Traveling Wave Solutions of the Kawahara Equation Joining Distinct Periodic Waves

Abstract

The Kawahara equation is a weakly nonlinear longwave model of dispersive waves that emerges when leading, third order, dispersive effects are in balance with the next, fifth-order correction. Traveling wave solutions of the Kawahara equation satisfy a fourth-order ordinary differential equation in which the traveling wave speed is a parameter. The fourth-order equation has Hamiltonian structure and admits a two-parameter family of single-phase periodic solutions with varying speed and Hamiltonian. A set of jump conditions is derived for pairs of distinct periodic solutions with equal speed and Hamiltonian. These are necessary conditions for the existence of traveling waves that asymptote to the periodic orbits at $\pm \infty$. Bifurcation theory and parameter continuation are used to construct multiple solution branches of the jump conditions. For example pairs of compatible periodic solutions, the heteroclinic orbit representing the traveling wave is constructed from the intersection of stable and unstable manifolds of the periodic orbits. Each branch terminates at an equilibrium-to-periodic solution in which the equilibrium is the background for a solitary wave that connects to the associated periodic solution.

Appendix A: Numerical Computation of Periodic Orbits

In this appendix, we discuss the numerical approximation of periodic solutions via a pseudospectral method similar to that used in Ehrnström and Kalisch (2009). We first set the constant of integration $A=0$ in the fourth-order ODE (1.12):

$$\begin{aligned} -c f + \frac{1}{2}f^2 + \alpha f'' + f'''' = 0. \end{aligned}$$

(A-1)

For each wavenumber k,  $2\pi /k$-periodic solutions f of (A-1) are approximated by a truncated Fourier series

$$\begin{aligned} f \approx F_N = \sum _{n = -{N}}^{N} \hat{f}_n e^{ink\xi }. \end{aligned}$$

(A-2)

Substituting into (A-1) gives the nonlinear equation

$$\begin{aligned} \frac{1}{2}F_N^2 + \sum _{n = -{N}}^{N} \left( -c + \alpha (nk)^2 + (nk)^4\right) \hat{f}_n e^{i n k \xi } = 0. \end{aligned}$$

(A-3)

The projection of this equation onto each Fourier mode $e^{ink\xi }$, $n = -N,\ldots , N$ results in a system of $2N+1$ equations for the $2N+1$ Fourier coefficients $\hat{f}_n,$ with constant c,  treated as a continuation parameter.

From linear theory (linearizing (A-1) about $f=0$), we have infinitesimal $2\pi /k$-periodic solutions with phase velocity $c_\textrm{p}(k) = - \alpha k^2 + k^4$. Approximate finite amplitude solutions are computed using Matlab’s nonlinear solver fsolve, choosing N large enough, depending on k,  to push the residual below $10^{-12}$. For example, for $k = 1$, $2^6$ Fourier modes are required, while $2^{12}$ Fourier modes are needed for $k = 0.005$. The solutions $f=\tilde{f}$ are found by continuation from the small amplitude solutions as c varies away from $c=c_\textrm{p}(k).$ This gives the periodic wave amplitude $a=a(c, k)$ and average $\tilde{u}(c,k)$ as functions of its velocity and wavenumber. Inverting the relation $a=\tilde{a}(c,k))$ for each k,  and interpolating using cubic splines, gives the velocity $c=\tilde{c}(a,k),$ and average $\overline{u}=\tilde{u}(a,k).$

In a final step, we can use the Galilean symmetry (1.9) to shift the mean of the periodic solutions to $\overline{u}=0,$ thereby modifying the wave velocity to $c(a,k)= \tilde{c}(a,k) - \overline{u}(a,k).$ In so doing, we obtain the two-parameter family of periodic solutions used throughout this manuscript.