On the soliton solutions of the Davey-Stewartson equation for long waves

Abstract

The Davey-Stewartson equations describe two-dimensional surface waves on water of finite depth. In the long wave limit, it is shown that these equations belong to the class derivable from operator equations in the manner of Zakharov & Shabat. The basic underlying linear system of equations is obtained and solutions to the original nonlinear system sought from the Gelfand-Levitan equations of Inverse Scattering Theory. Single soliton and multi-soliton solutions are deduced corresponding to the one-dimensional solutions already available. The solitons so obtained are pseudo one dimensional in that they have the same form as one-dimensional solitons but move at an angle to the main direction of propagation. The multi-soliton solution describes the interaction of many such solitons each propagating in different directions. For two solitons, it is shown that resonance occurs and a triple soliton structure is produced.