Integrable equations in 2+1-dimensions: deformations of dispersionless limits

Abstract

We classify integrable third order equations in 2+1 dimensions which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2+1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. %Conversely, the requirement of the existence of hydrodynamic reductions proves to be an efficient classification criterion. In this paper we adopt a novel perturbative approach to the classification problem. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2+1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third order equations, some of which are apparently new.

arXiv comment22 pages

References in corpus (11)

• On integrability of (2+1)-dimensional quasilinear systems
• Perturbative Symmetry Approach
• Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation
• On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour
• Hydrodynamic reductions and solutions of a universal hierarchy
• Classification of the line-soliton solutions of KPII
• The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type
• On Quasitriviality and Integrability of a Class of Scalar Evolutionary PDEs
• Dispersive deformations of hydrodynamic reductions of 2D dispersionless integrable systems
• On the structure of $(2+1)$--dimensional commutative and noncommutative integrable equations
• Darboux Transformations, Infinitesimal Symmetries and Conservation Laws for Nonlocal Two-Dimensional Toda Lattice

Cited by in corpus (9)

• New integrable (3+1)-dimensional systems and contact geometry
• Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D
• Integrable Dispersive Chains and Energy Dependent Schrodinger Operator
• On the classification of scalar evolutionary integrable equations in $2+1$ dimensions
• Multidimensional integrable systems from contact geometry
• Dispersive deformations of Hamiltonian systems of hydrodynamic type in 2+1 dimensions
• On a class of integrable Hamiltonian equations in 2+1 dimensions
• Integrable systems of the intermediate long wave type in 2+1 dimensions
• Gardner's deformations as generators of new integrable systems