2. Type I and II solutions: non-autonomous theory

The uniform stream and solitary wave solutions of the autonomous system (1.1) with no forcing are represented respectively by the fixed point at the origin and its homoclinic connection in figure 1(b). ‘Homoclinic’ means that branches of the stable and unstable manifolds of the fixed point coincide, and hence the associated wave approaches $({\it\eta},{\it\eta}_{x})=(0,0)$ as $x\rightarrow \pm \infty$ . Inclusion of the forcing term (i.e. ${\it\epsilon}\neq 0$ ) renders the system non-autonomous. To understand this, we adapt the recently developed theory of Balasuriya & Binder (Reference Balasuriya and Binder2014), a detailed discussion of which is provided in the supplementary material, appendix C. For $|{\it\epsilon}|\ll 1$ , it can be shown that there is only one solution that remains $\mathit{O}({\it\epsilon})$ -close to the uniform stream (i.e. the saddle fixed point in figure 1 b) over the entire unbounded domain (Balasuriya Reference Balasuriya2011; Balasuriya & Padberg-Gehle Reference Balasuriya and Padberg-Gehle2013, Reference Balasuriya and Padberg-Gehle2014), and this is given by

(2.1) $$\begin{eqnarray}{\it\eta}^{I}(x)=\frac{3{\it\epsilon}\cos (kx)}{k^{2}+6{\it\mu}}\end{eqnarray}$$

to $\mathit{O}({\it\epsilon}^{2})$ . The dotted curves in figure 2(a) are for this near-uniform solution (2.1), which is the non-autonomous type I solution.

Next, we consider solutions that are near-solitary. These are solutions that have one big hump with a height of approximately $2{\it\mu}$ , overlaid with smaller oscillations, and that decay in the far field to the near-uniform solution (2.1). This is the only natural far-field possibility for near-solitary waves in this non-autonomous setting, since the only solution that remains approximately uniform as $x\rightarrow \pm \infty$ is (2.1). The dynamical-systems interpretation is that these are homoclinic trajectories to (2.1) which are near to a corresponding homoclinic trajectory to the autonomous system. When ${\it\epsilon}=0$ , an uncountable number of one-humped solitary waves are obtained by taking arbitrary shifts of (1.3). When $|{\it\epsilon}|\ll 1$ but is non-zero, the non-autonomous theory of Balasuriya & Binder (Reference Balasuriya and Binder2014) can be adapted to characterise all one-humped solitary waves. As derived in the supplementary material, appendix C, there are exactly two possible types of such one-humped solitary waves: (i) those centred around crests and (ii) those centred around troughs of the topography. Near-solitary waves within each of these two families are simple shifts of each other by multiples of $k{\rm\pi}$ . For example, if ${\it\epsilon}>0$ , there is a crest of ${\it\sigma}$ at $x=0$ , resulting in a crest-centred near-solitary wave centred at $x=0$ . Other crest-centred near-solitary waves can be obtained by shifting this to the values $x=2n{\rm\pi}/k$ , $n\in \mathbb{Z}$ . Trough-centred near-solitary waves would therefore be centred at $x=(2n+1){\rm\pi}/k$ , $n\in \mathbb{Z}$ . In contrast to the autonomous situation, there are only a countable number of such one-humped near-solitary waves. As described in the supplementary material, appendix C, these near-solitary waves are the only ‘regular’ structures among ‘nearby’ solutions which are well known to be chaotic (Grimshaw & Tian Reference Grimshaw and Tian1994), exhibiting seemingly random $\mathit{O}(1)$ excursions.

