Nonlinear differential–difference equations and Fourier analysis

TLDR

Fourier analysis provides a conceptual framework for solving a class of nonlinear differential–difference equations. The authors employ inverse scattering techniques to solve the equations. They develop a systematic procedure based on the dispersion relation of the linearized equation, yielding new equations with explicit soliton solutions and conserved quantities.

Abstract

The conceptual analogy between Fourier analysis and the exact solution to a class of nonlinear differential–difference equations is discussed in detail. We find that the dispersion relation of the associated linearized equation is prominent in developing a systematic procedure for isolating and solving the equation. As examples, a number of new equations are presented. The method of solution makes use of the techniques of inverse scattering. Soliton solutions and conserved quantities are worked out.