Accurate approximation to the double sine-Gordon equation

In general, this paper deals with nonlinear double sine-Gordon equation with even potential energy which has arisen in many physical phenomena. The nonlinear dispersion problems without a small perturbation parameter are difficult to be solved analytically. Hence, the main concern is focused on solving the traveling wave of the double sine-Gordon equation. As commonly known, the perturbation method is for solving problems with small parameters, and the analytical representation thus derived has, in most cases, a small range of validity. For some nonlinear problems, although an exact analytical solution can be achieved, they often appear in terms of sophisticated implicit functions, and are not convenient for application. Although a variety of transformation methods has been developed for solving the nonlinear dispersion problems, such transformed equations still include nonlinear terms. To overcome these difficulties, a new approach for Newton-harmonic balance (NHB) method with Fourier–Bessel series is presented here. It is applied to solve the higher-order analytical approximations for dispersion relation in double sine-Gordon equation. The Fourier–Bessel series with the NHB method presents excellent improvement from lower-order to higher-order analytical approximations involving the nonlinear terms in the double sine-Gordon equation. Not restricted by the existence of a small perturbation parameter, the method is suitable for small as well as large amplitudes of wavetrains. Excellent agreement with exact solutions is presented in some practical examples.