The one-humped non-autonomous near-solitary waves can be expressed to $\mathit{O}({\it\epsilon})$ explicitly by adapting the wave-profile theory of Balasuriya & Binder (Reference Balasuriya and Binder2014) which builds on the stable/unstable manifold determination in Balasuriya (Reference Balasuriya2011) as described in detail in the supplementary material, appendix C. The near-solitary wave centred at $x=0$ is given by

(2.2) $$\begin{eqnarray}{\it\eta}^{II}(x)=\bar{{\it\eta}}(x)+{\it\epsilon}[{\it\eta}^{n}(x)+{\it\eta}^{t}(x)]+\mathit{O}({\it\epsilon}^{2}),\end{eqnarray}$$

in which $\bar{{\it\eta}}$ is the fundamental solitary wave solution (1.3),

(2.3) $$\begin{eqnarray}{\it\eta}^{n}(x)=-\frac{3\bar{{\it\eta}}_{xx}(x)}{\bar{{\it\eta}}_{x}(x)^{2}+\bar{{\it\eta}}_{xx}(x)^{2}}\int _{|x|}^{\infty }\bar{{\it\eta}}_{x}({\it\tau})\cos (k{\it\tau})\,\text{d}{\it\tau}\end{eqnarray}$$

and

(2.4) $$\begin{eqnarray}{\it\eta}^{t}(x)=-3\bar{{\it\eta}}_{x}(|x|)\int _{0}^{|x|}\frac{\bar{{\it\eta}}_{xx}({\it\tau})\cos (k{\it\tau})-{\it\Omega}({\it\tau})\displaystyle \int _{|x|}^{\infty }\bar{{\it\eta}}_{x}({\it\lambda})\cos [k({\it\lambda}+{\it\tau}-x)]\,\text{d}{\it\lambda}}{\bar{{\it\eta}}_{x}({\it\tau})^{2}+\bar{{\it\eta}}_{xx}({\it\tau})^{2}}\,\text{d}{\it\tau},\end{eqnarray}$$

where

(2.5) $$\begin{eqnarray}{\it\Omega}({\it\tau}):=\frac{[6(1+{\it\mu})-9\bar{{\it\eta}}({\it\tau})-5][\bar{{\it\eta}}_{x}({\it\tau})^{2}-\bar{{\it\eta}}_{xx}({\it\tau})^{2}]}{\bar{{\it\eta}}_{x}({\it\tau})^{2}+\bar{{\it\eta}}_{xx}({\it\tau})^{2}}.\end{eqnarray}$$

The solution (2.2) is crest-centred if ${\it\epsilon}>0$ and trough-centred if ${\it\epsilon}<0$ . Waves centred at other $x$ -values can be computed from these solutions by simply shifting the wave solutions. The theoretical waves computed from (2.2) are virtually identical to the numerically computed solutions shown by the solid curves in figure 2(b,d). To demonstrate this visually for curves that virtually fall on top of one another, we show the theoretical solution values computed from (2.2) on a coarse grid, shown by dots in figure 2(b,d). The crest-centred solutions have an upwelling at the central crest, whereas the trough-centred ones have dips, as shown respectively in figure 2(b,d).

The single-humped solitary waves are, as is clear from (2.2), symmetric about their central value. The presence of different single-humped solitary waves means that many-humped ones will exist as approximately the sum of several of the single-humped ones: figure 2(e) is composed of two trough-centred ones, while figure 2(f) has two trough-centred and one crest-centred one. Indeed, these figures were numerically obtained by using such a sum as an initial guess. Therefore, it is possible to form multi-humped solitary waves that are not symmetric.

While the above expressions are messy, the height at the central crest/trough at $x=0$ can be derived to be (see the supplementary material, appendix C)

(2.6) $$\begin{eqnarray}{\it\eta}^{II}(0)=2{\it\mu}+\frac{{\it\epsilon}}{{\it\mu}}\left(k\int _{0}^{\infty }\text{sech}^{2}\left(\sqrt{\frac{3{\it\mu}}{2}}t\right)\sin (kt)\,\text{d}t-1\right).\end{eqnarray}$$

The lower and upper broken straight lines in the panels of figure 3 correspond to values given by (2.1) and (2.6) respectively, demonstrating that the near-uniform and near-solitary waves obtained from the non-autonomous theory show excellent agreement with the numerics.

The near-solitary waves derived in this section are obtained explicitly in the unbounded $x$ -domain for small topography, with no additional assumptions, matching conditions or enforced spatial periodicity. Following a genuinely non-autonomous viewpoint helped to avoid such assumptions, which are often used in similar fKdV equations (e.g. Ockendon et al. Reference Ockendon, Ockendon and Johnson1986; Amundsen et al. Reference Amundsen, Cox and Mortell2007).

