On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking

摘要

We study the (n+1)-dimensional generalization of the dispersionlessKadomtsev-Petviashvili (dKP) equation, a universal equation describing thepropagation of weakly nonlinear, quasi one dimensional waves in n+1 dimensions,and arising in several physical contexts, like acoustics, plasma physics andhydrodynamics. For n=2, this equation is integrable, and it has been recentlyshown to be a prototype model equation in the description of the twodimensional wave breaking of localized initial data. We construct an exactsolution of the n+1 dimensional model containing an arbitrary function of onevariable, corresponding to its parabolic invariance, describing waves, constanton their paraboloidal wave front, breaking simultaneously in all points of it.Then we use such solution to build a uniform approximation of the solution ofthe Cauchy problem, for small and localized initial data, showing that such asmall and localized initial data evolving according to the (n+1)-dimensionaldKP equation break, in the long time regime, if and only if n=1,2,3; i.e., inphysical space. Such a wave breaking takes place, generically, in a point ofthe paraboloidal wave front, and the analytic aspects of it are givenexplicitly in terms of the small initial data.