F. Guil | Searcharxiv

arXiv1996

The Three-Wave Resonant Interaction: Deformation of the Plane-Wave Solutions and Darboux Transformations

The plane wave solutions of the three-wave resonant interaction in the plane are considered. It is shown that rank-one constraints over the right derivatives of invertible operators on an arbitrary linear space gives solutions of the three-wave resonant interaction that can be understood as a Darboux transformation of the plane wave solutions. The method is extended further to obtain general Darboux transformations: for any solution of the three-wave interaction problem and vector solutions of the corresponding Lax pair large families of new solutions, expressed in terms of Grammian type determinants of these vector solutions, are given.

solv-int↗

arXiv1993

Akns Hierarchy, Self-Similarity, String Equations and the Grassmannian

In this paper the Galilean, scaling and translational self--similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non--scaling limit analysis of the one--matrix model are shown to correspond to the Galilean self--similarity condition for this hierarchy. We describe, in terms of the initial data for the zero--curvature 1--form of the AKNS hierarchy, the moduli space of these self--similar solutions in the Sato Grassmannian. As a byproduct we characterize the points in the Segal--Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura--Hirota rational solutions of the NLS equation. An explicit 1--parameter family of Galilean self--similar solutions of the AKNS equation and the associated solution to the NLS equation is determined.

hep-th↗