Relationship between Hirota’s Method and the Inverse Spectral Method —The Korteweg-de Vries Equation’s Case—

Abstract

Recently, it is shown that for a number of soliton equations, their solutions expressing multiple solitons in a background of ripples, which may be called generalized soliton solutions, can be constructed using Hirota’s bilinear forms of these soliton equations (S. OISHI: submitted to J. Phy. Soc. Jpn.). In this letter, taking the KdV equation as an example, relationship between Hirota’s method and the inverse spectral method is clarified by showing that its generalized soliton solutions can be transformed into a form of Fredholm’s determinants of the Gel’fand-Levitan-Marcenko integral equation.