652 research outputs found

### 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

### New reductions of integrable matrix PDEs: $Sp(m)$-invariant systems

We propose a new type of reduction for integrable systems of coupled matrix
PDEs; this reduction equates one matrix variable with the transposition of
another multiplied by an antisymmetric constant matrix. Via this reduction, we
obtain a new integrable system of coupled derivative mKdV equations and a new
integrable variant of the massive Thirring model, in addition to the already
known systems. We also discuss integrable semi-discretizations of the obtained
systems and present new soliton solutions to both continuous and semi-discrete
systems. As a by-product, a new integrable semi-discretization of the Manakov
model (self-focusing vector NLS equation) is obtained.Comment: 33 pages; (v4) to appear in JMP; This paper states clearly that the
elementary function solutions of (a vector/matrix generalization of) the
derivative NLS equation can be expressed as the partial $x$-derivatives of
elementary functions. Explicit soliton solutions are given in the author's
talks at http://poisson.ms.u-tokyo.ac.jp/~tsuchida

### Soliton dynamics in deformable nonlinear lattices

We describe wave propagation and soliton localization in photonic lattices
which are induced in a nonlinear medium by an optical interference pattern,
taking into account the inherent lattice deformations at the soliton location.
We obtain exact analytical solutions and identify the key factors defining
soliton mobility, including the effects of gap merging and lattice imbalance,
underlying the differences with discrete and gap solitons in conventional
photonic structures.Comment: 5 pages, 4 figure

### 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

### 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

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