40 research outputs found
The Cauchy Problem on the Plane for the Dispersionless Kadomtsev - Petviashvili Equation
We 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
Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields
We review some results about the theory of integrable dispersionless PDEs
arising as commutation condition of pairs of one-parameter families of vector
fields, developed by the authors during the last years. We review, in
particular, the formal aspects of a novel Inverse Spectral Transform including,
as inverse problem, a nonlinear Riemann - Hilbert (NRH) problem, allowing one
i) to solve the Cauchy problem for the target PDE; ii) to construct classes of
RH spectral data for which the NRH problem is exactly solvable; iii) to
construct the longtime behavior of the solutions of such PDE; iv) to establish
if a localized initial datum breaks at finite time. We also comment on the
existence of recursion operators and Backl\"und - Darboux transformations for
integrable dispersionless PDEs.Comment: 17 pages, 1 figure. Written rendition of the talk presented by one of
the authors (PMS) at the PMNP 2013 Conference, in a special session dedicated
to the memory of S. V. Manakov. To appear in the Proceedings of the
Conference PMNP 2013, IOP Conference Serie
On the solutions of the dKP equation: nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking
We make use of the nonlinear Riemann Hilbert problem of the dispersionless
Kadomtsev Petviashvili equation, i) to construct the longtime behaviour of the
solutions of its Cauchy problem; ii) to characterize a class of implicit
solutions; iii) to elucidate the spectral mechanism causing the gradient
catastrophe of localized solutions, at finite time as well as in the longtime
regime, and the corresponding universal behaviours near breaking.Comment: 33 pages, 10 figures, few formulas update
Initial-Boundary Value Problems for Linear and Soliton PDEs
Evolution PDEs for dispersive waves are considered in both linear and
nonlinear integrable cases, and initial-boundary value problems associated with
them are formulated in spectral space. A method of solution is presented, which
is based on the elimination of the unknown boundary values by proper
restrictions of the functional space and of the spectral variable complex
domain. Illustrative examples include the linear Schroedinger equation on
compact and semicompact n-dimensional domains and the nonlinear Schroedinger
equation on the semiline.Comment: 18 pages, LATEX, submitted to the proccedings of NEEDS 2001 - Special
Issue, to be published in the Journal of Theoretical and Mathematical Physic
Dunajski generalization of the second heavenly equation: dressing method and the hierarchy
Dunajski generalization of the second heavenly equation is studied. A
dressing scheme applicable to Dunajski equation is developed, an example of
constructing solutions in terms of implicit functions is considered. Dunajski
equation hierarchy is described, its Lax-Sato form is presented. Dunajsky
equation hierarchy is characterized by conservation of three-dimensional volume
form, in which a spectral variable is taken into account.Comment: 13 page
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
Interpolating Dispersionless Integrable System
We introduce a dispersionless integrable system which interpolates between
the dispersionless Kadomtsev-Petviashvili equation and the hyper-CR equation.
The interpolating system arises as a symmetry reduction of the anti--self--dual
Einstein equations in (2, 2) signature by a conformal Killing vector whose
self--dual derivative is null. It also arises as a special case of the
Manakov-Santini integrable system. We discuss the corresponding Einstein--Weyl
structures.Comment: 11 pages. New title, some errors corrected, section 5 removed. Final
version, to appear in J. Phys.