Figure 2.

Free-surface profiles, ${\it\mu}=0.10$ , $k=3.0$ , $|{\it\epsilon}|=0.1$ . The solid curves are numerical solutions. The dotted curve in (a) is for the non-autonomous approximation (2.1). The dotted curves in (b,d) are for the non-autonomous approximation (2.2). The dotted curve in (c) is for the asymptotic theory, retaining only the first term in the regular expansion (3.2). (a) Perturbation of a uniform stream solution I, ${\it\eta}(0)=0.036$ , ${\it\epsilon}=0.1$ . (b) Perturbation of a solitary wave solution II, ${\it\eta}(0)=0.231$ , ${\it\epsilon}=0.1$ . (c) Perturbation of a uniform cnoidal stream solution III, ${\it\eta}(0)=0.167$ , ${\it\epsilon}=0.1$ . (d) Perturbation of a solitary wave solution II, ${\it\eta}(0)=0.151$ , ${\it\epsilon}=-0.1$ . (e,f) Additional solutions, ${\it\epsilon}=0.1$ .

Figure 3.

Bifurcation diagrams for the elevation of the free surface, ${\it\eta}(0)$ , versus the amplitude of forcing, ${\it\epsilon}$ , with ${\it\mu}=0.1$ . The solid curves are for the numerical solutions. The upper, middle and lower branches that emanate from the vertical axis correspond to perturbations of a uniform stream, solitary wave and uniform cnoidal stream solutions I–III. The dotted curves in (g–i) are additional numerical type III solutions, as indicated by the asymptotic theory. The lower and upper broken straight lines are for the non-autonomous approximations (2.1) and (2.6) respectively. The solid straight lines are for the asymptotic theory, retaining only the first term in the regular expansion (3.2). Wavenumber forcing: (a) $k=3$ , (b) $k=1$ , (c) $k=0.7746$ , (d) $k=0.69378$ , (e) $k=0.5$ , (f) $k=0.3878$ , (g) $k=0.387$ , (h) $k=0.346$ , (i) $k=0.3$ , (j) $k=0.25$ , (k) $k=0.1$ , (l) $k=0.05$ .

3. Type III solutions: asymptotic theory

As noted above, there exist unforced periodic cnoidal wave solutions to (1.1) when ${\it\epsilon}=0$ . To discuss the forced counterparts of such type III solutions we recast in terms of the shifted variable $N(x)={\it\eta}(x)-4{\it\mu}/3$ , in which case (1.1) becomes

(3.1) $$\begin{eqnarray}N_{xx}+{\textstyle \frac{9}{2}}N^{2}+{\it\nu}_{c}^{2}N=-3{\it\epsilon}\cos kx,\end{eqnarray}$$

where ${\it\nu}_{c}=\sqrt{6{\it\mu}}$ is the frequency of small-amplitude unforced cnoidal waves which arise when ${\it\epsilon}=0$ and $N\ll 1$ . It is now clear from (3.1) that resonance may occur for small-amplitude waves if $k={\it\nu}_{c}$ . Furthermore, nonlinear generation of higher harmonics via the second term on the left-hand side of (3.1) raises the prospect of secondary resonances at $k={\it\nu}_{c}/2^{n}$ for $n=2,3,4,\ldots$  . Grimshaw & Tian (Reference Grimshaw and Tian1994) discussed resonance for the fKdV equation in the presence of damping effects.

