3,522 research outputs found
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
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
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
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