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

TL;DR

This paper develops a spectral and dressing method for the dispersionless 2D Toda equation, analyzes wave breaking, long-term behavior, and constructs implicit solutions, revealing universal features of wave phenomena in multidimensional integrable PDEs.

Contribution

It introduces a nonlinear Riemann-Hilbert dressing for the 2D dispersionless Toda equation and analyzes wave breaking and long-term dynamics, extending the inverse spectral approach to this PDE.

Findings

Spectral mechanism for wave breaking identified

Longtime behavior described by dKP breaking formulas

Implicit solutions characterized via spectral data

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 $(exp(\phi))_{tt}=\phi_{\zeta_1\zeta_2}$, elucidating the spectral mechanism responsible for wave breaking; ii) we present the formal solution of the Cauchy problem for the wave form of it: $(exp(\phi))_{tt}=\phi_{xx}+\phi_{yy}$; 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.