3.1 Boussinesq approximation

The Boussinesq approximation is commonly adopted in the study of stratified fluids when the density variation is small. For the physical system under consideration, under the Boussinesq approximation, both $\unicode[STIX]{x1D6E5}_{1}=(\unicode[STIX]{x1D70C}_{2}-\unicode[STIX]{x1D70C}_{1})/\unicode[STIX]{x1D70C}_{2}$ and $\unicode[STIX]{x1D6E5}_{2}=(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2})/\unicode[STIX]{x1D70C}_{2}$ are assumed small ($\ll 1$) and of the same order of magnitude. Then, the coefficients of the KdV equation (3.3) are simplified to

(3.8)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6FC}=\frac{3}{2}c_{0}\frac{\displaystyle \frac{1}{H_{3}^{2}}\unicode[STIX]{x1D6FE}^{3}+\frac{1}{H_{2}^{2}}(1-\unicode[STIX]{x1D6FE})^{3}-\frac{1}{H_{1}^{2}}}{\displaystyle \frac{1}{H_{3}}\unicode[STIX]{x1D6FE}^{2}+\frac{1}{H_{2}}(1-\unicode[STIX]{x1D6FE})^{2}+\frac{1}{H_{1}}},\\ \displaystyle \unicode[STIX]{x1D6FD}=\frac{1}{6}c_{0}\frac{H_{3}\unicode[STIX]{x1D6FE}^{2}+H_{2}(1+\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D6FE}^{2})+H_{1}}{\displaystyle \frac{1}{H_{3}}\unicode[STIX]{x1D6FE}^{2}+\frac{1}{H_{2}}(1-\unicode[STIX]{x1D6FE})^{2}+\frac{1}{H_{1}}},\end{array}\right\}\end{eqnarray}$$

with $\unicode[STIX]{x1D6FE}$ now given by

(3.9)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}=1+\frac{H_{2}}{H_{1}}-\frac{g_{1}^{\prime }H_{2}}{c_{0}^{2}},\end{eqnarray}$$

or, equivalently,

(3.10)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\left[1+\frac{H_{2}}{H_{3}}-\frac{g_{2}^{\prime }H_{2}}{c_{0}^{2}}\right]^{-1},\end{eqnarray}$$

in order to satisfy the condition for the linear long wave speeds. Here, $g_{1}^{\prime }=g\unicode[STIX]{x1D6E5}_{1}$ and $g_{2}^{\prime }=g\unicode[STIX]{x1D6E5}_{2}$ denote reduced gravities. These coefficients coincide with those presented by Yang et al. (Reference Yang, Fang, Tang and Ramp2010), based on the work by Benney (Reference Benney1966) (see also Benjamin Reference Benjamin1966; Grimshaw Reference Grimshaw1981).

3.2 Criticality condition and polarity of internal solitary waves

Based on KdV theory for (3.3), solitary-wave solutions may have different polarities, according to the sign of the quadratic nonlinearity coefficient $\unicode[STIX]{x1D6FC}$ (notice that $\unicode[STIX]{x1D6FD}>0$), being of elevation (depression) when $\unicode[STIX]{x1D6FC}>0$ (${<}0$). The KdV model also predicts that no solitary-wave solutions exist in the critical case when the quadratic nonlinearity coefficient vanishes, i.e.

(3.11)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D70C}_{3}}{H_{3}^{2}}\unicode[STIX]{x1D6FE}^{3}+\frac{\unicode[STIX]{x1D70C}_{2}}{H_{2}^{2}}(1-\unicode[STIX]{x1D6FE})^{3}-\frac{\unicode[STIX]{x1D70C}_{1}}{H_{1}^{2}}=0,\end{eqnarray}$$

with $\unicode[STIX]{x1D6FE}$ defined by (3.5). Figures 3 and 4 display how the criticality condition (3.11) depends on the physical parameters considered (similar diagrams can be found in Kurkina et al. (Reference Kurkina, Kurkin, Rouvinskaya and Soomere2006) and Yuan, Grimshaw & Johnson (Reference Yuan, Grimshaw and Johnson2018), under the Boussinesq approximation). It must be stressed that, contrary to the two-layer case (with or without a top rigid lid, cf. Choi & Camassa (Reference Choi and Camassa1999) and Barros (Reference Barros2016)), criticality cannot be expressed in a polynomial fashion on the physical parameters for each one of the wave modes considered (see appendix B for further considerations).

Figure 3.

Criticality condition (3.11) for the slow mode (solid line). The shaded region represents the set of parameters for which the left-hand side of (3.11) is positive (i.e. $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE}^{-})>0$), and, therefore, KdV theory predicts convex mode-2 waves. The axes have been non-dimensionalized by the total depth $H=H_{1}+H_{2}+H_{3}$ and the diagrams are presented on the $(H_{1}/H,H_{2}/H)$-plane. The stratification is given by (2.12), with $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E5}_{2}$, and $\unicode[STIX]{x1D6E5}_{2}=0.01$. (a) $\unicode[STIX]{x1D6FF}=0.5$, (b) $\unicode[STIX]{x1D6FF}=1$, (c) $\unicode[STIX]{x1D6FF}=2$.

Figure 4.

Criticality condition (3.11) for the fast mode (solid line). The shaded region represents the set of parameters for which the left-hand side of (3.11) is positive (i.e. $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE}^{+})>0$), and, therefore, KdV theory predicts mode-1 waves of elevation. The axes have been non-dimensionalized by the total depth $H=H_{1}+H_{2}+H_{3}$ and the diagrams are presented on the $(H_{1}/H,H_{2}/H)$-plane. The stratification is given by (2.12), with $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E5}_{2}$, and $\unicode[STIX]{x1D6E5}_{2}=0.01$. (a) $\unicode[STIX]{x1D6FF}=0.5$, (b) $\unicode[STIX]{x1D6FF}=1$, (c) $\unicode[STIX]{x1D6FF}=2$.

4.1 Linearization at the origin

Clearly, the origin is a critical point of the dynamical system composed by (4.5)–(4.6). When linearized about the origin, the system can be cast into matrix form

(4.15)$$\begin{eqnarray}A\unicode[STIX]{x1D73B}^{\prime \prime }+B\unicode[STIX]{x1D73B}=0,\end{eqnarray}$$

by introducing $\unicode[STIX]{x1D73B}=(\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2})$ and symmetric matrices

(4.16)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle A=c^{2}\left[\begin{array}{@{}cc@{}}\displaystyle {\textstyle \frac{1}{3}}(\unicode[STIX]{x1D70C}_{1}H_{1}+\unicode[STIX]{x1D70C}_{2}H_{2}) & \displaystyle {\textstyle \frac{1}{6}}\unicode[STIX]{x1D70C}_{2}H_{2}\\ \displaystyle {\textstyle \frac{1}{6}}\unicode[STIX]{x1D70C}_{2}H_{2} & \displaystyle {\textstyle \frac{1}{3}}(\unicode[STIX]{x1D70C}_{2}H_{2}+\unicode[STIX]{x1D70C}_{3}H_{3})\end{array}\right],\\ \displaystyle B=\left[\begin{array}{@{}cc@{}}\displaystyle g(\unicode[STIX]{x1D70C}_{2}-\unicode[STIX]{x1D70C}_{1})-\left(\frac{\unicode[STIX]{x1D70C}_{1}}{H_{1}}+\frac{\unicode[STIX]{x1D70C}_{2}}{H_{2}}\right)c^{2} & \displaystyle c^{2}\frac{\unicode[STIX]{x1D70C}_{2}}{H_{2}}\\ \displaystyle c^{2}\frac{\unicode[STIX]{x1D70C}_{2}}{H_{2}} & \displaystyle g(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2})-\left(\frac{\unicode[STIX]{x1D70C}_{2}}{H_{2}}+\frac{\unicode[STIX]{x1D70C}_{3}}{H_{3}}\right)c^{2}\end{array}\right].\end{array}\right\}\end{eqnarray}$$

Looking for values of $\unicode[STIX]{x1D706}$ for which there are solutions of (4.15) of the form $\unicode[STIX]{x1D73B}=\boldsymbol{c}\text{e}^{\unicode[STIX]{x1D706}X}$, with $\boldsymbol{c}$ a non-zero vector, is equivalent to finding the eigenvalues of the system, given by the solutions of

(4.17)$$\begin{eqnarray}\operatorname{det}(B+\unicode[STIX]{x1D706}^{2}A)=0.\end{eqnarray}$$

Moreover, it can be easily checked that $A$ is a positive definite matrix. A classical result from linear algebra asserts that $\unicode[STIX]{x1D706}^{2}$ is a real root of the equation

(4.18)$$\begin{eqnarray}a_{0}\unicode[STIX]{x1D706}^{4}+a_{2}\unicode[STIX]{x1D706}^{2}+a_{4}=0,\end{eqnarray}$$

with $a_{0}>0$ and $a_{4}=\operatorname{det}B$, which is just a multiple of the right-hand side of (2.15). Collision of eigenvalues can only occur at the origin, i.e. when $c=c_{0}^{\pm }$. Moreover, their nature will change with different values of the wave speed. It can be shown that: the origin is a centre–centre equilibrium in the range $]\!0,c_{0}^{-}\![$, i.e. one has four pure imaginary eigenvalues $\pm \text{i}\unicode[STIX]{x1D706}_{1},\pm i\unicode[STIX]{x1D706}_{2}$; the origin is a saddle–centre in the range $]\!c_{0}^{-},c_{0}^{+}\! [$, i.e. one has two real eigenvalues $\pm \unicode[STIX]{x1D706}_{3}$ and two pure imaginary eigenvalues $\pm \text{i}\unicode[STIX]{x1D706}_{4}$; the origin is a saddle–saddle in the range $]\!c_{0}^{+},\infty \![$, i.e. one has four real eigenvalues $\pm \unicode[STIX]{x1D706}_{5},\pm \unicode[STIX]{x1D706}_{6}$, with $\unicode[STIX]{x1D706}_{i}$ all positive.

