real, oppositely signed eigenvalues.

Figure 4 depicts hyperbolicity for case (b) where we normalize the amplitude to $a = 1$ and require $\overline{w} \gt \overline{w}_{\rm min}$, $m_* \le m \lt 1$. We observe that the system is hyperbolic when $\overline{w}$ is sufficiently large. The depicted results are consistent with the solitary-wave limit (4.12) that is strictly hyperbolic for $\overline{w} \gt -\frac{1}{2}$. The upper dashed curve in Figure 4 where $\lambda_1 = \lambda_2$ limits to $\overline{w} = -\frac{1}{2}$ as $m \to 1$ but the approach is very sharp due to the logarithmic approach of $\overline{w}$ to $-\frac{1}{2}$ because $K(m) \sim -1/\log(1-m)$ in the mean equation (4.11) as $m \to 1$. In a similar manner, the lower dashed curve in Fig. 4 at the boundary of the admissible region also sharply limits to $\overline{w} = \overline{w}_{\rm min}(a=1,m=1) = -\frac{5}{6}$ as $m \to 1$. This demonstrates that the region in Fig. 4 where hyperbolicity is lost bifurcates from the segment $-\frac{5}{6} \lt \overline{w} \lt -\frac{1}{2}$ for $m = 1$, which is denoted by the red segment.

Figure 4.

Hyperbolicity of the regularized Boussinesq–Whitham equations for case III, quadratic interaction force with $\overline{\mathbf{A}} = (m,a=1,\overline{v}=0,\overline{w}) \in \overline{E}_a^{(b)}$: Strict hyperbolicity (white region). Greyscale is the magnitude of the imaginary part of $\lambda_2$ where the system is not hyperbolic. At $m = m_* \approx 0.96115$, we find that for $\overline{w} \gt -0.298$, the system is strictly hyperbolic. The dashed curves correspond to the boundary of the non-hyperbolic region where $\lambda_1 = \lambda_2$, so that $\mu_1 = \mu_2 = 0$ there also. Periodic travelling waves do not exist in the inadmissible pink region. The red segment at $m = 1$ corresponds to the region of non-hyperbolicity of the solitary-wave limiting system (4.12).

4.4. Cubic nonlinearity with $(\alpha,\beta) = (0,1)$

We now consider the case of cubic $f(w)$ with $(\alpha,\beta)=(0,1)$, to which, as we discussed in Section 2.1, case I in Section 3.1 can be reduced upon a proper rescaling. As before, we need to compute $\mathcal{W}$ and $\overline{w^2}$ that are given by

\begin{equation*} \mathcal{W} = \frac{1}{2\pi} \sqrt{\frac{6}{c^2}} \oint \sqrt{-G_4(w)}\, \mathrm{d} w, \quad \overline{w^2} = \frac{1}{2\pi} \sqrt{\frac{\omega^2}{6}} \oint \dfrac{w^2}{\sqrt{-G_4(w)}}\, \mathrm{d} w. \end{equation*}

Recall that in this case we have two families of periodic solutions related by the symmetry transformation (3.33) with $\alpha=0$ and $\beta=1$. Thus, it suffices to consider the first solution family (3.21). We have

(4.14)\begin{equation} \begin{split} &w_1 + w_2 + w_3 + w_4 = 0,\\ & A = \frac{1}{2}(w_1w_2w_3 + w_1 w_2 w_4+w_1 w_3 w_4+w_2 w_3 w_4),\\ &c^2 = 1 - \frac{1}{2} ( w_1 w_2 + w_1 w_3 + w_1 w_4 + w_2 w_3 + w_2 w_4 + w_3 w_4 ),\\ &k = \frac{\pi}{K(m)} \sqrt{\frac{3}{2c^2}(w_4-w_2)(w_3-w_1)}, \quad \omega=ck,\\ &m = \frac{(w_4-w_3)(w_2-w_1)}{(w_4-w_2)(w_3-w_1)}, \end{split} \end{equation}

where we used (3.7), (3.8), (3.11), (3.21) and (3.22).

To find $\mathcal{W}$ and $\overline{w^2}$ we use

\begin{equation*} \begin{split} \mathcal{W} &= \frac{1}{\pi} \sqrt{\frac{6}{c^2}} \int_{w_1}^{w_2} \sqrt{(w-w_1)(w_2-w)(w_3-w)(w_4-w)}\, \mathrm{d} w, \\ \overline{w^2} &= \frac{1}{\pi} \sqrt{\frac{\omega^2}{6}} \int_{w_1}^{w_2} \dfrac{w^2\,\mathrm{d} w}{\sqrt{(w-w_1)(w_2-w)(w_3-w)(w_4-w)}}. \end{split} \end{equation*}

