A (2+1)-dimensional quasilinear system is said to be 'integrable' if it can be decoupled in infinitely many ways into a pair of compatible n-component onedimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogs of n-gap solutions. It is demonstrated that the requirement of the existence of 'sufficiently many' n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.MSC: 35L40, 35L65, 37K10.
We obtain the necessary and sufficient conditions for a two-component (2+1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in involution, implying that the systems in question are locally parametrized by 15 arbitrary constants. It is proved that all such systems possess three conservation laws of hydrodynamic type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such systems are proved to possess a scalar pseudopotential which plays the role of the 'dispersionless Lax pair'. We demonstrate that the class of two-component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions, thus establishing the equivalence of the two possible definitions of the integrability. Explicit linearly degenerate examples are constructed.MSC: 35L40, 35L65, 37K10.
Let u be a function of n independent variables x 1 , ..., x n , and U = (u ij ) the Hessian matrix of u. The symplectic Monge-Ampére equation is defined as a linear relation among all possible minors of U . Particular examples include the equation det U = 1 governing improper affine spheres and the so-called heavenly equation, u 13 u 24 −u 23 u 14 = 1, describing self-dual Ricci-flat 4-manifolds. In this paper we classify integrable symplectic Monge-Ampére equations in four dimensions (for n = 3 the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of 'maximally singular' hyperplane sections of the Plücker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F (u ij ) = 0 in more than three dimensions is necessarily of the symplectic Monge-Ampére type.MSC: 35L70, 35Q58, 35Q75, 53A20, 53D99, 53Z05.
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