4.2 System reduction under Boussinesq approximation

The Hamiltonian structure is preserved under the Boussinesq approximation and the corresponding solitary-wave solutions are governed by the Euler–Lagrange equations for the following Lagrangian $\widetilde{{\mathcal{L}}}$:

(4.19)$$\begin{eqnarray}\widetilde{{\mathcal{L}}}(\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2},\unicode[STIX]{x1D701}_{1}^{\prime },\unicode[STIX]{x1D701}_{2}^{\prime })=\frac{1}{6}c^{2}\left(\frac{H_{1}^{2}}{h_{1}}+\frac{H_{2}^{2}}{h_{2}}\right)\unicode[STIX]{x1D701}_{1}^{\prime 2}+\frac{1}{6}c^{2}\frac{H_{2}^{2}}{h_{2}}\,\unicode[STIX]{x1D701}_{1}^{\prime }\unicode[STIX]{x1D701}_{2}^{\prime }+\frac{1}{6}c^{2}\left(\frac{H_{2}^{2}}{h_{2}}+\frac{H_{3}^{2}}{h_{3}}\right)\unicode[STIX]{x1D701}_{2}^{\prime 2}-\widetilde{V}(\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2}),\end{eqnarray}$$

with $\widetilde{V}$ defined by

(4.20)$$\begin{eqnarray}\widetilde{V}(\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2})=\frac{1}{2}\left\{-c^{2}\left(\frac{H_{1}^{2}}{h_{1}}+\frac{H_{2}^{2}}{h_{2}}+\frac{H_{3}^{2}}{h_{3}}\right)+g_{1}^{\prime }\,\unicode[STIX]{x1D701}_{1}^{2}+g_{2}^{\prime }\,\unicode[STIX]{x1D701}_{2}^{2}+c^{2}(H_{1}+H_{2}+H_{3})\right\}.\end{eqnarray}$$

Furthermore, the dynamical system

(4.21)$$\begin{eqnarray}\widetilde{{\mathcal{L}}}_{\unicode[STIX]{x1D701}_{i}}-\frac{\text{d}}{\text{d}X}(\widetilde{{\mathcal{L}}}_{\unicode[STIX]{x1D701}_{i}^{\prime }})=0,\quad i=1,2,\end{eqnarray}$$

can be cast into the form

(4.22a,b)$$\begin{eqnarray}T_{1}[\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2}]=0,\quad T_{2}[\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2}]=0,\end{eqnarray}$$

which correspond to a system of differential equations

(4.23)$$\begin{eqnarray}\displaystyle & & \displaystyle c^{2}\left\{\frac{1}{3}\left(\frac{H_{1}^{2}}{h_{1}}+\frac{H_{2}^{2}}{h_{2}}\right)\unicode[STIX]{x1D701}_{1}^{\prime \prime }+\frac{1}{6}\frac{H_{2}^{2}}{h_{2}}\,\unicode[STIX]{x1D701}_{2}^{\prime \prime }+\frac{1}{6}\left(\frac{H_{1}^{2}}{h_{1}^{2}}-\frac{H_{2}^{2}}{h_{2}^{2}}\right)\unicode[STIX]{x1D701}_{1}^{\prime 2}\right.\nonumber\\ \displaystyle & & \displaystyle \quad +\left.\frac{1}{3}\frac{H_{2}^{2}}{h_{2}^{2}}\,\unicode[STIX]{x1D701}_{1}^{\prime }\unicode[STIX]{x1D701}_{2}^{\prime }+\frac{1}{3}\frac{H_{2}^{2}}{h_{2}^{2}}\,\unicode[STIX]{x1D701}_{2}^{\prime 2}\right\}=\frac{1}{2}c^{2}\left(\frac{H_{1}^{2}}{h_{1}^{2}}-\frac{H_{2}^{2}}{h_{2}^{2}}\right)-g_{1}^{\prime }\unicode[STIX]{x1D701}_{1},\end{eqnarray}$$

(4.24)$$\begin{eqnarray}\displaystyle & & \displaystyle c^{2}\left\{\frac{1}{6}\frac{H_{2}^{2}}{h_{2}}\,\unicode[STIX]{x1D701}_{1}^{\prime \prime }+\frac{1}{3}\left(\frac{H_{2}^{2}}{h_{2}}+\frac{H_{3}^{2}}{h_{3}}\right)\unicode[STIX]{x1D701}_{2}^{\prime \prime }-\frac{1}{3}\frac{H_{2}^{2}}{h_{2}^{2}}\,\unicode[STIX]{x1D701}_{1}^{\prime 2}\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.\frac{1}{3}\frac{H_{2}^{2}}{h_{2}^{2}}\,\unicode[STIX]{x1D701}_{1}^{\prime }\unicode[STIX]{x1D701}_{2}^{\prime }+\frac{1}{6}\left(\frac{H_{2}^{2}}{h_{2}^{2}}-\frac{H_{3}^{2}}{h_{3}^{2}}\right)\unicode[STIX]{x1D701}_{2}^{\prime 2}\right\}=\frac{1}{2}c^{2}\left(\frac{H_{2}^{2}}{h_{2}^{2}}-\frac{H_{3}^{2}}{h_{3}^{2}}\right)-g_{2}^{\prime }\unicode[STIX]{x1D701}_{2}.\end{eqnarray}$$

We note here that the equations under the Boussinesq approximation are invariant under the transformation $g\rightarrow -g$ and $\unicode[STIX]{x1D6E5}_{1}\rightarrow -\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2}\rightarrow -\unicode[STIX]{x1D6E5}_{2}$. Thus one may look at the interfaces ‘upside down’. Under this condition, for every solution pair $(\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2})$ of a given configuration with densities $\unicode[STIX]{x1D70C}_{0}(1-\unicode[STIX]{x1D6E5}_{1}),\unicode[STIX]{x1D70C}_{0},\unicode[STIX]{x1D70C}_{0}(1+\unicode[STIX]{x1D6E5}_{2})$ and undisturbed thickness of layers $H_{1}$, $H_{2}$, $H_{3}$ (from top to bottom), $(-\unicode[STIX]{x1D701}_{2},-\unicode[STIX]{x1D701}_{1})$ is a solution of the modified physical configuration with densities $\unicode[STIX]{x1D70C}_{0}(1-\unicode[STIX]{x1D6E5}_{2}),\unicode[STIX]{x1D70C}_{0},\unicode[STIX]{x1D70C}_{0}(1+\unicode[STIX]{x1D6E5}_{1})$, where the undisturbed thickness of layers are $H_{3}$, $H_{2}$, $H_{1}$ (from top to bottom).

5.1 Case (i): small density difference between two adjacent layers

In this case, as $1-\unicode[STIX]{x1D70E}_{1}\ll 1$, or $\unicode[STIX]{x1D6E5}_{1}\equiv 1-\unicode[STIX]{x1D70E}_{1}\ll 1$, the top and middle layers have nearly identical densities. Then the long wave speeds can be approximated, from (2.15), by

(5.4)$$\begin{eqnarray}\displaystyle & \displaystyle (c_{0}^{+})^{2}=\frac{g(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2})H_{3}(H_{1}+H_{2})}{\unicode[STIX]{x1D70C}_{2}H_{3}+\unicode[STIX]{x1D70C}_{3}(H_{1}+H_{2})}[1+O(\unicode[STIX]{x1D6E5}_{1})], & \displaystyle\end{eqnarray}$$

(5.5)$$\begin{eqnarray}\displaystyle & \displaystyle (c_{0}^{-})^{2}=\frac{gH_{1}H_{2}}{H_{1}+H_{2}}\left[\unicode[STIX]{x1D6E5}_{1}+\frac{H_{2}(H_{1}+H_{2})(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2})-\unicode[STIX]{x1D70C}_{2}H_{1}^{2}}{(H_{1}+H_{2})^{2}(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2})}\,\unicode[STIX]{x1D6E5}_{1}^{2}+O(\unicode[STIX]{x1D6E5}_{1}^{3})\right]. & \displaystyle\end{eqnarray}$$

This being the case, the displacement ratios for the two wave modes can be approximated, from (2.16), by

(5.6a,b)$$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D701}_{2}}{\unicode[STIX]{x1D701}_{1}}\right)^{+}=\frac{H_{1}+H_{2}}{H_{1}}+O(\unicode[STIX]{x1D6E5}_{1}),\quad \left(\frac{\unicode[STIX]{x1D701}_{2}}{\unicode[STIX]{x1D701}_{1}}\right)^{-}=-\frac{\unicode[STIX]{x1D70C}_{2}H_{1}}{(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2})(H_{1}+H_{2})}\,\unicode[STIX]{x1D6E5}_{1}+O(\unicode[STIX]{x1D6E5}_{1}^{2}).\end{eqnarray}$$

As can be seen (5.4), the first baroclinic mode of a three-layer system in this limit is almost equivalent to that of a two-layer system, where the density and thickness are given, respectively, by $\unicode[STIX]{x1D70C}_{2}$ and $H_{1}+H_{2}$ for the upper layer and $\unicode[STIX]{x1D70C}_{3}$ and $H_{3}$ for the lower layer.

For the second baroclinic mode, since $(\unicode[STIX]{x1D701}_{2}/\unicode[STIX]{x1D701}_{1})^{-}=O(\unicode[STIX]{x1D6E5}_{1})$, the displacement of the lower interface can be neglected for $\unicode[STIX]{x1D6E5}_{1}\ll 1$, which implies that the second interface can be replaced by a rigid boundary for the second baroclinic mode. Therefore, once again, the second baroclinic mode of a three-layer system can be approximated by the internal wave mode of a two-layer system, whose density and thickness are given, respectively, by $\unicode[STIX]{x1D70C}_{1}$ and $H_{1}$ for the upper layer and $\unicode[STIX]{x1D70C}_{2}$ and $H_{2}$ for the lower layer. This observation is also valid for the nonlinear problem, as shown next.