Evaluation of the elliptic integrals yields [Reference Byrd and Friedman4]

\begin{equation*} \begin{split} \mathcal{W}&=\dfrac{(w_2 - w_1)^2 (w_3-w_1)^{1/2} (w_4 -w_1)^2}{4|c| \sqrt{6}\pi n^3 (1 + n)^2 (n + m)^2 (w_4-w_2)^{3/2}} \bigg\{n \bigg(3 n^4 + 3 m^2 + 4 n^3 (1 + m)\\ &+ 4 n m (1 + m) + 2 n^2 (2 + m + 2 m^2)\bigg) E(m) \\ &- (n + m) \bigg(3 n^4 - 4 n^2 (m-1) - 3 m^2 - 6 n m^2 + 2 n^3 (2 + m)\bigg) K(m) \\ &+ 3 (n^2 - m) (2 n + n^2 + m) (n^2 + m + 2 n m) \Pi(-n, m)\bigg\} \end{split} \end{equation*}

and

(4.15)\begin{equation} \begin{split} \overline{w^2}&=\dfrac{\sqrt{2}w_4^2|\omega|}{\sqrt{3}\pi \sqrt{(w_3 - w_1) (w_4 - w_2)}}\bigg\{K(m) + 2\left(\dfrac{w_1}{w_4}-1\right)\Pi(-n,m)\\ &+ \dfrac{(w_4 -w_1)^2}{2 (1 +n) (n + m) w_4^2} \bigg(n E(m) - (n + m) K(m) \\ &+ \bigg(n^2 + 3 m + 2 n (1 + m) \bigg) \Pi(-n, m)\bigg)\bigg\}, \end{split} \end{equation}

where we recall that $m$ and $n$ are given in (3.21) and satisfy (3.24) and (3.25) in the case of real and complex $w_{3,4}$, respectively.

4.4.1. Solitary-wave limit

Recall that in the solitary-wave limit $w_2 \to w_3$ ( $m \to 1$), the solutions approach the two-parameter family (3.28) with amplitude $a=w_3-w_1$, background $\overline{w}=w_3$ and velocity (3.29). In this limit, we obtain

\begin{equation*} \begin{split} \lim_{w_2 \to w_3}\dfrac{c}{12}\mathcal{W}&=\dfrac{\text{sgn}(c)}{\pi\sqrt{6}}\bigg\{\dfrac{1}{6}\sqrt{a(a-4\overline{w})}(a^2 - 4 a \overline{w} + 6 \overline{w}^2)\\ &+(a -2 \overline{w})^2 \overline{w} \arctan\sqrt{\dfrac{a}{a - 4\overline{w}}}\biggl\},\\ \lim_{w_2 \to w_3}c\dfrac{\overline{w^2}-\overline{w}^2}{k}&=\dfrac{c^2\text{sgn}(c)}{\pi\sqrt{6}} \bigg\{\sqrt{a(a-4\overline{w})}+4\overline{w}\arctan\sqrt{\dfrac{a}{a - 4\overline{w}}}\bigg\},\\ \lim_{w_2 \to w_3}\overline{w^2} &= \overline{w}^2, \quad \lim_{w_2 \to w_3}c^2=1+3\overline{w}^2+\dfrac{1}{2}a(a-4\overline{w}), \end{split} \end{equation*}

where we recall that either $\overline{w} \lt 0$ and $a \gt 0$ or $a \gt 4 \overline{w} \gt 0$ by (3.30) with $\alpha=0$. In this limit, the Whitham modulation equations (4.7) with $f(w)=w+w^3$ become

(4.16)\begin{equation} \begin{split} \overline{w}_T-\overline{v}_X&=0,\\ \overline{v}_T-(1+3\overline{w}^2)\overline{w}_X&=0,\\ a_T+c\,a_X+c\,\dfrac{g(a,\overline{w})}{d(a,\overline{w})}\overline{w}_X+\dfrac{h(a,\overline{w})}{d(a,\overline{w})}\overline{v}_X&=0, \end{split} \end{equation}

where

(4.17)\begin{equation} \begin{split} g(a,\overline{w})&=-\sqrt{a(a-4\overline{w})}(2 + 4 a^2 - 19 a \overline{w} + 24 \overline{w}^2)\\ &+2 (2 + (a - 6 \overline{w}) (a - 4 \overline{w})) (a - 2 \overline{w})\arctan\sqrt{\dfrac{a}{a - 4 \overline{w}}},\\ h(a,\overline{w})&=-\sqrt{a(a-4\overline{w})} (2 + 4 a^2 - 19 a \overline{w} + 24 \overline{w}^2)\\ &+ (4 + 3 (a - 4 \overline{w})^2) (a - 2 \overline{w}) \arctan\sqrt{\dfrac{a}{a - 4 \overline{w}}},\\ d(a,\overline{w})&=\sqrt{a(a-4\overline{w})} (1 + 2 a^2 - 8 a \overline{w} + 9 \overline{w}^2) +6 (a - 2 \overline{w})^2 \overline{w} \arctan\sqrt{\dfrac{a}{a - 4 \overline{w}}}. \end{split} \end{equation}