When $k\neq {\it\nu}_{c}/2^{n}$ , for $n\in \mathbb{N}$ , there is no resonance. In this case, for small-amplitude waves, $|{\it\epsilon}|\ll 1$ , we seek a solution as a regular expansion in the form

(3.2) $$\begin{eqnarray}N={\it\epsilon}N_{1}(x)+{\it\epsilon}^{2}N_{2}(x)+\cdots \,.\end{eqnarray}$$

Substituting into (3.1) and considering the linear problems obtained at each order of approximation, we may easily determine $N_{1}(x)$ , $N_{2}(x)$ , and so on. Indeed, we find

(3.3) $$\begin{eqnarray}N_{1}(x)=\frac{3\cos (kx)}{k^{2}-6{\it\mu}},\end{eqnarray}$$

which shows that small-amplitude waves are in phase with the topography for frequencies above resonance, $k>{\it\nu}_{c}$ , and are a half-period out of phase with the topography when $k<{\it\nu}_{c}$ (in contrast with the small-amplitude type I solution (2.1) which is always in phase with the topography). The dotted curve in figure 2(c) and the solid straight lines in the panels of figure 3 are for the asymptotic expansion (3.2), where we have retained only the first term in the expansion.

More interesting is the small-amplitude response at the resonant frequencies. At primary resonance, $k={\it\nu}_{c}$ , to obtain a bounded periodic solution, we expand in fractional powers of ${\it\epsilon}$ , writing

(3.4) $$\begin{eqnarray}N(x)={\it\epsilon}^{1/3}n_{1}(x)+{\it\epsilon}^{2/3}n_{2}(x)+{\it\epsilon}n_{3}(x)+\cdots \,.\end{eqnarray}$$

A key observation is that at second order, $\mathit{O}({\it\epsilon})$ , the first two terms in the expansion (3.4) will combine via the nonlinear term on the left-hand side of (3.1) to cancel the problematic forcing term on the right-hand side and thereby eliminate any secular terms. Indeed, at leading order $\mathit{O}({\it\epsilon}^{1/3})$ we find $n_{1}(x)=a_{1}\cos kx+b_{1}\sin kx$ , for constants $a_{1}$ and $b_{1}$ , and at first order $\mathit{O}({\it\epsilon}^{2/3})$ we obtain

(3.5) $$\begin{eqnarray}\displaystyle n_{2}(x) & = & \displaystyle a_{2}\cos kx+b_{2}\sin kx+\frac{3^{5/3}}{4}\frac{(a_{1}^{2}-b_{1}^{2})}{k^{2}}\cos 2kx\nonumber\\ \displaystyle & & \displaystyle +\,\frac{3^{5/3}}{2}\frac{a_{1}b_{1}}{k^{2}}\sin 2kx-\frac{3^{11/3}}{4}\frac{(a_{1}^{2}+b_{1}^{2})}{k^{2}},\end{eqnarray}$$

for constants $a_{2}$ and $b_{2}$ . At $\mathit{O}({\it\epsilon})$ we find that the term on the right-hand side of (3.1) is eliminated if

(3.6a,b ) $$\begin{eqnarray}a_{1}=\frac{2}{3^{2/3}5^{1/3}}k^{2/3},\quad b_{1}=0,\end{eqnarray}$$

and we obtain a periodic solution.

At the first secondary resonance, where $k={\it\nu}_{c}/2$ , we find that we must expand by writing

(3.7) $$\begin{eqnarray}N(x)={\it\epsilon}^{2/3}{\tilde{n}}_{1}(x)+{\it\epsilon}{\tilde{n}}_{2}(x)+{\it\epsilon}^{4/3}{\tilde{n}}_{3}(x)+{\it\epsilon}^{5/3}{\tilde{n}}_{4}(x)+{\it\epsilon}^{2}{\tilde{n}}_{5}(x)+\cdots \,.\end{eqnarray}$$

Working through the problems at successive orders of approximation, we find that ${\tilde{n}}_{1}(x)={\it\alpha}_{1}\cos 2kx+{\it\beta}_{1}\sin 2kx$ , for constants ${\it\alpha}_{1}$ and ${\it\beta}_{1}$ , and

(3.8) $$\begin{eqnarray}{\tilde{n}}_{2}(x)={\it\alpha}_{2}\cos 2kx+{\it\beta}_{2}\sin 2kx-\frac{1}{k^{2}}\cos kx,\end{eqnarray}$$

for constants ${\it\alpha}_{2}$ and ${\it\beta}_{2}$ . It is now evident that the quadratic nonlinearity in (3.1) will generate, via the second term in the expansion (3.7), a resonant forcing term proportional to $\cos {\it\nu}_{c}x\;(=\cos 2kx)$ at $\mathit{O}({\it\epsilon}^{2})$ . However, on noting that at next order we find