After writing $\unicode[STIX]{x1D70E}_{1}=1-\unicode[STIX]{x1D700}$ (or $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D700}\ll 1$), we assume from (5.5) that $F^{2}=\unicode[STIX]{x1D700}\,C$ with $C=O(1)$ for the slow mode. We seek approximate solutions of (5.1)–(5.2) by inserting into this set of equations the following expansions of the interface displacements: $\unicode[STIX]{x1D701}_{k}=\unicode[STIX]{x1D701}_{k,0}+\unicode[STIX]{x1D700}\,\unicode[STIX]{x1D701}_{k,1}+O(\unicode[STIX]{x1D700}^{2}),\,(k=1,2)$. The same representation will be adopted for each layer thickness $h_{i}=h_{i,0}+\unicode[STIX]{x1D700}h_{i,1}+O(\unicode[STIX]{x1D700}^{2}),\,(i=1,2,3)$. At leading order, it immediately follows from (5.2) that $\unicode[STIX]{x1D701}_{2,0}=0$, as expected from (5.6). We then conclude from (5.1) that $\unicode[STIX]{x1D701}_{1,0}$ is governed by

(5.7)$$\begin{eqnarray}C\left\{\frac{1}{3}\left(\frac{1}{h_{1,0}}+\frac{r_{2}^{2}}{h_{2,0}}\right)\unicode[STIX]{x1D701}_{1,0}^{\prime \prime }+\frac{1}{6}\left(\frac{\unicode[STIX]{x1D70E}_{1}}{h_{1,0}^{2}}-\frac{r_{2}^{2}}{h_{2,0}^{2}}\right)\unicode[STIX]{x1D701}_{1,0}^{\prime 2}\right\}=\frac{1}{2}C\left(\frac{1}{h_{1,0}^{2}}-\frac{r_{2}^{2}}{h_{2,0}^{2}}\right)-\unicode[STIX]{x1D701}_{1,0}.\end{eqnarray}$$

After integrating this once after multiplying an integrating factor, or, equivalently, setting to zero for homoclinic trajectories the Hamiltonian $\mathbb{H}$ defined by (4.13) with $\unicode[STIX]{x1D701}_{2,0}=0$, it can be shown that $\unicode[STIX]{x1D701}_{1,0}$ is a solution of

(5.8)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{1,0}^{\prime 2}=\frac{3}{C}\frac{\unicode[STIX]{x1D701}_{1,0}^{2}[\unicode[STIX]{x1D701}_{1,0}^{2}-(1-r_{2})\,\unicode[STIX]{x1D701}_{1,0}+C(1+r_{2})-r_{2}]}{(1+r_{2})[(1-r_{2})\unicode[STIX]{x1D701}_{1,0}+r_{2}]}.\end{eqnarray}$$

When converted back to dimensional variables, we find that the upper interface is approximated by the solution of the equation

(5.9)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{1}^{\prime 2}=\frac{3}{c^{2}}\frac{\unicode[STIX]{x1D701}_{1}^{2}[c^{2}(H_{2}+\unicode[STIX]{x1D701}_{1})+c^{2}(H_{1}-\unicode[STIX]{x1D701}_{1})-g\unicode[STIX]{x1D700}(H_{1}-\unicode[STIX]{x1D701}_{1})(H_{2}+\unicode[STIX]{x1D701}_{1})]}{c^{2}H_{1}^{2}(H_{2}+\unicode[STIX]{x1D701}_{1})+c^{2}H_{2}^{2}(H_{1}-\unicode[STIX]{x1D701}_{1})},\end{eqnarray}$$

which is simply the MCC equation for the interface of two liquid layers confined by two rigid walls placed at $z=H_{1}$ and $z=-H_{2}$, under the Boussinesq approximation. In this regime, we expect a large displacement of the interface $z=\unicode[STIX]{x1D701}_{1}(x,t)$ characterized by single-humped profile that broadens as the wave speed increases, while the displacement of the interface $z=-H_{2}+\unicode[STIX]{x1D701}_{2}(x,t)$ is almost imperceptible.

The reverse situation occurs when $\unicode[STIX]{x1D70E}_{3}=1+\unicode[STIX]{x1D700}$ ($\unicode[STIX]{x1D700}\ll 1$) while $1-\unicode[STIX]{x1D70E}_{1}=O(1)$. The solutions of the second baroclinic mode are characterized by a large displacement of the lower interface $z=-H_{2}+\unicode[STIX]{x1D701}_{2}$, with an almost imperceptible displacement of the upper interface $z=\unicode[STIX]{x1D701}_{1}$.

5.2 Case (ii): small density difference across all three layers

When both density increments are small and of the same order so that $\unicode[STIX]{x1D6E5}_{2}=O(\unicode[STIX]{x1D6E5}_{1})=O(\unicode[STIX]{x1D700})$, with $\unicode[STIX]{x1D700}\ll 1$, then the two linear long wave speeds become comparable and proportional to $\unicode[STIX]{x1D700}^{1/2}$, as can be seen from (5.4)–(5.5).

For simplicity, we consider the case when the density increments across the first and second interfaces are the same so that $\unicode[STIX]{x1D70E}_{1}=1-\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D70E}_{3}=1+\unicode[STIX]{x1D700}\;(\unicode[STIX]{x1D700}\ll 1)$. Therefore, we set $F^{2}=\unicode[STIX]{x1D700}\,C$ with $C=O(1)$ and let all other quantities be $O(1)$. As before, approximate solutions of (5.1)–(5.2) are sought by expanding the interface displacements in powers of $\unicode[STIX]{x1D700}$. Unfortunately, solving the leading-order system is as difficult as solving the full dynamical system. However, when the thicknesses of the top and bottom layers are almost the same ($r_{3}\approx 1$), some simplification can be made. As we will see, at leading order, mode-2 solutions are anti-symmetric and are governed by a Hamiltonian system with one degree of freedom.

To find such an approximate system, we assume the density increment across the layers is small and the same so that we can impose the Boussinesq approximation to the original system with $g_{1}^{\prime }=g_{2}^{\prime }=g^{\prime }$. Furthermore, if the thicknesses of the unperturbed top and bottom layers are assumed the same so that $H_{1}=H_{3}$ (known as the symmetric configuration), the system (4.22) enjoys the following property:

(5.10)$$\begin{eqnarray}T_{2}[\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2}]=-T_{1}[-\unicode[STIX]{x1D701}_{2},-\unicode[STIX]{x1D701}_{1}].\end{eqnarray}$$

This implies that, if $(\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2})$ is a solution, $(-\unicode[STIX]{x1D701}_{2},-\unicode[STIX]{x1D701}_{1})$ is also a solution. In particular, anti-symmetric solutions of the form $\unicode[STIX]{x1D701}_{2}=-\unicode[STIX]{x1D701}_{1}$ are worth being considered, as the original system given by (4.5)–(4.6) or (5.1)–(5.2) can be reduced to a single equation. We remark that such constraint is in agreement with the predictions of the weakly nonlinear theory for the mode-2 ISW, given that the proportionality constant $\unicode[STIX]{x1D6FE}^{-}$ between the displacements of the two interfaces is simply $-1$, according to (3.9) with $(c_{0}^{-})^{2}=g^{\prime }H_{1}H_{2}/(2H_{1}+H_{2})$.

The anti-symmetric solutions can be found by solving the following leading-order equation, in dimensional variables, for $\unicode[STIX]{x1D701}_{1}$:

(5.11)$$\begin{eqnarray}c^{2}\left\{\left(\frac{1}{3}\frac{H_{1}^{2}}{h_{1}}+\frac{1}{6}\frac{H_{2}^{2}}{h_{2}}\right)\unicode[STIX]{x1D701}_{1}^{\prime \prime }+\frac{1}{6}\left(\frac{H_{1}^{2}}{h_{1}^{2}}-\frac{H_{2}^{2}}{h_{2}^{2}}\right)\unicode[STIX]{x1D701}_{1}^{\prime 2}\right\}=\frac{1}{2}c^{2}\left(\frac{H_{1}^{2}}{h_{1}^{2}}-\frac{H_{2}^{2}}{h_{2}^{2}}\right)-g^{\prime }\unicode[STIX]{x1D701}_{1},\end{eqnarray}$$

which can be integrated once to yield the Hamiltonian

(5.12)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{1}^{\prime 2}=\frac{6}{c^{2}}\frac{\unicode[STIX]{x1D701}_{1}^{2}[(2H_{1}+H_{2})c^{2}-g^{\prime }h_{1}h_{2}]}{2H_{1}^{2}h_{2}+H_{2}^{2}h_{1}},\end{eqnarray}$$

where $h_{1}=H_{1}-\unicode[STIX]{x1D701}_{1}$ and $h_{2}=2\unicode[STIX]{x1D701}_{1}+H_{2}$. This may be rewritten as

(5.13)$$\begin{eqnarray}\frac{1}{2}\unicode[STIX]{x1D701}_{1}^{\prime 2}=\left[\frac{6g^{\prime }}{(4H_{1}^{2}-H_{2}^{2})c^{2}}\right]\frac{\unicode[STIX]{x1D701}_{1}^{2}(\unicode[STIX]{x1D701}_{1}-a_{-})(\unicode[STIX]{x1D701}_{1}-a_{+})}{\unicode[STIX]{x1D701}_{1}-a_{\ast }}\equiv -K(\unicode[STIX]{x1D701}_{1}).\end{eqnarray}$$