The ordered characteristic velocities (for the fast waves with $c \gt 0$) are

(4.18)\begin{equation} \lambda_1 = -\sqrt{1+3\overline{w}^2}, \quad \lambda_2 = \sqrt{1+3\overline{w}^2}, \quad \lambda_3 = c = \sqrt{1+3\overline{w}^2+\dfrac{1}{2}a(a-4\overline{w})} . \end{equation}

The associated eigenvectors are

(4.19)\begin{equation} \mathbf{z}_j = \begin{bmatrix} 1 \\ -\lambda_j \\ \frac{\lambda_j h(a,\overline{w}) - \lambda_3 g(a,\overline{w})}{d(a,\overline{w})(\lambda_3 - \lambda_j)} \end{bmatrix} , \quad j = 1,2, \quad \mathbf{z}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} . \end{equation}

In the limit $w_2 \to w_3 \to w_4$ the solitary waves reduce to the one-parameter family (3.31) with background $\overline{w}=w_4$, amplitude $a=w_4-w_1=4\overline{w}$, where the second equality follows from the first equation in (4.14), and velocity (3.32), yielding

\begin{equation*} \begin{split} \lim_{w_2 \to w_3 \to w_4}\dfrac{c}{12}\mathcal{W}&=\dfrac{\text{sgn}(c)}{32\sqrt{6}}a^3, \quad \lim_{w_2 \to w_3 \to w_4}c\dfrac{\overline{w^2}-\overline{w}^2}{k}=\dfrac{c^2\text{sgn}(c)}{2\sqrt{6}}a,\\ \lim_{w_2 \to w_3 \to w_4}\overline{w^2} &= \overline{w}^2, \quad \lim_{w_2 \to w_3 \to w_4}c^2=1+\dfrac{3}{16}a^2=c_s^2(\overline{w}). \end{split} \end{equation*}

In this case, the third equation in (4.16) decouples from the first two and becomes $a_T+c\,a_X=0$, which for $c \gt 0$ is equivalent to the fast characteristic family in the first two equations of (4.16), i.e., $\lambda_2 = \lambda_3$. Thus, the algebraic solitary wave (3.31) with $c=c_s(\overline{w}) \gt 0$ ( $c=-c_s(\overline{w}) \lt 0$) moves with the fast (slow) characteristic velocity of mean evolution.

4.4.2. Convexity

Here, we examine the convexity of the Whitham modulation equations using various means. First, we consider convexity of the solitary-wave modulation system (4.16) analytically and then proceed to assess the convexity of the full modulation equations (4.7) by numerically evaluating the eigenstructure of the modulation matrix $\mathbb{A}$ in (4.9).

The solitary-wave existence conditions $\overline{w} \lt 0$ and $a \gt 0$ or $a \gt 4\overline{w} \gt 0$ in eq. (3.30) with $\alpha=0$ for the first solitary-wave family (3.28) imply real characteristic velocities (4.18). When $a = 4 \overline{w}$, either $\lambda_2 = \lambda_3$ (for $c \gt 0$) or $\lambda_1 = c = \lambda_2 = -\sqrt{1+3\overline{w}^2}$ (for $c \lt 0$). This case corresponds to the algebraic solitary wave, whose modulation is equivalent to the fast or slow mean modulation. Consequently, the solitary-wave modulation system (4.16) with (4.17) is strictly hyperbolic.

From (4.18) and (4.19) it follows that the genuine nonlinearity conditions (4.10) are

\begin{align*} \mu_j &= \nabla \lambda_j \cdot \mathbf{z}_j = \partial_{\overline{w}} \lambda_j = \frac{3 \overline{w}}{\lambda_j} \ne 0, \quad j = 1,2, \\ \mu_3 &= \nabla \lambda_3 \cdot \mathbf{z}_3 = \partial_a \lambda_3(a,\overline{w}) = \frac{a - 2\overline{w}}{2 \lambda_3} \ne 0 . \end{align*}

Consequently, the solitary-wave modulation system (4.16) with (4.17) loses genuine nonlinearity in both of the mean modulation characteristic families at an inflection point of the interaction force $f''(\overline{w}) = 3 \overline{w} = 0$. Since either $\overline{w} \lt 0$ and $a \gt 0$ or $a \gt 4 \overline{w} \gt 0$ for solitary wave existence, we have $\mu_3 \gt 0$.

Thus, the modulation system in the solitary-wave limit is strictly hyperbolic for all parameter values and loses genuine nonlinearity when $\overline{w} = 0$. Recall that the same is true for the harmonic limit in this case. Since $\overline{w} = w_3$, $a = w_3 - w_1$ and $w_4 = -2w_3-w_1$, the loss of genuine nonlinearity occurs precisely when $n = (w_3-w_1)/(w_4-w_3) = 1$.

