6 - The Periodic Hirota Method

Abstract

This chapter discusses some of the aspects of using the Hirota method for periodic boundary conditions. The Hirota method for testing the integrability of nonlinear wave equations is important for deriving the infinite-line, N-soliton solutions. However, for the purposes of most of the work in the field of physical oceanography, periodic boundary conditions are required. This requirement is in complete analogy with the linear Fourier series, which is a periodic algorithm. Use of the fast Fourier transform (FFT) means that periodic boundary conditions are assumed for data analysis, for data assimilation, or for modeling purposes. A preliminary analysis using the Hirota method often provides with the insight necessary to begin a full analysis of the problem using the inverse scattering transform (IST). The dependent variable transformation of Hirota, for nonlinear, integrable wave equations, results in an almost miraculous cancellation of opposing forces, such as nonlinearity and dispersion. This procedure therefore also provides a physical interpretation for soliton solutions. The chapter describes the Burgers equation, Korteweg–de Vries (KdV) equation, Kadomtsev–Petviashvili (KP) Equation, nonlinear Schrödinger equation, KdV–Burgers equation, modified KdV equation, Boussinesq equation, 2 + 1 Boussinesq equation, and 2 + 1 Gardner equation.