Exact Solutions of Some Nonlinear Evolution Equations

A different approach to finding solutions of certain diffusive‐dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, which is carried to all terms, followed by a summation of the resulting infinite series. Sometimes this is done directly and other times in terms of inverses of operators in an appropriate space. We first illustrate the method with Burgers's and Thomas's equations, and show how it quickly leads to the Cole‐Hopf and Thomas transformations which linearize these equations. The method is described in detail with the Korteweg‐de Vries equation and then applied to the modified KdV, sine‐Gordon, nonlinear (cubic) Schrödinger, complex modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained, and new expressions for some of them follow. More generally, the Marčenko integral equations, together with the inverse problem that originates them, follow naturally from the approach. A method for modifying known solutions (in a way different from the known Backlund transformations) is also developed. Thus, for example, formulas for the interaction of solitons with an arbitrary given solution are obtained. Other equations tractable by this approach are presented. These include the vector‐valued cubic Schrödinger equation and a two‐dimensional nonlinear Schrödinger equation. Higher‐order and matrix‐valued equations with nonscalar dispersion functions are also included.