For the full Whitham modulation system in the case of real roots $w_j$, we find that the equations are strictly hyperbolic for $n \gt 0$, $0 \lt m \lt 1$. Figure 5 depicts curves in the $m$- $n$ plane with fixed $a$ where the modulation system loses genuine nonlinearity in one of the first two characteristic families. These curves bifurcate from the $m = 1$ solitary-wave limit at $n = 1$, the inflection point of the interaction force $\overline{w} = 0$ where the solitary wave limiting system (4.16) loses genuine nonlinearity. Since the modulated periodic travelling wave amplitude is held constant, the limit $m \to 0$ corresponds to the trigonometric wave limit where $w_3 \to w_4$.

Figure 5.

Convexity of the Whitham modulation equations for $(\alpha,\beta) = (0,1)$ (case I) with real roots $w_j$. For each fixed amplitude $a$, the solid (dashed) curves correspond to the loss of genuine nonlinearity $\mu_1 = 0$ ( $\mu_2 = 0$) bifurcating from the point $(n,m) = (1,1)$ where the solitary-wave limiting system loses genuine nonlinearity at $\overline{w} = 0$.

Figure 5 Long description

The plots depict n versus m for different fixed values of a, illustrating the loss of genuine nonlinearity. The x-axis is labeled m, ranging from 0 to 1. The y-axis is labeled n, ranging from 0 to 2.5. Units are not specified. Each plot contains a solid and a dashed curve. The solid curve represents one characteristic family, while the dashed curve represents another. For a equals 0.2, the dashed curve starts at n approximately 0.85 at m equals 0, dips slightly and ends at n 1.0 at m equals 1. The solid curve appears from m approximately 0.75, decreasing from n 2.5 to n 1.3 at m equals 1. For a equals 0.5, the dashed curve decreases from n 1.6 at m equals 0 to n 1.1 at m equals 1. The solid curve starts around m 0.6, decreasing from n 2.5 to n 1.2 at m equals 1. For a equals 1, the dashed curve decreases from n 2.5 at m equals 0 to n 1.2 at m equals 1. The solid curve starts around m 0.45, decreasing to n 1.1 at m equals 1. For a equals 2, the dashed curve decreases from n 2.5 at m approximately 0.25 to n 1.2 at m equals 1. The solid curve starts around m 0.5, decreasing to n 1.1 at m equals 1. The plots show how the curves shift and converge as a increases, with the solid curve truncated at higher m values.

On the other hand, modulations of periodic travelling waves with complex conjugate roots $w_{3,4}$ exhibit strict hyperbolicity in a limited range of the $n_r$- $n_i$ plane. This is shown in Figure 6 for differing, fixed amplitudes $a$. Accompanying the loss of hyperbolicity for some parameter regions is the loss of genuine nonlinearity in the first, second and third characteristic fields. The second characteristic field is linearly degenerate along a curve $\mu_2 = 0$ that bifurcates from the trigonometric limit $n_i = 0$. For example, when $a = 0.2$ in Figure 5, the level curve $\mu_2 = 0$ meets the trigonometric limit when $(n,m) \approx (0.878,0)$. Similarly, in Figure 6 with $a = 0.2$, the curve $\mu_2 = 0$ enters the domain from $(n_r,n_i) \approx (0.878,0)$. As $a$ varies, the level curves exhibit somewhat complex deformations. The level curve $\mu_1 = 0$ appears within the parameter window of consideration when $a = 2$. The boundary of the non-hyperbolic region is where $\lambda_2 = \lambda_3$, so $\mu_2 = \mu_3 = 0$ there.

Figure 6.

Convexity of the Whitham modulation equations for $(\alpha,\beta) = (0,1)$ (case I), cubic interaction force with complex conjugate roots $w_{3,4}$, $\tilde{\mathbf{A}} = (n_r,n_i,a,\overline{v}=0) \in \tilde{E}_a$: The system is strictly hyperbolic in the white region. Greyscale is $\mathrm{Im}(\lambda_3)$ where the system is not hyperbolic. Periodic travelling waves do not exist in the inadmissible pink region. For each fixed amplitude $a$, the solid (dashed) curves correspond to the loss of genuine nonlinearity $\mu_2 = 0$ ( $\mu_3 = 0$). For $a = 2$, the dash-dotted curve corresponds to $\mu_1 = 0$.

Figure 6 Long description

The figure contains four contour plots illustrating the imaginary part of lambda 3 versus n r and n i for different values of parameter a. The x-axis is labeled n r, ranging from 0 to 1 and the y-axis is labeled n i, ranging from 0 to minus 1.5. Both axes are unitless. The plots show how the imaginary part of lambda 3 changes with these parameters. The color scale varies for each plot, indicating different ranges of the imaginary part of lambda 3. The solid and dashed curves represent boundaries of stability and loss of genuine nonlinearity, respectively. The masked peach regions indicate areas where periodic traveling waves do not exist. For a equals 0.2, the color scale ranges from 0 to 0.03. The solid curve encloses a region near n r equals 0.5 and n i equals 0. The dashed curves extend from n r equals 0 to n r equals 1, showing a loss of nonlinearity. For a equals 0.5, the color scale ranges from 0 to 0.08. The solid curve is positioned higher on the n i axis compared to the previous plot, indicating a shift in stability. For a equals 1, the color scale ranges from 0 to 0.1. The solid curve encloses a larger area and the dashed curves show a more pronounced downward slope, indicating changes in stability boundaries. For a equals 2, the color scale ranges from 0 to 0.15. The solid curve encloses the largest area and the dashed curves are positioned lower on the n i axis, indicating further changes in stability. The plots collectively show how increasing a affects the stability and nonlinearity regions, with the imaginary part of lambda 3 increasing in magnitude and the stability regions expanding.

