SOLUTIONS OF NONLINEAR PARTIAL-DIFFERENTIAL EQUATIONS IN PHASE-SPACE

In this paper we consider the problem of constructing solutions of sev eral well known nonlinear partial differential equations (p.d.e.s) in phase space (i.e. the Fourier transform domain). We seek solutions rep resenting travelling focussed pulses. As such, based on a technique us ed to construct such solutions (so called Localized Wave solutions) of linear p.d.e.s, we look for phase space solutions consisting of a gen eralized function whose support is a particular line or surface, toget her with a suitable weighting function. The support of the phase space solution must be such that it regenerates itself after the appropriat e nonlinear operation. In one spatial dimension we construct the usual well known soliton solutions of several equations. For the case of hi gher spatial dimensions we construct a travelling ''slab'' pulse solut ion of the nonlinear Schrodinger equation. We also discuss some issues involved with the extra freedom one has for the phase space support, leading perhaps to more exotic spacetime domain solutions.