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| It has long been recognised that waves and their interactions play a fundamental role in the theory of integrability of partial differential equations. On the one hand, the presence of solitonic interactions and Lax pair representations characterise integrability of dispersive equations. On the other hand, the presence of sufficiently many interacting Riemann waves can be used as a criterion for detecting integrability of dispersionless equations, leading to solutions in implicit form through the method of generalized hodograph. Remarkably, integrable dispersionless equations also arise as the compatibility condition of nonlinear pseudo-potentials.
In this contribution, we propose a new approach to the classification of multidimensional integrable dispersive systems of PDEs based on the requirement that the quasi-classical limit of such a system should be generated by a polynomial pseudo-potential. An integrability-preserving perturbative procedure based on the method of hydrodynamic reductions then allows to reconstruct the corresponding dispersive systems along with associated Lax pairs. We illustrate this approach by providing a classification of systems of the Davey-Stewartson type and new integrable systems within this class are presented. |
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