(3.9) $$\begin{eqnarray}{\tilde{n}}_{3}(x)={\it\alpha}_{3}\cos 2kx+{\it\beta}_{3}\sin 2kx+\frac{3^{1/3}}{5^{2/3}4k^{10/3}}(\cos 4kx-3),\end{eqnarray}$$

where ${\it\alpha}_{3}$ and ${\it\beta}_{3}$ are constants, it becomes clear that the problematic resonant term at $\mathit{O}({\it\epsilon}^{2})$ may be removed by virtue of the nonlinear interaction of the first and third terms in the expansion (3.7) and a judicious choice of the constants ${\it\alpha}_{1}$ and ${\it\beta}_{1}$ . Indeed, we find that the resonant term is eliminated if we choose

(3.10a,b ) $$\begin{eqnarray}{\it\alpha}_{1}=\frac{2}{3^{1/3}5^{1/3}k^{2/3}},\quad {\it\beta}_{1}=0,\end{eqnarray}$$

to yield a periodic solution. Other secondary resonances with $k={\it\nu}_{c}/4$ , and so on, can be dealt with similarly.

Figure 4.

Analysis of cnoidal waves, ${\it\mu}=0.10$ . (a,b) Comparison of the type III computed solution branches (solid curves) with the asymptotic theory (dotted curves) for forcing wavenumbers $k=0.7746$ and $k=0.3873$ respectively. In (a) the first two terms in the expansion (3.4) have been used and in (b) the first three terms in the expansion (3.7) have been used. (c,d) Unforced cnoidal wave maxima/minima against wavenumber ${\it\nu}$ , ${\it\epsilon}=0$ . The wavenumber occupies $0\leqslant {\it\nu}\leqslant \sqrt{6{\it\mu}}=0.7746$ . (c) Wave maximum, with $4/3\leqslant \max ({\it\eta})\leqslant 0.2$ . (d) Wave minimum, with $0\leqslant \min ({\it\eta})\leqslant 4/3$ .

In figure 4 the expansions (3.4) and (3.7) are compared with numerical results at the primary resonance and first secondary resonance for ${\it\mu}=0.10$ . The asymptotic solutions (3.4) and (3.7) are shown as dotted curves in figure 4(a,b) respectively. In figure 4(a) the slope of the solution curve is infinite at ${\it\epsilon}=0$ , as is predicted by the asymptotic expansion (3.4), and in figure 4(b) the cusp predicted by (3.7) at ${\it\epsilon}=0$ is evident. The close overlap between the solid and the broken curves confirms good agreement between the asymptotic theory and the numerical computations. Furthermore, we find that the region of overlap between the solid lines and the broken lines broadens on including higher-order terms in the expansions.

To interpret the numerical results, it is helpful to consider the cnoidal wave solutions that are compatible with a given forcing wavenumber $k$ . In fact, for a chosen wavenumber $k$ , we would expect to obtain branches of periodic wave solutions that emerge continuously (as $|{\it\epsilon}|$ is increased) from unforced cnoidal wave solutions at ${\it\epsilon}=0$ if the cnoidal wavenumber, $m$ , is a rational multiple of $k$ , that is if $m=rk$ for $r\in \mathbb{Q}$ . If we restrict our numerical computations to a truncated domain containing just one wavelength, then we instead require either $r\in \mathbb{N}$ , giving superharmonic cnoidal waves, or $1/r\in \mathbb{N}$ , giving subharmonic cnoidal waves. To this end, it will be useful to consider the possible unforced cnoidal wave frequencies for given values of ${\it\mu}$ and $k$ . We restrict attention to superharmonic cnoidal waves, that is waves with wavelength a fraction of the forcing period, and assume that $r\in \mathbb{N}$ . To inform the discussion in the next section, we present in figure 4(c,d) the maxima and minima of the family of cnoidal waves obtained as periodic solutions to the fully nonlinear problem (3.1) for the case ${\it\mu}=0.1$ . The curves are plotted against the wavenumber ${\it\nu}$ . In the limit of small-amplitude cnoidal waves, corresponding to circuits tight around the point III in figure 1(b), $\max ({\it\eta})$ and $\min ({\it\eta})$ both approach the value $4{\it\mu}/3=1.333$ , and the wavenumber ${\it\nu}$ approaches the critical value ${\it\nu}_{c}=0.7746$ .

