On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

Abstract

We study the bifurcation of periodic travelling waves of the capillary–gravity Whitham equation. This is a nonlinear pseudo-differential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillary–gravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov–Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and, therefore, positive and convex; the threshold value for this to happen turns out to be $T = \frac{4}{\pi ^2}$, not the critical Bond number $\frac{1}{3}$), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in $L^2(\mathbb {S})$ (but not in $L^\infty$) towards one of the two constant solution curves.

Appendix A: Bifurcation formulas

This appendix contains higher order expansions of the quantities in Theroem 3.1 and Theorem 4.1. We start with the first- and second-order terms in the expansion for the speed c(t) in the one-dimensional bifurcation case, which is required by the proof of the global extension in Theorem 3.10. We then proceed to study the first-order terms for the expansions of the functions r and p in the two-dimensional bifurcation case.

1.1 One-Dimensional Bifurcation Case

We begin by determining the derivatives $\dot{c}(0)$ and $\ddot{c}(0)$ associated to the bifurcation curve constructed in Theorem 3.1. This can be done either directly using the Lyapunov–Schmidt reduction carried out in the proof of Theorem 3.1 or by the means of bifurcation formulas given for example in [26]. The latter requires an identification between the bifurcation function $\phi (u,c) = \Pi F(u + \psi (u,c), c)$ used in [26] and the functions v and r used in the proof of Theorem 3.1. This relation is given by $v(t) = \psi (t \cos (k x), c(t))$.

Here, start from the Lyapunov–Schmidt representation

$$\begin{aligned} \begin{aligned} 0 =&F(t\cos (kx) + v(t) , c_0 + r(t))\\ =&t\cos (kx) + v(t)\\&+ L \left[ (t\cos (kx) + v(t))^2 -(c_0 + r(t)) (t\cos (kx) + v(t)) \right] . \end{aligned} \end{aligned}$$

(A.1)

Here, it is understood that for each t small the function v(t) is a $2\pi /k$-periodic function of x. Differentiating (A.1) once with respect to t, evaluating at $t=0$ and using that $v(0) = \dot{v}(0) = r(0) = 0$ yields the equation

$$\begin{aligned} \left( 1-c_0L\right) \cos (kx)=0, \end{aligned}$$

which holds by our choice of $c_0$. Similarly, differentiating (A.1) twice with respect to t and evaluating at $t=0$ yields

$$\begin{aligned} \begin{aligned} (1-c_0L)\ddot{v}(0)&=2\dot{r}(0)L\cos (kx)-2L\cos ^2(kx)\\&=2\dot{r}(0)l(k)\cos (kx)-\left( 1+l(2k)\cos (2kx)\right) . \end{aligned} \end{aligned}$$

(A.2)

Since $\int _{-\pi }^\pi v(t)\cos (kx)dx=0$ for all $|t|\ll 1$, the above implies that $\dot{r}(0)=0$. Returning to (A.2), it now follows that

$$\begin{aligned} \ddot{v}(0)=\frac{1}{c_0-1}+\frac{l(2k)\cos (2kx)}{c_0l(2k)-1}. \end{aligned}$$

(A.3)

Continuing, we observe that taking the third derivative of (A.1) with respect to t and evaluating at $t=0$ yield

$$\begin{aligned} \left( 1-c_0L\right) \dddot{v}(0)=3\ddot{r}(0)L\cos (kx)-6L\left( \ddot{v}(0)\cos (kx)\right) . \end{aligned}$$

Using (A.3), we compute that

$$\begin{aligned} L\left( \ddot{v}(0)\cos (kx)\right) =\frac{l(k)\cos (kx)}{c_0-1}+\frac{l(2k)\left( l(k)\cos (kx)+l(3k)\cos (3kx)\right) }{2(c_0l(2k)-1)}. \end{aligned}$$

Using again that $\int _{-\pi }^\pi v(t)\cos (kx)\,\,{\mathrm {d}}x=0$ for all $|t|\ll 1$, it follows that

$$\begin{aligned} \ddot{r}(0)=\frac{3}{c_0-1}+\frac{l(2k)}{c_0l(2k)-1}=\frac{3c_0l(2k)-l(2k)-2}{(c_0-1)(c_0l(2k)-1)}, \end{aligned}$$

which is the expression (3.10) for $\ddot{c}(0)$ given in Theorem 3.6. Note that the above procedure could be continued to obtain asymptotic expansions of r(t) and v(t) to arbitrarily high order in t. We also note that the above result is consistent with the asymptotic formulas in [23].

1.2 Two-Dimensional Bifurcation Case

We now consider the case of a two-dimensional bifurcation as considered in Section 4 above. Recall that the solutions constructed in Theorem 4.1 can be written as

$$\begin{aligned} u(t_1, t_2)&= t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + v(t_1, t_2), \\ c(t_1, t_2)&= c_0 + r(t_1, t_2), \\ \kappa (t_1, t_2)&= \kappa _0 + p(t_1, t_2), \end{aligned}$$

with v of order $\mathcal {O}(|(t_1, t_2)|^2)$ and r, p of order $\mathcal {O}(|(t_1,t_2)|)$. We now characterize the order of vanishing of the functions r and p at the origin.