4.5. Cubic nonlinearity with $(\alpha,\beta) = (0,-1)$

Next, we discuss the case of cubic $f(w)$ with $(\alpha,\beta)=(0,-1)$, which corresponds to case II in Section 3.2 after an appropriate rescaling discussed in Section 2.1. In this setting, we have

\begin{equation*} \begin{split} \mathcal{W} &= \frac{1}{2\pi} \sqrt{\frac{6}{c^2}} \oint \sqrt{G_4(w)}\, \mathrm{d} w = \frac{1}{\pi} \sqrt{\frac{6}{c^2}} \int_{w_2}^{w_3} \sqrt{(w-w_1)(w-w_2)(w_3-w)(w_4-w)}\, \mathrm{d} w, \\ \overline{w^2} &= \frac{1}{2\pi} \sqrt{\frac{\omega^2}{6}} \oint \dfrac{w^2}{\sqrt{G_4(w)}}\, \mathrm{d} w =\frac{1}{\pi} \sqrt{\frac{\omega^2}{6}} \int_{w_2}^{w_3} \dfrac{w^2\,\mathrm{d} w}{\sqrt{(w-w_1)(w-w_2)(w_3-w)(w_4-w)}}. \end{split} \end{equation*}

Recall that in this case, there are two families of periodic solutions related by symmetry (3.33) with $\alpha=0$, $\beta=-1$ and that differ by a $\pi$ phase shift, so it suffices to consider one of them given by (3.41). We have

\begin{equation*} \begin{split} &w_1 + w_2 + w_3 + w_4 = 0,\\ & A = -\frac{1}{2}(w_1w_2w_3 + w_1 w_2 w_4+w_1 w_3 w_4+w_2 w_3 w_4),\\ &c^2 = 1 + \frac{1}{2} ( w_1 w_2 + w_1 w_3 + w_1 w_4 + w_2 w_3 + w_2 w_4 + w_3 w_4 ),\\ &k = \frac{\pi}{K(m)} \sqrt{\frac{3}{2c^2}(w_4-w_2)(w_3-w_1)}, \quad \omega=ck,\\ &m = \frac{(w_4-w_1)(w_3-w_2)}{(w_4-w_2)(w_3-w_1)}, \end{split} \end{equation*}

where we used (3.7), (3.8), (3.11), (3.41) and (3.43), and obtain [Reference Byrd and Friedman4]

(4.20)\begin{equation} \begin{split} \mathcal{W}&=\dfrac{(w_3 - w_2)^2 (w_4-w_2)^{1/2} (w_2 -w_1)^2}{4|c| \sqrt{6}\pi n^3 (1 - n)^2 (m-n)^2 (w_3-w_1)^{3/2}} \bigg\{n \bigg(3 n^4 + 3 m^2 - 4 n^3 (1 + m)\\ &- 4 n m (1 + m) + 2 n^2 (2 + m + 2 m^2)\bigg) E(m) \\ &+(m-n) \bigg(3 n^4 - 4 n^2 (m-1) - 3 m^2 + 6 n m^2 - 2 n^3 (2 + m)\bigg) K(m) \\ &+ 3 (m-n^2) (-2 n + n^2 + m) (n^2 + m - 2 n m) \Pi(n, m)\bigg\} \end{split} \end{equation}

and

(4.21)\begin{equation} \begin{split} \overline{w^2}&=\dfrac{\sqrt{2}w_1^2|\omega|}{\sqrt{3}\pi \sqrt{(w_3 - w_1) (w_4 - w_2)}}\bigg\{K(m) + 2\left(\dfrac{w_2}{w_1}-1\right)\Pi(n,m)\\ &+ \dfrac{(w_2 -w_1)^2}{2 (n-1) (m-n) w_1^2} \bigg(n E(m) + (m-n) K(m) \\ &- \bigg(n^2 + 3 m - 2 n (1 + m)\bigg) \Pi(n, m)\bigg)\bigg\}, \end{split} \end{equation}

where we recall that $m$ and $n$ are given in (3.41) and satisfy (3.42).

