The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behavior, implicit solutions and wave breaking

Abstract

We have recently solved the inverse spectral 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 in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev - Petviashvili (dKP) equation, arising as commutation of vector fields. In this paper we make use of the above theory i) to construct the nonlinear Riemann-Hilbert dressing for the so-called two dimensional dispersionless Toda equation $(exp(\varphi ){)}_{tt}={\varphi }_{{\zeta }_1{\zeta }_2}$, elucidating the spectral mechanism responsible for wave breaking; ii) we present the formal solution of the Cauchy problem for the wave form of it: $(exp(\varphi ){)}_{tt}={\varphi }_{xx}+{\varphi }_{yy}$; iii) we obtain the longtime behaviour of the solutions of such a Cauchy problem, showing that it is essentially described by the longtime breaking formulae of the dKP solutions, confirming the expected universal character of the dKP equation as prototype model in the description of the gradient catastrophe of two-dimensional waves; iv) we finally characterize a class of spectral data allowing one to linearize the RH problem, corresponding to a class of implicit solutions of the PDE.

22 pages

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• On the solutions of the second heavenly and Pavlov equations
• Numerical Study of the semiclassical limit of the Davey-Stewartson II equations
• On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking
• Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
• Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy
• Duality of positive and negative integrable hierarchies via relativistically invariant fields
• Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities
• On the dispersionless Davey-Stewartson system: Hamiltonian vector fields Lax pair and relevant nonlinear Riemann-Hilbert problem for dDS-II system
• Dunajski-Tod equation and reductions of the generalized dispersionless 2DTL hierarchy
• An integral geometry lemma and its applications: the nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects
• Particle-like, dyx-coaxial and trix-coaxial Lie algebra structures for a multi-dimensional continuous Toda type system
• The Cauchy problem for the Pavlov equation
• The Cauchy problem for the Pavlov equation with large data
• Einstein Metrics, Projective Structures and the $SU(\infty )$ Toda Equation