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We have recently solved the inverse spectral problem for one-parameterfamilies of vector fields, and used this result to construct the formalsolution of the Cauchy problem for a class of integrable nonlinear partialdifferential equations in multidimensions, including the second heavenlyequation of Plebanski and the dispersionless Kadomtsev - Petviashvili (dKP)equation, arising as commutation of vector fields. In this paper we make use ofthe above theory i) to construct the nonlinear Riemann-Hilbert dressing for theso-called two dimensional dispersionless Toda equation$(exp(\phi))_{tt}=\phi_{\zeta_1\zeta_2}$, elucidating the spectral mechanismresponsible for wave breaking; ii) we present the formal solution of the Cauchyproblem 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 ofthe dKP solutions, confirming the expected universal character of the dKPequation as prototype model in the description of the gradient catastrophe oftwo-dimensional waves; iv) we finally characterize a class of spectral dataallowing one to linearize the RH problem, corresponding to a class of implicitsolutions of the PDE.

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