Nonlinear differential–difference equations and Fourier analysis

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.