7 research outputs found
On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking
We study the (n+1)-dimensional generalization of the dispersionless
Kadomtsev-Petviashvili (dKP) equation, a universal equation describing the
propagation of weakly nonlinear, quasi one dimensional waves in n+1 dimensions,
and arising in several physical contexts, like acoustics, plasma physics and
hydrodynamics. For n=2, this equation is integrable, and it has been recently
shown to be a prototype model equation in the description of the two
dimensional wave breaking of localized initial data. We construct an exact
solution of the n+1 dimensional model containing an arbitrary function of one
variable, corresponding to its parabolic invariance, describing waves, constant
on their paraboloidal wave front, breaking simultaneously in all points of it.
Then we use such solution to build a uniform approximation of the solution of
the Cauchy problem, for small and localized initial data, showing that such a
small and localized initial data evolving according to the (n+1)-dimensional
dKP equation break, in the long time regime, if and only if n=1,2,3; i.e., in
physical space. Such a wave breaking takes place, generically, in a point of
the paraboloidal wave front, and the analytic aspects of it are given
explicitly in terms of the small initial data.Comment: 20 pages, 10 figures, few formulas adde
Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
We have recently solved the inverse spectral problem for integrable PDEs in
arbitrary dimensions arising as commutation of multidimensional vector fields
depending on a spectral parameter . The associated inverse problem, in
particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on
a given contour of the complex plane. The most distinguished examples
of integrable PDEs of this type, like the dispersionless
Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional
dispersionless Toda equations, are real PDEs associated with Hamiltonian vector
fields. The corresponding NRH data satisfy suitable reality and symplectic
constraints. In this paper, generalizing the examples of solvable NRH problems
illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct
solvable NRH problems for integrable real PDEs associated with Hamiltonian
vector fields, allowing one to construct implicit solutions of such PDEs
parametrized by an arbitrary number of real functions of a single variable.
Then we illustrate this theory on few distinguished examples for the dKP and
heavenly equations. For the dKP case, we characterize a class of similarity
solutions, a class of solutions constant on their parabolic wave front and
breaking simultaneously on it, and a class of localized solutions breaking in a
point of the plane. For the heavenly equation, we characterize two
classes of symmetry reductions.Comment: 29 page
On the solutions of the second heavenly and Pavlov equations
We have recently solved the inverse scattering 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 connected with the commutation of multidimensional
vector fields, like the heavenly equation of Plebanski, the dispersionless
Kadomtsev - Petviashvili (dKP) equation and the two-dimensional dispersionless
Toda (2ddT) equation, as well as with the commutation of one dimensional vector
fields, like the Pavlov equation. We also showed that the associated
Riemann-Hilbert inverse problems are powerfull tools to establish if the
solutions of the Cauchy problem break at finite time,to construct their
longtime behaviour and characterize classes of implicit solutions. In this
paper, using the above theory, we concentrate on the heavenly and Pavlov
equations, i) establishing that their localized solutions evolve without
breaking, unlike the cases of dKP and 2ddT; ii) constructing the longtime
behaviour of the solutions of their Cauchy problems; iii) characterizing a
distinguished class of implicit solutions of the heavenly equation.Comment: 16 pages. Submitted to the: Special issue on nonlinearity and
geometry: connections with integrability of J. Phys. A: Math. and Theor., for
the conference: Second Workshop on Nonlinearity and Geometry. Darboux day