Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking

Abstract

We have recently solved the inverse spectral problem for integrable PDEs in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter $\lambda$. The associated inverse problem, in particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on a given contour of the complex $\lambda$ plane. The most distinguished examples of integrable PDEs of this type, like the dispersionless Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional dispersionless Toda equations, are real PDEs associated with Hamiltonian vector fields. The corresponding NRH data satisfy suitable reality and symplectic constraints. In this paper, generalizing the examples of solvable NRH problems illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct solvable NRH problems for integrable real PDEs associated with Hamiltonian vector fields, allowing one to construct implicit solutions of such PDEs parametrized by an arbitrary number of real functions of a single variable. Then we illustrate this theory on few distinguished examples for the dKP and heavenly equations. For the dKP case, we characterize a class of similarity solutions, a class of solutions constant on their parabolic wave front and breaking simultaneously on it, and a class of localized solutions breaking in a point of the $(x,y)$ plane. For the heavenly equation, we characterize two classes of symmetry reductions.

29 pages

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Cited by in corpus (10)

• New integrable (3+1)-dimensional systems and contact geometry
• Whitham modulation theory for the Kadomtsev-Petviashvili equation
• On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking
• On the Generation of Dispersive Shock Waves
• Multidimensional Toda Lattices: Continuous and Discrete Time
• Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields
• Shock formation in the dispersionless Kadomtsev-Petviashvili equation
• The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited
• $\mathrm{GL}(2)$-structures in dimension four, $H$-flatness and integrability
• The Cauchy problem for the Pavlov equation