On the solutions of the second heavenly and Pavlov equations

Abstract

We have recently solved the inverse scattering problem for one parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, like the heavenly equation of Plebanski, the dispersionless Kadomtsev - Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one dimensional vector fields, like the Pavlov equation. We also showed that the associated Riemann-Hilbert inverse problems are powerfull tools to establish if the solutions of the Cauchy problem break at finite time,to construct their longtime behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; ii) constructing the longtime behaviour of the solutions of their Cauchy problems; iii) characterizing a distinguished class of implicit solutions of the heavenly equation.

16 pages. Submitted to the: Special issue on nonlinearity and geometry: connections with integrability of J. Phys. A: Math. and Theor., for the conference: Second Workshop on Nonlinearity and Geometry. Darboux days

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• The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited
• On the dispersionless Davey-Stewartson system: Hamiltonian vector fields Lax pair and relevant nonlinear Riemann-Hilbert problem for dDS-II system
• The integrable heavenly type equations and their Lie-algebraic structure
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• Second-order PDEs in 4D with half-flat conformal structure
• Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property
• The Cauchy problem for the Pavlov equation
• Characteristic integrals in 3D and linear degeneracy
• On linear degeneracy of integrable quasilinear systems in higher dimensions