The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behaviour, 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 , 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(φ))tt = φxx + φyy; (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.