A Fourier-Series Method for Solving Soliton Problems

Abstract

A Fourier–Galerkin method with an earlier proposed complete orthonormal system of functions in $L^2 ( - \infty ,\infty )$ as the set of trial functions is developed and displayed for the problem of calculating the shape of the one-soliton solution of the Korteweg–de Vries equation. The convergence of the method is investigated through comparison with the analytic solution, which appears to be very good. The truncation and discretization errors are assessed pointwise. The technique developed is also applied to the soliton problem for the so-called Kuramoto–Sivashinsky equation and the obtained soliton shape is compared to the existing difference solution. The quantitative agreement between the Fourier-series-method result and the numerical one is good. In the present paper, however, the soliton solution is obtained for a significantly wider range of phase velocities, which suggests that the spectrum might be continuous. The new technique can also be applied to a variety of other problems involving identification of homoclinic solutions.