In (5.13), $a_{\ast }$ is given by

(5.14)$$\begin{eqnarray}a_{\ast }=-\frac{H_{1}H_{2}}{2H_{1}-H_{2}},\end{eqnarray}$$

and $a_{\pm }$ are the two roots of a quadratic equation,

(5.15)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{1}^{2}+q_{1}\unicode[STIX]{x1D701}_{1}+q_{2}=0,\end{eqnarray}$$

with $q_{1}$ and $q_{2}$ defined by

(5.16a,b)$$\begin{eqnarray}q_{1}=-\frac{1}{2}(2H_{1}-H_{2}),\quad q_{2}=\frac{1}{2}H_{1}H_{2}\left(\frac{c^{2}}{(c_{0}^{-})^{2}}-1\right).\end{eqnarray}$$

A homoclinic orbit for (5.13) is possible when $K$ is negative through $\unicode[STIX]{x1D701}_{1}=0$ and $\unicode[STIX]{x1D701}_{1}=a_{\pm }$. For this to happen, the leading order of $K$ must be negative in a neighbourhood of the origin. We use Taylor series to expand $K$ around $\unicode[STIX]{x1D701}_{1}=0$ as

(5.17)$$\begin{eqnarray}K(\unicode[STIX]{x1D701}_{1})=-\frac{3g^{\prime }}{(2H_{1}+H_{2})c^{2}}\left(\frac{c^{2}}{(c_{0}^{-})^{2}}-1\right)\unicode[STIX]{x1D701}_{1}^{2}+O(\unicode[STIX]{x1D701}_{1}^{3}),\end{eqnarray}$$

from which it follows that solitary waves exist only if $c^{2}>(c_{0}^{-})^{2}$. Furthermore, when the roots $a_{\pm }$ coincide, the dynamical system admits a heteroclinic, rather than homoclinic, solution, connecting the origin $\unicode[STIX]{x1D701}_{1}=0$ to the critical point $\unicode[STIX]{x1D701}_{1}=a_{m}\equiv (2H_{1}-H_{2})/4$. This peculiar scenario occurs at the speed $c_{m}^{2}=g^{\prime }(2H_{1}+H_{2})/8$, and the corresponding solution is known as a front wave solution. Different wave polarities are possible depending on the sign of $2H_{1}-H_{2}$. More precisely, if $H_{2}<2H_{1}$, then $\unicode[STIX]{x1D701}_{1}(X)$ is a wave of elevation. Otherwise, it is a wave of depression. In the latter case, a curious feature holds: the value of $c_{m}$ can actually exceed the linear long wave speed $c_{0}^{+}$, provided $H_{2}>6H_{1}$. Such peculiar case will be further discussed in § 5.4 (see figure 16).

Figure 5.

Anti-symmetric solutions governed by (5.13). These approximate the mode-2 solitary waves in a weakly stratified flow with the same density increment across the layers and nearly identical depths in equilibrium for the top and bottom layers. The upper and lower interfaces are plotted in dimensionless variables, i.e. $\unicode[STIX]{x1D702}_{2}/H_{1}$ and $\unicode[STIX]{x1D702}_{3}/H_{1}$ are plotted as functions of $(x-ct)/H_{1}$. (a) $H_{2}/H_{1}=0.5$ and $a/H_{1}=0.1$, $0.3$, $0.374$. The corresponding wave speeds for $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6E5}_{2}=10^{-3}$ are $c^{2}/g^{\prime }H_{1}\approx 0.252,0.308,0.313$. (b) $H_{2}/H_{1}=5$ and $a/H_{1}=-0.1$, $-0.5$, $-0.74$. The corresponding wave speeds for $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6E5}_{2}=10^{-3}$ are $c^{2}/g^{\prime }H_{1}\approx 0.754,0.857,0.875$.

Equation (5.15) shows that the wave speed $c$ can be written in terms of amplitude $a$ as

(5.18)$$\begin{eqnarray}\frac{c^{2}}{(c_{0}^{-})^{2}}=\frac{(H_{1}-a)(H_{2}+2a)}{H_{1}H_{2}},\end{eqnarray}$$

which can be convenient to plot the solutions of (5.13) as in figure 5.

Finally, and remarkably, the dynamical system obtained in this limit is precisely the same as in the MCC (two-layer) model, under the Boussinesq approximation, when the layers in equilibrium have thicknesses $H_{1}$ and $H_{2}/2$ (see Gavrilov & Liapidevskii Reference Gavrilov and Liapidevskii2010).

5.3 Case (iii): thin transition layer with weak stratification

Another interesting limit is case (iii), where the middle layer is much thinner than the top and bottom layers, i.e. $r_{2}\equiv H_{2}/H_{1}\ll 1$ while $H_{3}/H_{1}=O(1)$. In this case, the linear long wave speeds can be approximated by

(5.19a,b)$$\begin{eqnarray}(c_{0}^{+})^{2}=\frac{g(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{1})H_{1}H_{3}}{\unicode[STIX]{x1D70C}_{1}H_{3}+\unicode[STIX]{x1D70C}_{3}H_{1}}\left[1+O(r_{2})\right],\quad (c_{0}^{-})^{2}=\frac{g(\unicode[STIX]{x1D70C}_{2}-\unicode[STIX]{x1D70C}_{1})(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2})H_{1}}{\unicode[STIX]{x1D70C}_{2}(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{1})}[r_{2}+O(r_{2}^{2})].\end{eqnarray}$$

Then, from (2.16), the displacement ratios for the two wave modes can be estimated as

(5.20a,b)$$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D701}_{2}}{\unicode[STIX]{x1D701}_{1}}\right)^{+}=1+\frac{\unicode[STIX]{x1D70C}_{1}(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2})H_{3}+\unicode[STIX]{x1D70C}_{3}(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})H_{1}}{\unicode[STIX]{x1D70C}_{2}(\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{1})H_{3}}\,r_{2}+O(r_{2}^{2}),\quad \left(\frac{\unicode[STIX]{x1D701}_{2}}{\unicode[STIX]{x1D701}_{1}}\right)^{-}=\frac{\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2}}{\unicode[STIX]{x1D70C}_{3}-\unicode[STIX]{x1D70C}_{2}}+O(r_{2}).\end{eqnarray}$$

To find the solitary-wave solution of the first baroclinic mode, the thin middle layer can be neglected at leading order. Therefore, for the first baroclinic mode, the solitary-wave solutions of the three-layer system are expected to be in good agreement with those of the two-layer MCC model, as shown in appendix D. On the other hand, for the second baroclinic mode, more elaborate solutions can be found as discussed in the following.

Inspired by the work of Gavrilov et al. (Reference Gavrilov, Liapidevskii and Liapidevskaya2013), who first considered a variant of the strongly nonlinear multi-layer model (Choi Reference Choi, Goda, Ikehata and Suzuki2000) to study the propagation of mode-2 waves in a three-layer flow with a thin intermediate layer, we consider the case of a weakly stratified flow in which $\unicode[STIX]{x1D70E}_{1}=1-\unicode[STIX]{x1D700}^{m}$ and $\unicode[STIX]{x1D70E}_{3}=1+\unicode[STIX]{x1D700}^{m}$, where $m$ is a positive integer and $\unicode[STIX]{x1D700}\ll 1$ measures the thickness of the thin intermediate layer, i.e. $r_{2}=O(\unicode[STIX]{x1D716})$. By applying the Boussinesq approximation to (5.19), the linear long wave speeds can be approximated as

(5.21a,b)$$\begin{eqnarray}(c_{0}^{-})^{2}=\frac{1}{2}g^{\prime }H_{1}[r_{2}+O(r_{2}^{2})],\quad (c_{0}^{+})^{2}=\frac{2g^{\prime }H_{1}H_{3}}{H_{1}+H_{3}}[1+O(r_{2})],\end{eqnarray}$$

where $g^{\prime }=g\unicode[STIX]{x1D700}^{m}$. Therefore, regardless the value of $m$, one can see that the linear long wave speeds have the following orders of magnitude: $(c_{0}^{-})^{2}/gH_{1}=O(\unicode[STIX]{x1D700}^{1+m})$ and $(c_{0}^{+})^{2}/gH_{1}=O(\unicode[STIX]{x1D700}^{m})$.

Following the strategy employed for the previous cases, one would assume, from (5.19), $F^{2}=O(\unicode[STIX]{x1D700}^{1+m})$ for mode-2 waves and seek asymptotic solutions to (5.1)–(5.2) by expanding the interface displacements in powers of $\unicode[STIX]{x1D700}$ as $\unicode[STIX]{x1D701}_{k}=\unicode[STIX]{x1D701}_{k,0}+\unicode[STIX]{x1D700}\,\unicode[STIX]{x1D701}_{k,1}+O(\unicode[STIX]{x1D700}^{2}),\,(k=1,2)$, while letting all other quantities to be $O(1)$. However, given that $\unicode[STIX]{x1D701}_{k,p}\equiv 0$ at any order $O(\unicode[STIX]{x1D716}^{p})$, this would yield the null solution. This implies that no finite amplitude solutions ($\unicode[STIX]{x1D701}_{k,0}\neq 0$) exist under the assumption of $F^{2}=O(\unicode[STIX]{x1D700}^{1+m})$. Note that this does not preclude the existence of solitary waves under the assumption of $\unicode[STIX]{x1D701}_{k}/H_{1}=O(r_{2})=O(\unicode[STIX]{x1D700})$, for a small range of wave speeds. For example, in the special case when $H_{3}=H_{1}$, anti-symmetric solutions can be found from (5.13) under the condition of $H_{2}/H_{1}\ll 1$.

