  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.