Proposition A.1

Let the functions r and p be as in Theorem 4.1. If $k_2/k_1 \notin \mathbb {N}_0$, then

$$\begin{aligned} \nabla r(0,0) = 0, \qquad \nabla p(0,0) = 0 \end{aligned}$$

so that, in particular, r and p are of order $\mathcal {O}(|(t_1,t_2)|^2)$ near the origin. If instead $k_2/k_1 \in \mathbb {N}_0$, then for any $\delta >0$ small we have that, in polar coordinates,

$$\begin{aligned} r_\varrho \left( 0,\vartheta \right) = 0, \qquad p_\varrho \left( 0,\vartheta \right) = 0 \end{aligned}$$

if and only if either $k_2 \notin \{0, 2k_1 \}$ or $(k_2,\vartheta )=\left( 2k_1,\frac{\pi }{2}\right)$.

Proof

We begin the non-resonant case, $k_2/k_1 \notin \mathbb {N}_0$. From the proof of Theorem 4.1, we know for all $0<|(t_1,t_2)|\ll 1$ the functions r and p satisfy

$$\begin{aligned} \Psi _i (t_1, t_2, r(t_1, t_2), p(t_1, t_2)) = 0\quad {\mathrm{for}}~~ i=1,2, \end{aligned}$$

where the $\Psi _i$ are defined in (4.9) and (4.13). Fixing $j\in \{1,2\}$ we find that differentiating the above with respect to $t_j$ and evaluating at $(t_1,t_2)=(0,0)$ gives the system of equations

$$\begin{aligned} \left( \begin{array}{cc}\Psi _{1,r}(\mathbf{0}) &{} \Psi _{1,p}(\mathbf{0})\\ \Psi _{2,r}(\mathbf{0}) &{} \Psi _{2,p}(\mathbf{0})\end{array}\right) \left( \begin{array}{c}r_{t_j}(0,0)\\ p_{t_j}(0,0)\end{array}\right) = -\left( \begin{array}{c}\Psi _{1,t_j}(\mathbf{0})\\ \Psi _{2,t_j}(\mathbf{0})\end{array}\right) , \end{aligned}$$

(A.4)

where here $\mathbf{0}$ denotes the origin in $\mathbb {R}^4$. Since the above system matrix is invertible by (4.15), it remains to determine the values of $\Psi _{i,t_j}(\mathbf{0})$ for $i=1,2$. This can be accomplished by recalling (4.9) and (4.13) and noting that (4.7) implies that

$$\begin{aligned} \frac{\partial ^2 Q_i}{\partial t_j^2}(\mathbf{0}) = -\frac{2}{\pi }\,l(\kappa _0 k_i) \int _{-\pi }^\pi \cos ^3(k_i x) \;{\mathrm {d}}x \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 Q_i}{\partial t_1 \partial t_2}(\mathbf{0}) = \left\{ \begin{aligned} -\frac{2}{\pi } l(\kappa _0 k_2) \int _{-\pi }^\pi \cos ^2(k_1 x) \cos (k_2 x) \;{\mathrm {d}}x,\quad i=1,\\ -\frac{2}{\pi } l(\kappa _0 k_1) \int _{-\pi }^\pi \cos ^2(k_2 x) \cos (k_1 x) \;{\mathrm {d}}x,\quad i=2 \end{aligned}\right. \end{aligned}$$

Consequently, since $k_2/k_1\notin \mathbb {N}_0$ it follows that $\Psi _{i,t_j}(\vec {0})=0$ for $i=1,2$ and hence (A.4) implies that $r_{t_j}(0,0)=p_{t_j}(0,0)=0$ as claimed. Since $j\in \{1,2\}$ was arbitrary, this proves the proposition in the non-resonant case.

Now, consider the resonant case when $k_2/k_1\in \mathbb {N}_0$ and fix $\delta >0$ small. In this case, for each $\delta<|\vartheta |<\pi -\delta$ and $0<\varrho \ll 1$ the functions $r(\varrho ,\vartheta )$ and $p(\varrho ,\vartheta )$ satisfy the system

$$\begin{aligned} {\widetilde{\Psi }}_i\left( \varrho ,\vartheta ,r(\varrho ,\vartheta ),p(\varrho ,\vartheta )\right) =0\quad {\mathrm{for}}~~i=1,2, \end{aligned}$$

where here the ${\widetilde{\Psi }}_i$ are as in (4.20) and (4.18). Differentiating this system with respect to $\varrho$ at $\varrho =0$ gives the system of equations

