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

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

1996-01-11

arxiv.org/abs/solv-int/9601002v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

16 pages, AMSLaTeX

Scientific paper

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.

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