Figure 5.

Free-surface profiles for ${\it\mu}=0.10$ : (a–c) $k=0.7746$ , ${\it\epsilon}=-0.0065$ ; (a) ${\it\eta}(0)=0.171$ , (b) ${\it\eta}(0)=0.120$ , (c) ${\it\eta}(0)=0.043$ ; (d–f) $k=0.69378$ , ${\it\epsilon}=0.0005$ ; (d) ${\it\eta}(0)=0.187$ , (e) ${\it\eta}(0)=0.120$ , (f) ${\it\eta}(0)=0.070$ ; (g–i) $k=0.3878$ , ${\it\epsilon}=0.005$ ; (g) ${\it\eta}(0)=0.173$ , (h) ${\it\eta}(0)=0.131$ , (i) ${\it\eta}(0)=0.029$ ; (j–l) $k=0.3$ ; (j) ${\it\epsilon}=0.005$ , ${\it\eta}(0)=0.089$ , (k) ${\it\epsilon}=0.005$ , ${\it\eta}(0)=0.057$ , (l) ${\it\epsilon}=-0.0097$ , ${\it\eta}(0)=0.256$ ; (m–q) $k=0.1$ , ${\it\epsilon}=0.006$ ; (m) ${\it\eta}(0)=0.103$ , (n) ${\it\eta}(0)=0.0796$ , (o) ${\it\eta}(0)=0.0626$ , (p) ${\it\eta}(0)=0.044$ , (q) ${\it\eta}(0)=0.1225$ ; (r) $k=0.1$ , ${\it\epsilon}=-0.006$ , ${\it\eta}(0)=0.2495$ .

5. Concluding remarks

We have studied solutions to the forced Korteweg–de Vries equation under periodic forcing as a model for the flow of water over a spatially periodic bed. We have computed new near-solitary wave solutions, which have periodic tails in the far field. Solutions with a peak of the solitary wave over each crest in the bed and solutions with a peak of the solitary wave over each trough in the bed are both possible. We have also found many bound-state solutions in the same flow which are approximately the superposition of several of these solitary-type waves. It is worth emphasising that in the unforced case (flat bed) the translational invariance of the KdV equation means that the classical $\text{sech}^{2}$ solution, given by (1.3), may be translated by an arbitrary amount to construct infinitely many acceptable solutions; however, it is not permissible to superpose two such solutions to form a multi-humped solution.

It is interesting to draw a connection between the new solutions discussed in the present work and known behaviour in the problem of resonant sloshing in a horizontally oscillating water tank (e.g. Ockendon et al. Reference Ockendon, Ockendon and Johnson1986; Amundsen et al. Reference Amundsen, Cox and Mortell2007). In the latter problem the fKdV equation arises in the limit of small-amplitude oscillations and shallow water. For the driving frequency close to resonance, Ockendon et al. (Reference Ockendon, Ockendon and Johnson1986) used an asymptotic analysis to obtain periodic solutions with very rapid and short-lived spiked bursts occurring at regular intervals in the time signal. Local solutions were constructed inside each of the rapid-burst regions; these solutions had a solitary-type character which was not dissimilar to those we have presented in figure 2 (see their figure 4, for example). We emphasise, however, that these local solutions form part of an overall periodic solution. Our solutions are of genuine solitary-type character, are non-periodic and have a clear physical interpretation in the context of flow over a corrugated bed.

We have left open here the issue of whether the solutions we have computed are stable or unstable, and work on this question is currently underway. The physical relevance of the new solitary-type solutions to fKdV that we have constructed must also be tested by seeking solutions of this type to the fully nonlinear water wave problem. This will also form the subject of a future investigation.