4.5.1. Solitary-wave limit

In the limit $w_3 \to w_4$ ( $m \to 1$), the solutions approach the two-parameter family (3.45) of solitary waves with amplitude $a=w_4-w_2$, background $\overline{w}=w_4$ and velocity (3.46). In this limit, we obtain

(4.22)\begin{equation} \begin{split} \lim_{w_3 \to w_4}\dfrac{c}{12}\mathcal{W}&=\dfrac{\text{sgn}(c)}{\pi\sqrt{6}}\bigg\{\dfrac{1}{6}\sqrt{a(4\overline{w}-a)}(a^2 - 4 a \overline{w} + 6 \overline{w}^2)\\ &-(a -2 \overline{w})^2 \overline{w}\, \text{arctanh}\sqrt{\dfrac{a}{4\overline{w}-a}}\biggl\},\\ \lim_{w_3 \to w_4}c\dfrac{\overline{w^2}-\overline{w}^2}{k}&=\dfrac{c^2\text{sgn}(c)}{\pi\sqrt{6}} \bigg\{-\sqrt{a(4\overline{w}-a)}+4\overline{w}\, \text{arctanh}\sqrt{\dfrac{a}{4\overline{w}-a}}\bigg\},\\ \lim_{w_3 \to w_4}\overline{w^2} &= \overline{w}^2, \quad \lim_{w_3 \to w_4}c^2=1-3\overline{w}^2+\dfrac{1}{2}a(4\overline{w}-a), \end{split} \end{equation}

where we recall that $0 \lt a \lt 2\overline{w}$ by (3.47) with $\alpha = 0$. The Whitham modulation equations (4.7) with $f(w)=w-w^3$ therefore become

(4.23)\begin{equation} \begin{split} \overline{w}_T-\overline{v}_X&=0,\\ \overline{v}_T-(1-3\overline{w}^2)\overline{w}_X&=0,\\ a_T+c\,a_X+c\,\dfrac{g(a,\overline{w})}{d(a,\overline{w})}\overline{w}_X+\dfrac{h(a,\overline{w})}{d(a,\overline{w})}\overline{v}_X&=0, \end{split} \end{equation}

where

(4.24)\begin{equation} \begin{split} g(a,\overline{w})&=\sqrt{a(4\overline{w}-a)}(-2 + 4 a^2 - 19 a \overline{w} + 24 \overline{w}^2)\\ &+2 \bigg(-2 + (a - 6 \overline{w}) (a - 4 \overline{w})\bigg) (a - 2 \overline{w})\,\text{arctanh}\sqrt{\dfrac{a}{4 \overline{w}-a}},\\ h(a,\overline{w})&=\sqrt{a(4\overline{w}-a)} (-2 + 4 a^2 - 19 a \overline{w} + 24 \overline{w}^2)\\ &+\bigg(-4 + 3 (a - 4 \overline{w})^2\bigg) (a - 2 \overline{w})\, \text{arctanh}\sqrt{\dfrac{a}{4 \overline{w}-a}},\\ d(a,\overline{w})&=\sqrt{a(4\overline{w}-a)} (-1 + 2 a^2 - 8 a \overline{w} + 9 \overline{w}^2) -6 (a - 2 \overline{w})^2 \overline{w}\, \text{arctanh}\sqrt{\dfrac{a}{4 \overline{w}-a}}. \end{split} \end{equation}

The ordered characteristic velocities (for the fast-wave family with $c \gt 0$) are

(4.25)\begin{equation} \lambda_1 = -\sqrt{1-3\overline{w}^2}, \quad \lambda_2 = \sqrt{1-3\overline{w}^2}, \quad \lambda_3 = c = \sqrt{1-3\overline{w}^2+\dfrac{1}{2}a(4\overline{w}-a)} . \end{equation}

Note that the existence conditions (3.47) for $(\alpha,\beta)=(0,-1)$ amount to satisfying one of the following inequality pairs

(4.26a)\begin{equation} \mathrm{either} \quad 0 \lt a \lt 2\overline{w} \quad \mathrm{and} \quad 0 \lt \overline{w} \le \frac{1}{\sqrt{3}}, \end{equation}

(4.26b)\begin{equation} \mathrm{or} \quad 2\overline{w} - \sqrt{2(1-\overline{w}^2)} \lt a \lt 2\overline{w} \quad \mathrm{and} \quad \frac{1}{\sqrt{3}} \lt \overline{w} \lt 1 . \end{equation}

The associated eigenvectors are given by (4.19).

4.5.2. Kink limit

The limit $a \to 2\overline{w}$ corresponds to the supersonic kink solution (3.48). To obtain its modulation, it is tempting to send $a \to 2\overline{w}$ in eq. (4.23). But this results in a singularity. Instead, we return to the wave-action equation (4.7c) with the solitary-wave limiting density and flux in eq. (4.22). We denote the limit as $a \to 2\hat{w}$ in order to distinguish the kink $\hat{w}$ from the mean $\overline{w}$. An expansion of the density and flux results in