One question that remains to be answered is if there exists any mode-2 solitary-wave solution of finite amplitude in a three-layer system with a thin transition layer so that $\unicode[STIX]{x1D701}_{k,0}\neq 0$ although the solution might not be connected smoothly with small amplitude solutions. More specifically, we are looking for solutions whose amplitudes are comparable to $H_{1}$, which have been observed experimentally. To find such a solution, one might need to adopt a different approach, which will be further investigated in detail in § 6.

5.4 Numerical solutions

This section is not intended to address an exhaustive description of all possible internal solitary-wave solutions to the strongly nonlinear long wave model given by (4.5)–(4.6). Instead, we will focus on the second baroclinic mode for the cases of weak stratification, in particular, cases (i) and (ii), for which the asymptotic solutions have been obtained here.

After fixing the density ratios ($\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}$ and $\unicode[STIX]{x1D70C}_{3}/\unicode[STIX]{x1D70C}_{2}$) and the depth ratios ($H_{2}/H_{1}$ and $H_{3}/H_{1}$), the wave profiles are calculated using a shooting method as in Barros & Gavrilyuk (Reference Barros and Gavrilyuk2007). Figure 6 illustrates solutions obtained for case (i), where the top and intermediate liquid layers have close density values ($\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}=0.9$), while a greater density difference is allowed for the bottom liquid layer ($\unicode[STIX]{x1D70C}_{3}/\unicode[STIX]{x1D70C}_{2}=2.0$). As expected from the asymptotic analysis in § 5.1, the deformation of the lower interface is almost imperceptible, while the displacement of the top interface is large and starts broadening as the wave speed increases.

Figure 6.

Mode-2 internal solitary-wave solutions computed numerically for the dynamical system (4.5)–(4.6) for case (i). The upper and lower interfaces are plotted in dimensionless variables, i.e. $\unicode[STIX]{x1D701}_{1}/H_{1}$ and $\unicode[STIX]{x1D701}_{2}/H_{1}$ are plotted as functions of $(x-ct)/H_{1}$. We set $\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}=0.9$, $\unicode[STIX]{x1D70C}_{3}/\unicode[STIX]{x1D70C}_{2}=2.0$, $H_{2}/H_{1}=0.5$, $H_{3}/H_{1}=1.0$ and consider different values for the wave speed. (a) $c^{2}/g_{1}^{\prime }H_{1}\approx 0.37$. (b) $c^{2}/g_{1}^{\prime }H_{1}\approx 0.387698$.

In figure 7, we consider case (ii), where the stratification is weak everywhere. Here we assume that the density increment across the layers is constant ($\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}=0.999$, $\unicode[STIX]{x1D70C}_{3}/\unicode[STIX]{x1D70C}_{2}=1.001$) and, in equilibrium, the top and bottom layers have the same depth, or $H_{3}/H_{1}=1$. Because the Boussinesq approximation is not being imposed here, a slight break of symmetry of the dynamical system occurs. However, as predicted in § 5.2, solutions may still be well approximated by anti-symmetric solutions. Our numerical tests have revealed that when the intermediate layer is thick ($H_{2}/H_{1}=5$), solutions can be easily computed for any given wave speed. Waves with a single-hump profile seem to persist up to large values of the wave speed, and become broader as the wave speed increases, as shown in figure 7(a,b).

When the intermediate layer is thinner than the other two layers, e.g. $H_{2}/H_{1}=0.5$, solutions with one single-hump profile become more difficult to compute. While we could find for $H_{2}/H_{1}=0.5$ asymptotic solutions with large amplitudes approaching a front, as shown in figure 5(a), we are able to find numerical solutions with wave amplitudes relatively smaller than the maximum amplitude predicted by the asymptotics. The solution presented in figure 7(c) is close to the largest amplitude wave solution we can compute with the current numerical method. It is not so clear if the solitary-wave solutions of the original system cease to exist at a smaller wave amplitude than the asymptotic prediction, or if an improved numerical method is required to larger amplitude waves of the original dynamical system with two degrees of freedom.

As the asymptotics in § 5.3 suggests, when the thickness of the middle layer decreases further, finding the large amplitude solitary-wave solutions of the original system given by (4.5)–(4.6) is a delicate subject. For this reason we reserve the following section for a detailed analysis for the case of a thin transition layer, complemented by numerics.

Figure 7.

Mode-2 internal solitary-wave solutions computed numerically for the dynamical system (4.5)–(4.6) for case (ii) with $H_{1}=H_{3}$. The upper and lower interfaces are plotted in dimensionless variables, i.e. $\unicode[STIX]{x1D701}_{1}/H_{1}$ and $\unicode[STIX]{x1D701}_{2}/H_{1}$ are plotted as functions of $(x-ct)/H_{1}$. In all cases we have set $\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}=0.999$, $\unicode[STIX]{x1D70C}_{3}/\unicode[STIX]{x1D70C}_{2}=1.001$, $H_{3}/H_{1}=1.0$. (a) $H_{2}/H_{1}=5$, $c^{2}/g^{\prime }H_{1}\approx 0.86$. (b) $H_{2}/H_{1}=5$, $c^{2}/g^{\prime }H_{1}\approx 0.874999$. (c) $H_{2}/H_{1}=0.5$, $c^{2}/g^{\prime }H_{1}\approx 0.281$.

6.1 Leading-order asymptotic solutions for a thin transition layer

Here we revisit case (iii) considered in § 5.3, where the thickness of the middle transition layer is assumed thin so that $r_{2}=O(\unicode[STIX]{x1D700})$ with $\unicode[STIX]{x1D70E}_{1}=1-\unicode[STIX]{x1D700}^{m}$ and $\unicode[STIX]{x1D70E}_{3}=1+\unicode[STIX]{x1D700}^{m}$. When $F^{2}=\unicode[STIX]{x1D700}^{m}C$ is assumed (instead of $F^{2}=\unicode[STIX]{x1D700}^{m+1}C$) with $m$ positive integer and $C=O(1)$, the leading-order equations of (5.1)–(5.2) can be found, after dimensional variables are recovered, as

(6.1)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle (\unicode[STIX]{x1D701}_{1}-\unicode[STIX]{x1D701}_{2})^{2}\left\{2h_{1}\unicode[STIX]{x1D701}_{1}^{\prime \prime }+\unicode[STIX]{x1D701}_{1}^{\prime 2}-3+6g^{\prime }\frac{\unicode[STIX]{x1D701}_{1}h_{1}^{2}}{c^{2}H_{1}^{2}}\right\}=0,\\ \displaystyle (\unicode[STIX]{x1D701}_{1}-\unicode[STIX]{x1D701}_{2})^{2}\left\{2h_{3}\unicode[STIX]{x1D701}_{2}^{\prime \prime }-\unicode[STIX]{x1D701}_{2}^{\prime 2}+3+6g^{\prime }\frac{\unicode[STIX]{x1D701}_{2}h_{3}^{2}}{c^{2}H_{3}^{2}}\right\}=0,\end{array}\right\}\end{eqnarray}$$

where $H_{3}/H_{1}=O(1)$ and $\unicode[STIX]{x1D701}_{k}/H_{1}=O(1)$ ($k=1,2$) have been assumed. Notice that the same system can be obtained directly from (4.22) with $g_{1}^{\prime }=g_{2}^{\prime }=g^{\prime }$, and $H_{2}=0$. Therefore, $\unicode[STIX]{x1D701}_{1}=\unicode[STIX]{x1D701}_{2}$, or the following decoupled system of differential equations is satisfied:

(6.2)$$\begin{eqnarray}\displaystyle & \displaystyle 2h_{1}\unicode[STIX]{x1D701}_{1}^{\prime \prime }+\unicode[STIX]{x1D701}_{1}^{\prime 2}=3-6g^{\prime }\frac{\unicode[STIX]{x1D701}_{1}h_{1}^{2}}{c^{2}H_{1}^{2}}, & \displaystyle\end{eqnarray}$$

(6.3)$$\begin{eqnarray}\displaystyle & \displaystyle 2h_{3}\unicode[STIX]{x1D701}_{2}^{\prime \prime }-\unicode[STIX]{x1D701}_{2}^{\prime 2}=-3-6g^{\prime }\frac{\unicode[STIX]{x1D701}_{2}h_{3}^{2}}{c^{2}H_{3}^{2}}. & \displaystyle\end{eqnarray}$$

Both equations can be solved analytically in terms of elliptic functions. We start with the first of these two equations. By multiplying both sides of (6.2) by $h_{1}^{-1/2}$, one may integrate it once to have the following Hamiltonian:

(6.4)$$\begin{eqnarray}\frac{1}{2}\unicode[STIX]{x1D701}_{1}^{\prime 2}=\frac{3}{2c^{2}H_{1}^{2}}\unicode[STIX]{x1D701}_{1}(c^{2}H_{1}-g^{\prime }h_{1}\unicode[STIX]{x1D701}_{1}),\end{eqnarray}$$

which can be rewritten as

(6.5)$$\begin{eqnarray}\frac{1}{2}\unicode[STIX]{x1D701}_{1}^{\prime 2}=\frac{3g^{\prime }}{2c^{2}H_{1}^{2}}\unicode[STIX]{x1D701}_{1}(\unicode[STIX]{x1D701}_{1}-a_{-})(\unicode[STIX]{x1D701}_{1}-a_{+}),\end{eqnarray}$$

where $a_{\pm }$ are the roots of the equation

(6.6)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{1}^{2}-H_{1}\unicode[STIX]{x1D701}_{1}+\frac{c^{2}H_{1}}{g^{\prime }}=0.\end{eqnarray}$$

Notice that (6.5) can be also obtained from (5.13) with $H_{2}=0$. Although no homoclinic orbits for solitary-wave solutions are possible, periodic orbits starting from rest at zero exist provided $c^{2}<(1/4)g^{\prime }H_{1}$, under which $a_{\pm }$ are real and positive. As shown later, when these (inner) solutions are connected to zero (outer) solutions, they will be the leading-order approximate solution to (5.1)–(5.2). When (6.5) is integrated once, one can obtain