$$\begin{aligned} \left( \begin{array}{cc}{\widetilde{\Psi }}_{1,r}(0,\vartheta ,0,0) &{} {\widetilde{\Psi }}_{1,p}(0,\vartheta ,0,0)\\ {\widetilde{\Psi }}_{2,r}(0,\vartheta ,0,0) &{} {\widetilde{\Psi }}_{2,p}(0,\vartheta ,0,0)\end{array}\right) \left( \begin{array}{c}r_\varrho (0,\vartheta )\\ p_\varrho (0,\vartheta )\end{array}\right) = -\left( \begin{array}{c}{\widetilde{\Psi }}_{1,\varrho }(0,\vartheta ,0,0)\\ {\widetilde{\Psi }}_{2,\varrho }(0,\vartheta ,0,0)\end{array}\right) . \end{aligned}$$

(A.5)

As in the non-resonant case, the above system matrix is invertible, this time thanks to (4.21), and hence it remains to determine the values of ${\widetilde{\Psi }}_{i,\varrho }(0,\vartheta ,0,0)$ for $i=1,2$. Let us begin by determining the value in the case $i=1$. From (4.20) and the preceding discussion, we know we can write

$$\begin{aligned} {\widetilde{\Psi }}_1(\varrho ,\vartheta ,0,0)=\int _0^1\frac{\partial {\widetilde{Q}}_1}{\partial \varrho }(z\varrho ,\vartheta ,0,0)\,\,{\mathrm {d}}z \end{aligned}$$

where, using (4.7), we have explicitly

$$\begin{aligned} \widetilde{Q}_1(\varrho ,\vartheta ,0,0)&=Q_1(\varrho \cos (\vartheta ),\varrho \sin (\vartheta ),0,0)\\&=-\frac{2\varrho ^2 l(k_0k_1)\cos (\vartheta )\sin (\vartheta )}{\pi }\int _{-\pi }^\pi \cos ^2(k_1 x)\cos (k_2x)\,\,{\mathrm {d}}x. \end{aligned}$$

Clearly then, $\widetilde{Q}_{2,\varrho \varrho }(0,\vartheta ,0,0)$ is equal to zero if and only if either $\vartheta =\frac{\pi }{2}$ or $k_2\notin \{0,2k_1\}$. Since

$$\begin{aligned} {\widetilde{\Psi }}_{1,\varrho }(0,\vartheta ,0,0)=\frac{1}{2}\frac{\partial ^2\widetilde{Q}_1}{\partial \varrho ^2}(0,\vartheta ,0,0) \end{aligned}$$

by above, we have shown that ${\widetilde{\Psi }}_{1,\varrho }(0,\vartheta ,0,0)=0$ if and only if either of the conditions $\vartheta =\frac{\pi }{2}$ or $k_2\notin \{0,2k_1\}$ hold.

Similarly, we have

$$\begin{aligned} {\widetilde{\Psi }}_{2,\varrho }(0,\vartheta ,0,0)=\frac{1}{2}\frac{\partial ^2\widetilde{Q}_2}{\partial \varrho ^2}(0,\vartheta ,0,0) \end{aligned}$$

where, using (4.16), we have

$$\begin{aligned} \widetilde{Q}_2(\varrho ,\vartheta ,0,0)=&-\frac{\varrho ^2 l(k_0k_2)}{\pi }\\&\times \int _{-\pi }^\pi \cos (k_2 z)\left[ \cos ^2(\vartheta )\cos ^2(k_1 x)+\sin ^2(\vartheta )\cos ^2(k_2 x)\right] \,{\mathrm {d}}x. \end{aligned}$$

Clearly, $\widetilde{Q}_{2,\varrho \varrho }(0,\vartheta ,0,0)$ vanishes whenever $k_2\notin \{0,2k_1\}$. When $k_2=0$, $\widetilde{Q}_{2,\varrho \varrho }(0,\vartheta ,0,0)$ does not vanish for any $\vartheta$, and when $k_2=2k_1$ it only vanishes when $\vartheta =\frac{\pi }{2}$. Consequently, ${\widetilde{\Psi }}_{2,\varrho }(0,\vartheta ,0,0)$ vanishes only when either $k_2\notin \{0,2k_1\}$ or $\left( k_2,\vartheta \right) =(2k_1,\frac{\pi }{2})$. Together with the results concerning ${\widetilde{\Psi }}_{1,\varrho }$, this completes the proof. $\square$

Remark A.2

The special case $k_2 = 2k_1$ has been found also in the Euler equations (with gravity and vorticity) by the authors of [1]. The special case $k_2 = 0$ is instead due to the transcritical double bifurcation allowed by the capillary–gravity Whitham equation.

Remark A.3

An explicit example where $r_\varrho (0,\vartheta ) \ne 0$ can be seen in [34, Figure 6], where the branch of nontrivial solutions has a non-vertical tangent at the bifurcation point in the speed–height plane.