\begin{equation*} \begin{split} \frac{c}{12} \mathcal{W} + c \frac{\overline{w^2}-\overline{w}^2}{k} &= -\frac{2\mathrm{sgn}(c)}{\pi\sqrt{6}}f(\hat{w}) \log(2\hat{w}-a) + \mathcal{O}(1) , \\ c^2 \frac{\overline{w^2}-\overline{w}^2}{k} &= -\frac{2|c|}{\pi\sqrt{6}}f(\hat{w}) \log(2\hat{w}-a) + \mathcal{O}(1) , \end{split} \end{equation*}

as $a \to 2\hat{w}$ where $f(\hat{w}) = \hat{w}-\hat{w}^3$ is the interaction force and $c^2 = 1-\hat{w}^2$ is the squared kink velocity (3.49). Inserting this into the conservation of wave action (4.7c), we obtain the expansion

\begin{equation*} \frac{f(\hat{w})}{2\hat{w}-a}\left [ (2\hat{w}-a)_T + c (2\hat{w}-a)_X \right ] + \log(2\hat{w}-a) \left [ f(\hat{w})_T + \left ( c f(\hat{w}) \right )_X \right ] + \cdots = 0 . \end{equation*}

Multiplying this expression by $2\hat{w}-a$ and equating each order to zero as $a \to 2\hat{w}$, we obtain the kink conservation law

\begin{equation*} f(\hat{w})_T + \left [ c f(\hat{w}) \right]_X = 0, \quad f(\hat{w}) = \hat{w}-\hat{w}^3, \quad c = \pm \sqrt{1-\hat{w}^2} . \end{equation*}

This equation is distinct from the mean equations for $(\overline{w},\overline{v})$ in (4.23) and admits the discontinuous travelling wave solution

(4.27)\begin{equation} \hat{w}(X,T) = \pm 2w_0 \,\mathrm{sgn}(X-cT), \quad c^2 = 1-w_0^2, \quad 0 \lt w_0 \lt 1 , \end{equation}

whose velocity $s = c$ results from the Rankine–Hugoniot jump condition $-s[f(\hat{w})] + [cf(\hat{w})] = 0$ where $[\cdot]$ is the difference between the state $\cdot$ to the right and left of the discontinuity. The weak solution (4.27) is the zero-dispersion limit of the kink solution (3.48) and thus is an admissible shock. Since the mean field propagates along characteristics with velocities $\lambda_j = \mp \sqrt{f'(\overline{w})} = \mp \sqrt{1-3\overline{w}^2}$, they are unchanged on either side of the shock ( $f'(w_0) = f'(-w_0)$), i.e., the characteristics pass through the shock. It is because of this that the solution (4.27) can be interpreted as a non-classical, undercompressive shock [Reference El, Hoefer and Shearer17, Reference Herrmann and Rademacher23] and is an example of a superkink in the classification of steady transition fronts in mechanical systems [Reference Gorbushin, Vainchtein and Truskinovsky22].

4.5.3. Convexity

We start by considering the modulation system (4.23) with (4.24) for the solitary-wave family (3.45), with eigenvalues (4.25) and eigenvectors (4.19). The existence condition for this family is (4.26). In this case, we obtain

\begin{equation*} \mu_j=-\dfrac{3\overline{w}}{\lambda_j},\quad j=1,2, \quad \mu_3=\dfrac{2\overline{w}-a}{2\lambda_3}. \end{equation*}

Thus, the system is strictly hyperbolic when $0 \lt \overline{w} \lt 1/\sqrt{3}$ and genuinely nonlinear when $a \neq 2\overline{w}$, which ensures that $ \mu_3 \neq 0$. Since a nonzero mean is required for the existence of a solitary wave, $\mu_j \ne 0$ for $j = 1,2$. We recall that the harmonic limit is strictly hyperbolic when $|\overline{w}| \lt 1/\sqrt{3}$ and genuinely nonlinear when $\overline{w} \neq 0$ and $\omega \neq 0$.

To investigate the convexity of the full Whitham modulation system (4.7) with (4.20) and (4.21), we will present our results in terms of $\mathbf{A}$ in (3.34). For several choices of the periodic travelling wave amplitude $a$, we plot the existence region and imaginary part of the second eigenvalue in Figure 7. Within the existence region, two islands nucleate where hyperbolicity is lost. The islands take up a larger percentage of the existence region as the wave amplitude is increased. To further understand these islands, we note that the depression solitary-wave limit with $m \to 1$ admits the following relation between the solitary-wave parameters $(a,\overline{w})$ and the parameter $n$ in (3.45)

(4.28)\begin{equation} n = \frac{a}{4\overline{w}-a} , \quad 0 \lt a \lt 2 \overline{w} , \quad \overline{w} \gt 0 , \end{equation}

by $a = w_4 - w_2$, $\overline{w} = w_4$ and $w_1 + w_2 + 2w_4 = 0$. According to (4.25), we expect the nucleation of non-hyperbolic regions in the second characteristic family from $\overline{w} \ge 1/\sqrt{3}$ in the solitary-wave limit. Indeed, the red segment in Fig. 7 along $m = 1$ corresponds to the values of $n$ in (4.28) satisfying the inequalities (4.26b) that yield $1/\sqrt{3} \lt \overline{w} \lt a/3 + \sqrt{2(6-a^2)}/6$.