(6.7)$$\begin{eqnarray}\frac{cH_{1}}{\sqrt{3g^{\prime }}}\int _{0}^{\unicode[STIX]{x1D701}_{1}(X)}[(\unicode[STIX]{x1D701}_{1}-a_{+})(\unicode[STIX]{x1D701}_{1}-a_{-})\unicode[STIX]{x1D701}_{1}]^{-1/2}\,\text{d}\unicode[STIX]{x1D701}_{1}=X,\end{eqnarray}$$

with $X\equiv x-ct$. This integral can be computed explicitly with recourse to elliptic functions (Abramowitz & Stegun Reference Abramowitz, Stegun, Abramowitz and Stegun1965):

(6.8)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{1}(X)=a_{-}\sin ^{2}\left[{\mathcal{F}}^{-1}\left(\left.\frac{\sqrt{3g^{\prime }a_{+}}}{2cH_{1}}X\,\right|\,\frac{a_{-}}{a_{+}}\right)\right].\end{eqnarray}$$

Here, ${\mathcal{F}}$ is the elliptic integral of the first kind and ${\mathcal{F}}^{-1}$ is known as the Jacobi amplitude function. It is worth pointing out that near the origin, the solution has the following behaviour:

(6.9)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{1}(X)=\frac{3}{4H_{1}}X^{2}+O(X^{4}).\end{eqnarray}$$

Similar steps can be taken to solve (6.3). In this case we need to solve

(6.10)$$\begin{eqnarray}\frac{cH_{3}}{\sqrt{3g^{\prime }}}\int _{0}^{-\unicode[STIX]{x1D701}_{2}(X)}[(\overline{\unicode[STIX]{x1D701}}_{2}+b_{+})(\overline{\unicode[STIX]{x1D701}}_{2}+b_{-})\overline{\unicode[STIX]{x1D701}}_{2}]^{-1/2}\,\text{d}\overline{\unicode[STIX]{x1D701}}_{2}=X,\end{eqnarray}$$

with $\overline{\unicode[STIX]{x1D701}}_{2}=-\unicode[STIX]{x1D701}_{2}$, and where $b_{\pm }$ are the roots of the equation

(6.11)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{2}^{2}+H_{3}\unicode[STIX]{x1D701}_{2}+\frac{c^{2}H_{3}}{g^{\prime }}=0.\end{eqnarray}$$

Periodic orbits starting from rest at zero exist provided $c^{2}<(1/4)g^{\prime }H_{3}$, under which $b_{\pm }$ are real and negative ($b_{+}<b_{-}<0$). Their analytical expression is provided by

(6.12)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{2}(X)=b_{-}\sin ^{2}\left[{\mathcal{F}}^{-1}\left(\left.\frac{\sqrt{-3g^{\prime }b_{+}}}{2cH_{3}}X\,\right|\,\frac{b_{-}}{b_{+}}\right)\right],\end{eqnarray}$$

which has the following behaviour near the origin:

(6.13)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{2}(X)=-\frac{3}{4H_{3}}X^{2}+O(X^{4}).\end{eqnarray}$$

These periodic solutions given by (6.8)–(6.12) exist when $c^{2}<c_{m}^{2}$, where $c_{m}^{2}=\min \{{\textstyle \frac{1}{4}}g^{\prime }H_{1},{\textstyle \frac{1}{4}}g^{\prime }H_{3}\}$. Moreover, one can see that these solutions represent mode-2 waves as $c_{m}<c_{0}^{+}$, where $c_{0}^{+}$ is the linear long wave speed of mode-1 waves. For this particular stratification, under the Boussinesq approximation, the linear long wave speeds are, in the limit when $H_{2}\rightarrow 0$, given by

(6.14a,b)$$\begin{eqnarray}(c_{0}^{-})^{2}=0,\quad (c_{0}^{+})^{2}=g^{\prime }H_{1}\left(\frac{2H_{3}}{H_{1}+H_{3}}\right).\end{eqnarray}$$

Consider $c<c_{m}$ and let $L_{1}$ and $L_{2}$ be half-periods of $\unicode[STIX]{x1D701}_{1}$ and $\unicode[STIX]{x1D701}_{2}$, i.e.

(6.15)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \begin{array}{@{}rcl@{}}L_{1}\, & =\, & \displaystyle \frac{cH_{1}}{\sqrt{3g^{\prime }}}\int _{0}^{a_{-}}[(\unicode[STIX]{x1D701}_{1}-a_{+})(\unicode[STIX]{x1D701}_{1}-a_{-})\unicode[STIX]{x1D701}_{1}]^{-1/2}\,\\ \,\\ \text{d}\unicode[STIX]{x1D701}_{1}\, & =\, & \displaystyle \frac{cH_{1}}{\sqrt{3g^{\prime }}}\frac{2}{\sqrt{a_{+}}}{\mathcal{F}}\left(\left.\frac{\unicode[STIX]{x03C0}}{2}\,\right|\,\frac{a_{-}}{a_{+}}\right).\end{array}\\ \displaystyle \begin{array}{@{}rcl@{}}L_{2}\, & =\, & \displaystyle \frac{cH_{3}}{\sqrt{3g^{\prime }}}\int _{0}^{-b_{-}}[(\tilde{\unicode[STIX]{x1D701}}_{2}+b_{+})(\tilde{\unicode[STIX]{x1D701}}_{2}+b_{-})\tilde{\unicode[STIX]{x1D701}}_{2}]^{-1/2}\\ \,\\ \text{d}\tilde{\unicode[STIX]{x1D701}}_{2}\, & =\, & \displaystyle \frac{cH_{3}}{\sqrt{3g^{\prime }}}\frac{2}{\sqrt{-b_{+}}}{\mathcal{F}}\left(\frac{\unicode[STIX]{x03C0}}{2}\frac{b_{-}}{b_{+}}\right).\end{array}\end{array}\right\}\end{eqnarray}$$

Suppose we look for the two wave profiles that touch at some of their troughs, which requires the period of $\unicode[STIX]{x1D701}_{1}$ to be a rational multiple $N$ of that of $\unicode[STIX]{x1D701}_{2}$, which yields

(6.16)$$\begin{eqnarray}{\mathcal{K}}\left(\frac{a_{-}}{a_{+}}\right)=\frac{NH_{3}}{H_{1}}\sqrt{\frac{a_{+}}{-b_{+}}}{\mathcal{K}}\left(\frac{b_{-}}{b_{+}}\right),\end{eqnarray}$$

where ${\mathcal{K}}$ is the complete elliptic integral of the first kind. Notice that when $H_{1}=H_{3}$ (symmetric configuration) we have $b_{-}=-a_{-}$ and $b_{+}=-a_{+}$. As a consequence, equation (6.16) is only satisfied for $N=1$, regardless the value for the wave speed. However, if symmetry is slightly broken, say by setting $H_{3}/H_{1}=1.1$, then (6.16) is met at other integer values of $N$, namely $N=1,2,3$ (see figure 8a).

Figure 8.

Set of parameters verifying condition (6.16). The diagrams are presented on the $(N,c^{2}/g^{\prime }H_{1})$-plane for different depth ratios. (a) $H_{3}/H_{1}=0.2$. (b) $H_{3}/H_{1}=1.1$. These parameters will be used in § 6.2 to find numerically exotic solutions for a three-layer flow with a thin (finite thickness) intermediate layer.

Figure 9.

Leading-order asymptotic solutions (compactons) ruled by (6.2)–(6.3). The upper and lower interfaces given by (6.8) and (6.12) are plotted in dimensionless variables, i.e. $\unicode[STIX]{x1D701}_{1}/H_{1}$ and $\unicode[STIX]{x1D701}_{2}/H_{1}$ are plotted as functions of $(x-ct)/H_{1}$. We set $H_{3}/H_{1}=1.1$ and consider different values for the wave speed. (a) $c^{2}/g^{\prime }H_{1}\approx 0.19751417344421549$. (b) $c^{2}/g^{\prime }H_{1}\approx 0.24998489280008013$. (c) $c^{2}/g^{\prime }H_{1}\approx 0.2499999853496341$. These solutions correspond to the cases when (6.16) holds for $N=1$, $2$, $3$, respectively. The ratio $(c/c_{0}^{+})^{2}$ for the three cases is given respectively as $(c/c_{0}^{+})^{2}\approx 0.188536;0.238622;0.238636$.

If $N$ is a rational number, there exist unique integer values $p,q$ such that $N=p/q$ and $\operatorname{gcd}(p,q)=1$. If (6.16) holds for for such value of $N$, there is a periodic solution characterized by $q$ humps for the upper interface over $p$ humps for the lower interface, within any period between two common zero values of $\unicode[STIX]{x1D701}_{1}$, $\unicode[STIX]{x1D701}_{2}$. Let $\unicode[STIX]{x1D706}^{\star }$ be such period. Then it can be easily established that

(6.17)$$\begin{eqnarray}\unicode[STIX]{x1D706}^{\star }=2q\frac{2cH_{1}}{\sqrt{3g^{\prime }a_{+}}}{\mathcal{K}}\left(\frac{a_{-}}{a_{+}}\right).\end{eqnarray}$$

We may now select the wave profiles over any such period and connect its edges with the trivial null solution $\unicode[STIX]{x1D701}_{1}=\unicode[STIX]{x1D701}_{2}=0$. Then the solutions for $\unicode[STIX]{x1D701}_{1}$ and $\unicode[STIX]{x1D701}_{2}$ given by (6.8) and (6.12) for $0<X<\unicode[STIX]{x1D706}^{\star }$ are referred to as compactons (Rosenau Reference Rosenau2005). Notice that both interface displacements have a jump in their second derivatives at $X=0$ and $X=\unicode[STIX]{x1D706}^{\star }$. However, it follows from (6.2)–(6.3) (6.9) and (6.13) that

(6.18)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle (\unicode[STIX]{x1D701}_{1}-\unicode[STIX]{x1D701}_{2})^{2}(H_{1}-\unicode[STIX]{x1D701}_{1})\,\unicode[STIX]{x1D701}_{1}^{\prime \prime }\backsim X^{4}\times H(X),\\ \displaystyle (\unicode[STIX]{x1D701}_{1}-\unicode[STIX]{x1D701}_{2})^{2}(\unicode[STIX]{x1D701}_{2}+H_{3})\,\unicode[STIX]{x1D701}_{2}^{\prime \prime }\backsim X^{4}\times H(X),\end{array}\right\}\end{eqnarray}$$

which both converge to zero as $X$ goes to zero, where $H(X)$ is the Heaviside function. Hence, it satisfies our equation.

