445 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
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
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
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
A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy
In the present paper we begin studies on the large time asymptotic behavior
for solutions of the Cauchy problem for the Novikov--Veselov equation (an
analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are
focused on a family of reflectionless (transparent) potentials parameterized by
a function of two variables. In particular, we show that there are no isolated
soliton type waves in the large time asymptotics for these solutions in
contrast with well-known large time asymptotics for solutions of the KdV
equation with reflectionless initial data
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