Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian

Abstract

We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.

arXiv comment32 pages

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Cited by in corpus (5)

• Integrable (2+1)-dimensional systems of hydrodynamic type
• Bi-Hamiltonian representation, symmetries and integrals of mixed heavenly and Husain systems
• Second order quasilinear PDEs and conformal structures in projective space
• Conformal geometry of surfaces in the Lagrangian--Grassmannian and second order PDE
• Integrable pseudopotentials related to generalized hypergeometric functions