Some examples are depicted in figures 9 and 10. Multi-humped solutions can easily be obtained in this special limit and it is seen how small deviations of the wave speed can radically alter the solution behaviour. In particular, there is a countable number of these peculiar waves and their support can be made large, as intended.

Figure 10.

Leading-order asymptotic solutions (compactons) ruled by (6.2)–(6.3). The upper and lower interfaces given by (6.8), (6.12) are plotted in dimensionless variables, i.e. $\unicode[STIX]{x1D701}_{1}/H_{1}$ and $\unicode[STIX]{x1D701}_{2}/H_{1}$ are plotted as functions of $(x-ct)/H_{1}$. We set $H_{3}/H_{1}=1.1$ and consider different values for the wave speed. (a) $c^{2}/g^{\prime }H_{1}\approx 0.23738395840085139$. (b) $c^{2}/g^{\prime }H_{1}\approx 0.2467648492972218$. These solutions correspond to the cases when (6.16) holds for $N=11/10$, $5/4$, respectively.

Finally, we point out that these results can be easily extended to the general case when $g_{1}^{\prime }\neq g_{2}^{\prime }$, for which condition (6.16) should be modified to

(6.19)$$\begin{eqnarray}{\mathcal{K}}\left(\frac{a_{-}}{a_{+}}\right)=\frac{NH_{3}}{H_{1}}\sqrt{\unicode[STIX]{x1D6FF}}\sqrt{\frac{a_{+}}{-b_{+}}}{\mathcal{K}}\left(\frac{b_{-}}{b_{+}}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6E5}_{1}/\unicode[STIX]{x1D6E5}_{2}$ and $a_{\pm }$ and $b_{\pm }$ are roots of the quadratic equations (6.6) and (6.11), replacing $g^{\prime }$ by $g_{1}^{\prime }$ and $g_{2}^{\prime }$, respectively.

It is important to stress that compactons are not solutions to the coupled system of equations (4.5)–(4.6), but simply leading-order asymptotic solutions (near-field periodic solutions connected to far-field trivial solutions) for a thin transition layer. Unlike the compacton, solutions to the original dynamical system are smooth.

Figure 11.

Numerical solutions of the dynamical system (4.5)–(4.6) for multi-humped solitary waves in case (iii). The upper and lower interfaces are plotted in dimensionless variables, i.e. $\unicode[STIX]{x1D701}_{1}/H_{1}$ and $\unicode[STIX]{x1D701}_{2}/H_{1}$ are plotted as functions of $(x-ct)/H_{1}$. We set $\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}=1-\unicode[STIX]{x1D700}^{2}$, $\unicode[STIX]{x1D70C}_{3}/\unicode[STIX]{x1D70C}_{2}=1+\unicode[STIX]{x1D700}^{2}$, $H_{2}/H_{1}=\unicode[STIX]{x1D700}$, $H_{3}/H_{1}=1.1$, with $\unicode[STIX]{x1D700}=0.01$, and consider different values for the wave speed. (a) $c^{2}/g^{\prime }H_{1}\approx 0.19751417344421549$. (b) $c^{2}/g^{\prime }H_{1}\approx 0.25130330695239749509$. For comparison with the results in figure 9, the ratio $(c/c_{0}^{+})^{2}$ for the two cases is given here respectively as $(c/c_{0}^{+})^{2}\approx 0.188533;0.239877$.

To validate the asymptotic theory leading to multi-humped solitary wave solutions, we use the shooting method adopted in § 5.4 to compute solitary-wave solutions of (4.5)–(4.6) for $H_{2}/H_{1}=0.01$ and $H_{3}/H_{1}=1.1$. According to the asymptotic theory, we should be able to find solutions of large amplitudes that exhibit multi-hump profiles. As shown in figure 9, if the upper interface profile has a single hump, we expect to find wave profiles for the lower interface with one, two, or three humps for very specific (discrete) wave speed values. Numerical solutions are shown in figure 11. The computed wave speeds at which multi-humped solutions exist are slightly greater than those predicted by the asymptotic theory, but the agreement between asymptotic and numerical solutions in figures 9 and 11, respectively, is particularly striking.

As the thickness of the transition layer increases from $H_{2}/H_{1}=0.01$, it is found difficult to compute multi-humped solitary-wave solutions using the shooting method. Therefore, a different numerical method is adopted in the following section.

6.2 Numerical solutions with Boussinesq approximation

As the density increment is assumed small to find leading-order asymptotic solutions, we solve the dynamical system (4.5)–(4.6) under the Boussinesq approximation, as a direct extension of § 6.1. In addition, for simplicity, equal density jumps between the layers are assumed.

To seek multi-humped solutions we consider the case when the intermediate layer is thin (but finite) relative to the other layers. Based on the results in § 6.1, classical solitary-wave solutions may not exist for a continuum set of wave speeds, for a given layer configuration. In this scenario numerical computations are more delicate as one generically expects generalized solitary waves, which are solutions that do not decay to zero in the far field, but instead have oscillations whose amplitude asymptotes to a finite amplitude at infinity. These far-field oscillations correspond to first baroclinic mode ‘tails’ of the second baroclinic mode ‘core’ of the generalized solitary wave. True solitary waves may exist along branches of generalized solitary waves at some wave speeds, where the far-field oscillation has zero amplitude. These are called embedded solitary waves and usually occur only at discrete values of the wave speed on the branch of generalized solitary waves (Yang Reference Yang2010). We present here numerical evidence that embedded solitary-wave solutions exist in our three-layer system. The computational method consists on using the intermediate layer thickness as a parameter to perform a direct continuation of solutions in § 6.1. More precisely, starting from the compactons found in § 6.1, we find single and multi-humped solitary-wave solutions as we vary continuously the thickness intermediate layer to a finite value.

All solutions in this section are computed by using a collocation method on (4.22) resulting in a system of nonlinear algebraic equations that is solved by Newton’s method. Collocation methods perform well in computing homoclinic orbits (solitary waves) of dynamical systems which are often too ill conditioned for shooting methods (Liu, Moore & Russell Reference Liu, Moore and Russell1997). The equations (4.22) are discretized using second-order finite differences on a domain with $n$ uniformly distributed mesh points on $X\in [-L,L]$, resulting in $2n+1$ unknowns: $\unicode[STIX]{x1D701}_{1,j},\unicode[STIX]{x1D701}_{2,j}$ approximating $\unicode[STIX]{x1D701}_{1}(x_{j}),\unicode[STIX]{x1D701}_{2}(x_{j})$ for $j=1,\ldots ,n$ and $c$. Evaluating the equation at the interior mesh points results in $2n-4$ equations. The Jacobian matrix is calculated analytically and Newton iterations are halted when the $l_{2}$-norm of the residual is less than $10^{-10}$. Various mesh sizes and domain sizes were tested to ensure convergence and typical solutions presented here use $10^{3}$ points. We use either periodic boundary conditions or fix the solution at the boundaries such that the interface is at the undisturbed interface level. Periodic boundary conditions are used in the computations of generalized solitary-wave solutions. For embedded solitary-wave solutions, the period is taken large enough such that the solution has decayed sufficiently. Once four boundary conditions are imposed, the remaining system has one degree of freedom. We use the degree of freedom either by imposing a last condition such as a particular amplitude of the crest of one of the interfaces, or, in the cases of embedded solitary waves, as a parameter in an optimization step in the algorithm to determine their wave speeds. This optimization step minimizes the $l_{2}$-norm of the solution on a fixed set of $m$ points (typically $m=n/8$) in the far field, seeking a true solitary wave along a continuous branch of generalized solitary waves. The objective function for the optimization step is thus given by

(6.20)$$\begin{eqnarray}Q=\mathop{\sum }_{j=1}^{m}(\unicode[STIX]{x1D701}_{1,j}^{2}+\unicode[STIX]{x1D701}_{2,j}^{2}+\unicode[STIX]{x1D701}_{1,N+1-j}^{2}+\unicode[STIX]{x1D701}_{2,N+1-j}^{2}),\end{eqnarray}$$

which is minimized over a range of speeds. Denoting $\unicode[STIX]{x1D701}_{1,j},\unicode[STIX]{x1D701}_{2,j},j=1,\ldots ,N+1$ by $\boldsymbol{Z}_{\mathbf{1}},\boldsymbol{Z}_{\mathbf{2}}$ and the discretized version of the Boussinesq equations (4.22) as $\boldsymbol{P}(\boldsymbol{Z}_{\mathbf{1}},\boldsymbol{Z}_{\mathbf{2}},c)=0$, the discrete minimization problem may be stated as

(6.21)$$\begin{eqnarray}(\boldsymbol{Z}_{\mathbf{1}}^{\ast },\boldsymbol{Z}_{\boldsymbol{ 2}}^{\ast },c^{\ast })=\underset{(\boldsymbol{Z}_{\boldsymbol{ 1}},\boldsymbol{Z}_{\mathbf{2}},c)\;\text{s.t.}\;\boldsymbol{P}=0}{\operatorname{argmin}}Q(\boldsymbol{Z}_{\mathbf{1}},\boldsymbol{Z}_{\mathbf{2}}).\end{eqnarray}$$

We accept the solution as an approximation to a solitary wave when the numerical optimization algorithm (MATLAB fminbnd) has achieved $Q<10^{-6}$. We also increase the computational domain to confirm numerical convergence.

