170 research outputs found
Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
No full textWe have recently solved the inverse spectral problem for integrable partial differential equations (PDEs) in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter λ. The associated inverse problem, in particular, can be formulated as a nonlinear Riemann-Hilbert (NRH) problem on a given contour of the complex λ plane. The most distinguished examples of integrable PDEs of this type, like the dispersionless Kadomtsev-Petviashivili (dKP), the heavenly and the two-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 Manakov and Santini (2009 J. Phys. A: Math. Theor. 42 095203; 2008 J. Phys. A: Math. Theor. 41 055204; 2009 J. Phys. A: Math. Theor. 42 404013), we present a general procedure to construct solvable NRH problems for integrable real PDEs associated with Hamiltonian vector fields, allowing one to construct exact 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, of solutions constant on their parabolic wave front and breaking simultaneously on it, of localized solutions whose breaking point travels with constant speed along the wave front, and of localized solutions breaking in a point of the (x, y) plane. For the heavenly equation, we characterize two classes of symmetry reductions. © 2011 IOP Publishing Ltd
On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking
No full textWe 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, such as acoustics, plasma physics and hydrodynamics. For n = 2, this equation is integrable, and 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 a 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 1 <= n <= 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
On the solutions of the dKP equation: nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking
No full textInverse scattering problem for vector fields and the Cauchy problem for the heavenly equation
No full textWe solve the inverse scattering problem for multidimensional vector fields and we use this result to construct the formal solution of the Cauchy problem for the second heavenly equation of Plebanski, a scalar nonlinear partial differential equation in four dimensions underlying self-dual vacuum solutions of the Einstein equations, which arises from the commutation of multidimensional Hamiltonian vector fields. (c) 2006 Elsevier B.V. All rights reserved
The Cauchy Problem on the Plane for the Dispersionless Kadomtsev - Petviashvili Equation
No full textWe construct the formal solution of the Cauchy problem for the dispersionless
Kadomtsev - Petviashvili equation as application of the Inverse Scattering
Transform for the vector field corresponding to a Newtonian particle in a
time-dependent potential. This is in full analogy with the Cauchy problem for
the Kadomtsev - Petviashvili equation, associated with the Inverse Scattering
Transform of the time dependent Schroedinger operator for a quantum particle in
a time-dependent potential.Comment: 10 pages, submitted to JETP Letter
Wave breaking in solutions of the dispersionless kadomtsev-petviashvili equation at a finite time
No full textWe discuss some interesting aspects of the wave breaking in localized solutions of the dispersionless Kadomtsev-Petviashvili equation, an integrable partial differential equation describing the propagation of weakly nonlinear, quasi-one-dimensional waves in 2+1 dimensions, which arise in several physical contexts such as acoustics, plasma physics, and hydrodynamics. For this, we use an inverse spectral transform for multidimensional vector fields that we recently developed and, in particular, the associated inverse problem, a nonlinear Riemann-Hilbert problem on the real axis. In particular, we discuss how the derivative of the solution blows up at the first breaking point in any direction of the plane (x, y) except in the transverse breaking direction and how the solution becomes three-valued in a compact region of the plane (x, y) after the wave breaking
On the solutions of the second heavenly and Pavlov equations
No full textWe 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, such as 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, such as the Pavlov equation. We also showed that the associated Riemann-Hilbert inverse problems are powerful 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 long-time behaviour of the solutions of their Cauchy problems; (iii)characterizing a distinguished class of implicit solutions of the heavenly equation
A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields
No full textWe introduce a hierarchy of integrable partial differential equations in 2+1 dimensions arising from the commutation of one-para,meter families of vector fields, and we construct the lormal solution of the associated Cauchy problems using the inverse scattering method for one-parameter families of vector fields. Because the space of eigenfunctions is a ring, the inverse problem can be formulated in three distinct ways. In particular, one formulation corresponds to a linear integral equation for a Jost eigenfunction, and another formulation is a scalar nonlinear niemann problein for suitable analytic eigenlunctions
Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields
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