Figure 7.

Hyperbolicity of the Whitham modulation equations for $(\alpha,\beta) = (0,-1)$ (case II). For each fixed amplitude $a$, the greyscale contour plot conveys the imaginary part of a characteristic velocity. Periodic waves do not exist in the inadmissible pink region. The solid (dashed) black curves correspond to the loss of genuine nonlinearity $\mu_1 = \mu_4 = 0$ ( $\mu_2 = 0$). The red segment at $m = 1$ corresponds to the region of non-hyperbolicity of the solitary-wave limiting system (4.23).

Figure 7 Long description

A equals 0.15 A contour plot with a vertical color bar labeled Im lambda 2. No units are shown for m, n, or Im lambda 2. The horizontal axis is labeled m, with labeled ticks 0.2, 0.4, 0.6, 0.8, 1. The vertical axis is labeled n, with labeled ticks 0, 0.2, 0.4, 0.6, 0.8, 1. The color bar has labeled values 0, 0.05, 0.1, 0.15, 0.2, 0.25. A white region is enclosed by a solid black curve and dashed black curves. The solid curve passes from about m equals 0.2, n equals 0.1 to m equals 1, n equals 1. A dashed curve forms an upper boundary from about m equals 0.2, n equals 0.1 to m equals 1, n equals 1. A dashed curve forms a lower boundary close to n equals 0 from about m equals 0.2 to m equals 1. a equals 0.5 A contour plot with a vertical color bar labeled Im lambda 2. No units are shown for m, n, or Im lambda 2. The horizontal axis is labeled m, with labeled ticks 0.5, 0.6, 0.7, 0.8, 0.9, 1. The vertical axis has labeled ticks 0.2, 0.4, 0.6, 0.8, 1. The color bar has labeled values 0, 0.1, 0.2, 0.3, 0.4, 0.5. A white region is enclosed by a solid black curve and dashed black curves. The solid curve passes from about m equals 0.5, n equals 0.3 to m equals 1, n equals 1. A dashed curve forms an upper boundary from about m equals 0.5, n equals 0.3 to m equals 1, n equals 1. A dashed lower boundary runs from about m equals 0.5, n equals 0.3 down toward about n equals 0.2 around m equals 0.8 to 0.9, then continues toward m equals 1 near about n equals 0.25. Shading appears within the bounded region near m close to 1 and n about 0.2 to 0.3. a equals 1 A contour plot with a vertical color bar labeled Im lambda 2. No units are shown for m, n, or Im lambda 2. The horizontal axis is labeled m, with labeled ticks 0.8, 0.85, 0.9, 0.95, 1. The vertical axis is labeled n, with labeled ticks 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1. The color bar has labeled values 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6. A white region is enclosed by a solid black curve and dashed black curves. The solid curve passes from about m equals 0.8, n equals 0.55 to m equals 1, n equals 1. A dashed curve forms an upper boundary approaching m equals 1, n equals 1. A dashed lower boundary dips from about m equals 0.8, n equals 0.55 down toward about n equals 0.45 around m equals 0.95, then turns upward toward m equals 1. Shading occupies the bounded region for m close to 1 and n about 0.45 to 0.6. a equals 1.3 A contour plot with a vertical color bar labeled Im lambda 2. No units are shown for m, n, or Im lambda 2. The horizontal axis is labeled m, with labeled ticks 0.9, 0.92, 0.94, 0.96, 0.98, 1. The vertical axis has labeled ticks 0.5, 0.6, 0.7, 0.8, 0.9, 1. The color bar has labeled values 0, 0.2, 0.4, 0.6. A white region is enclosed by a solid black curve and dashed black curves. The solid curve passes from about m equals 0.9, n equals 0.68 to m equals 1, n equals 1. A dashed curve forms an upper boundary approaching m equals 1, n equals 1. A dashed lower boundary dips from about m equals 0.9, n equals 0.68 down toward about n equals 0.6 around m equals 0.96, then rises toward m equals 1. Shading fills much of the bounded region for m close to 1, spanning n about 0.6 up to near 1. Across the four plots, the shaded area associated with nonzero Im lambda 2 is concentrated at m close to 1 and expands to a wider n range as a increases from 0.15 to 1.3, while the color bar maximum increases from 0.25 to 0.6.

Figure 7 also depicts the level curves $\mu_j = 0$. The solid black curve corresponds to both $\mu_1 = 0$ and $\mu_4 = 0$. It bifurcates from the kink limit $m \to 1$, $n \to 1$. The dashed curves correspond to regions where $\mu_2 = 0$. Necessarily, these surround the non-hyperbolic greyscale regions where $\lambda_1 = \lambda_2$ because the coalescence of two characteristics (non-strict hyperbolicity) coincides with loss of genuine nonlinearity, so that $\mu_1 = 0$ also there.