Figure 12.

Numerical solutions obtained for the Boussinesq model (4.22) with $g_{1}^{\prime }=g_{2}^{\prime }=g^{\prime }$. The displacements of each interface are plotted in dimensionless variables, i.e. $\unicode[STIX]{x1D701}_{1}/H$ and $\unicode[STIX]{x1D701}_{2}/H$ are plotted as functions of $(x-ct)/H$, where $H=H_{1}+H_{2}+H_{3}$ with $H_{3}/H_{1}=1.1$. Starting with the original compact solution valid for the zero thickness, the middle layer is opened up gradually, with $H_{1}$ and $H_{3}$ decreasing equally. (a) Solitary waves for $c^{2}/g^{\prime }H_{1}\approx 0.258067$ with $H_{2}/H_{1}\approx 0.065$ ($H_{2}/H=0.03$) and $c^{2}/g^{\prime }H_{1}\approx 0.263769$ with $H_{2}/H_{1}\approx 0.11$ ($H_{2}/H=0.05$). The ratio $(c/c_{0}^{+})^{2}$ for the two cases is given respectively as $(c/c_{0}^{+})^{2}\approx 0.246319;0.251749$. (b) Generalized solitary waves for $c^{2}/g^{\prime }H\approx 0.11904$ and $H_{2}/H=0.01$, 0.06, 0.12. We stress that the solitary-wave solutions shown in (a) can be found at special values of $c$ found by the optimization step discussed in § 6.2.

Figure 12 shows numerical solutions with multiple humps by extending the branch into finite values of $H_{2}$ from the compacton solution in § 6.1 with $q=1$, $p=2$, and $H_{3}/H_{1}=1.1$, reminiscent of the waves observed experimentally by Gavrilov et al. (Reference Gavrilov, Liapidevskii and Liapidevskaya2013). Both solitary waves and generalized solitary waves are found and their nature depends sensitively on the value of the speed $c$. Although generalized solitary waves are prevalent, it is shown in panel (a) that solutions decaying to zero at infinity can be obtained for special values of $c$. Panel (b) shows that whenever the wave speed is not finely tuned, oscillations tend to arise at the far field. To illustrate this claim, we fix the wave speed and vary the thickness of the intermediate layer by preserving the depth ratio $H_{3}/H_{1}=1.1$. The amplitude of such oscillations can vary significantly according to different parameters. Although oscillations may be clearly visible in some cases (e.g. $H_{2}/H=0.12$), they can be almost imperceptible in other cases (e.g. $H_{2}/H=0.01$).

Figure 13.

Similar to figure 12, but with a single hump. Solitary waves obtained in the case when $H_{3}/H_{1}=1.1$ and different values of $H_{2}/H_{1}$: $c^{2}/g^{\prime }H_{1}\approx 0.2021$ with $H_{2}/H_{1}\approx 0.0429$ ($H_{2}/H=0.02$); $c^{2}/g^{\prime }H_{1}\approx 0.2064$ with $H_{2}/H_{1}\approx 0.0875$ ($H_{2}/H=0.04$); $c^{2}/g^{\prime }H_{1}\approx 0.2105$ with $H_{2}/H_{1}\approx 0.134$ ($H_{2}/H=0.06$); $c^{2}/g^{\prime }H_{1}\approx 0.2145$ with $H_{2}/H_{1}\approx 0.1826$ ($H_{2}/H=0.08$).

Figure 14.

A mode-2 internal solitary wave observed in a laboratory experiment with the parameters $H_{1}=0.133~\text{m}$, $H_{2}=0.025~\text{m}$, $H_{3}=0.142~\text{m}$ (courtesy of Magda Carr). Even though the density varies continuously in the salt-stratified experiment, it can be well approximated by a three-layer system with densities $\unicode[STIX]{x1D70C}_{1}=1025~\text{Kg}~\text{m}^{-3}$, $\unicode[STIX]{x1D70C}_{3}=1048~\text{Kg}~\text{m}^{-3}$, $\unicode[STIX]{x1D70C}_{2}=(\unicode[STIX]{x1D70C}_{1}+\unicode[STIX]{x1D70C}_{3})/2=1036.5~\text{Kg}~\text{m}^{-3}$. Here, $H_{3}/H_{1}=1.067$, $H_{2}/H_{1}\approx 0.1880$ and the density increment is constant across the layers ($\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6E5}_{2}\approx 0.0111$), which is close to the scenario depicted in figure 13.

Figure 13 shows numerical solutions for single-hump solitary waves corresponding to $q=1$ and $p=1$ in the previous geometrical configuration ($H_{3}/H_{1}=1.1$). If the top and bottom symmetry had not been broken, or $H_{3}/H_{1}=1$, anti-symmetric waves decaying to zero at infinity at a fixed value of $H_{2}$ would exist for any value of $c$ within the range $c_{0}^{-}<c<c_{m}$, as discussed in § 5.2. The slight break of symmetry leads to a very different behaviour of the solutions to the dynamical system. In particular, true solitary waves of finite amplitudes no longer exist for a continuum set of values of $c$, and $H_{2}$ must be varied in order to obtain a branch of solutions. In figure 13, solitary-wave solutions for four different values of $H_{2}/H_{1}$ are presented. As shown in figure 14, a mode-2 solitary wave was observed in a laboratory experiment with a parameter set close to that of the solitary wave with the largest value of $H_{2}/H_{1}$ in figure 13. While no quantitative comparison between the numerical solution and the experiment has been made, it is clearly shown that large amplitude mode-2 solitary waves in this parameter regime can be observed experimentally.

Next we consider the case when the centre of the pycnocline is considerably displaced with respect to the midline ($H_{3}/H_{1}=0.2$). As long as the thickness of the intermediate layer is thin enough, the asymptotics in § 6.1 predicts the existence of solutions with $q$ humps on the upper interface over $p$ humps on the lower interface, provided $p/q$ is roughly within the range $]\!0.5,2.2\![$ (see figure 8). As shown in figure 15, we were able to find numerically such a solution from the compacton of § 6.1 with $q=2$, $p=3$, and $H_{3}/H_{1}=0.2$, by extending the branch up to the value $H_{2}/H_{1}\approx 0.00603$.

Based on the asymptotic theory, it has been shown in § 5.2 that the wave speeds of mode-2 solitary waves could be comparable to those of mode-1 waves, in particular, when the density difference across the layers is small and the thickness of the intermediate layer is greater than those of the top and bottom layers. Therefore, it is of interest to examine numerically the non-unicity of solutions of the original system under the Boussinesq approximation at a given wave speed. The particular situation depicted here is the critical case when $H_{3}=H_{1}$ and the intermediate layer is thick enough ($H_{2}>6H_{1}$). According to § 5.2, this guarantees the existence of a branch of true solitary mode-2 waves, whose speeds may exceed the linear long wave speed of the first baroclinic mode. The KdV theory fails in describing mode-1 solitary waves for the system (see § 3.2 and appendix B). However, the Gardner equation could potentially be used to show the co-existence of waves of depression and elevation of the first baroclinic mode (see Kurkina et al. Reference Kurkina, Kurkin, Rouvinskaya and Soomere2006). As shown in figure 16, for the same fixed value of $c$, three distinct solitary-wave solutions are found numerically. This feature is highlighted in figure 16(c), where the speed–amplitude relationship for waves of the two baroclinic modes is presented.

Figure 15.

Exotic mode-2 solitary wave obtained for the physical parameters $H_{3}/H_{1}=0.2$, $H_{2}/H_{1}\approx 0.00603$ ($H_{2}/H=0.005$). A solution with two humps on the upper interface over three humps on the lower interface is obtained when $c^{2}/g^{\prime }H_{1}\approx 0.0471449$.

Figure 16.

Numerical solutions obtained in the critical case when $H_{3}=H_{1}$ under the Boussinesq approximation. The upper and lower interfaces are plotted in non-dimensional variables, i.e. $\unicode[STIX]{x1D701}_{1}/H$ and $\unicode[STIX]{x1D701}_{2}/H$ are plotted as functions of $(x-ct)/H$. We have set $H_{2}/H_{1}=7.0$, $H_{3}/H_{1}=1.0$, and $c^{2}/g^{\prime }H$ is increased from its bifurcation values $c_{0}^{-}/\sqrt{g^{\prime }H}=\sqrt{7}/9$ for the mode-2 wave and $c_{0}^{+}/\sqrt{g^{\prime }H}=1/3$ for the mode-1 wave. (a) Mode-2 solution. (b) Mode-1 solution. (c) Speed–amplitude relationship for mode-1 (solid line) and mode-2 (dashed line) waves. The amplitude is measured as $\max (\Vert \unicode[STIX]{x1D701}_{1}\Vert _{\infty },\Vert \unicode[STIX]{x1D701}_{2}\Vert _{\infty })$. For a range of $c/\sqrt{g^{\prime }H}>1/3$ the mode-1 and 2 solutions co-exist and three distinct solutions travelling at the same speed can be determined (the third solution is the reflection of (b) about the midline). Mode-2 and mode-1 solutions in (a) and (b), respectively, are obtained for the wave speed $c/\sqrt{g^{\prime }H}=0.3492$.