Nonlinear Evolution Equations—Continuous and Discrete

An important recent advance in nonlinear wave motion has been the discovery of a method of solution to a class of nonlinear evolution equations. The technique relies on a relation between the evolution equation, and an associated linear eigenvalue (scattering) problem. The initial value solution is found by the method of inverse scattering. In this paper, procedures are outlined which systematically isolate certain nonlinear evolution equations which fit into the above methodology. As examples of the ideas, the partial differential nonlinear Schrodinger and a differential-difference nonlinear Schrodinger equation are considered in detail. Some of the other physically important evolution equations are enumerated and an analogy to Fourier analysis is brought out.