Traveling wave solutions to Kawahara and related equations

TL;DR

This paper introduces an elliptic function method for finding traveling wave solutions to Kawahara, transmission line, and KdV equations, offering a more general approach than traditional methods by utilizing Weierstrass functions.

Contribution

It develops a novel elliptic function method that simplifies the solution process and applies to higher-order dispersive equations like Kawahara and KdV.

Findings

Derives solitary wave solutions (solitons) for zero boundary conditions.

Finds cnoidal wave solutions for nonzero boundary conditions.

Provides solutions propagating in both directions with arbitrary velocities.

Abstract

Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg-de Vries (KdV) equation are found by using an elliptic function method which is more general than the $\mathrm{tanh}$-method. The method works by assuming that a polynomial ansatz satisfies a Weierstrass equation, and has two advantages: first, it reduces the number of terms in the ansatz by an order of two, and second, it uses Weierstrass functions which satisfy an elliptic equation for the dependent variable instead of the hyperbolic tangent functions which only satisfy the Riccati equation with constant coefficients. When the polynomial ansatz in the traveling wave variable is of first order, the equation reduces to the KdV equation with only a cubic dispersion term, while for the KE which includes a fifth order dispersion term the polynomial ansatz must necessary be of quadratic type. By solving the elliptic equation with coefficients that depend on the boundary conditions, velocity of the traveling waves, nonlinear strength, and dispersion coefficients, in the case of KdV equation we find the well-known solitary waves (solitons) for zero boundary conditions, as well as wave-trains of cnoidal waves for nonzero boundary conditions. Both solutions are either compressive (bright) or rarefactive (dark), and either propagate to the left or right with arbitrary velocity.