A HEURISTIC WAY OF FINDING LINEAR PROBLEMS FROM SOLITON SOLUTIONS OF NONLINEAR WAVE EQUATIONS

Abstract

This chapter presents a heuristic way of finding linear problems from soliton solutions of nonlinear wave equations. Soliton problem has long been investigated both analytically and numerically in various fields of mathematical physics. Hirota's direct method is one of the analytical methods to obtain soliton solutions of nonlinear wave equations. In this method, nonlinear equations are transformed into bilinear forms through dependent variable transformations. Then N-soliton solutions describing multiple collisions of solitons are systematically calculated from the bilinear form of equations. The two-soliton solutions generated from a certain form of the Gel’fand–Levitan equation have a peculiar structure, and the linear problem for a given nonlinear equation can be deduced by comparing the two-soliton solution with that obtained through Hirota’ method. The procedure is rather heuristic, but it may give a simple way of finding linear problems from special soliton solutions. This chapter discusses the intermediate long wave equation as an example.