Javier Villarroel

A novel class of solutions of the non-stationary Schrödinger and the Kadomtsev–Petviashvili I equations

Abstract A new class of real, non-singular and rationally decaying potentials and eigenfunctions of the non-stationary Schrodinger equation and solutions of the KP I equation are constructed via binary Darboux transformations. These solutions are classified by the pole structure of the corresponding meromorphic eigenfunction and a set of integers including a quantity called the charge. The properties of the potential, eigenfunction and their relationship to the inverse scattering transform are discussed.

The Cauchy Problem for the Kadomtsev–Petviashili II Equation with Nondecaying Data along a Line

The initial value problem for the Kadomstev-Petviashili II (KPII) equation is considered with given data that are nondecaying along a line. The associated direct and inverse scattering of the two-dimensional heat equation is constructed. The direct problem is formulated in terms of a bounded Greens function. The inverse data are decomposed into scattering data along the line and a data from the decaying portion of the potential. The solution of the KPII equation is then obtained via coupled linear integral equations.

Monotonic continuous-time random walks with drift and stochastic reset events

In this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes. We focus our attention on monotonic continuous-time random walks with a constant drift: The process increases between the reset events, either by the effect of the random jumps, or by the action of the deterministic drift. As a result of all these combined factors interesting properties emerge, like the existence (for any drift strength) of a stationary transition probability density function, or the faculty of the model to reproduce power-law-like behavior. General formulas for two extreme statistics, the survival probability, and the mean exit time are also derived. To corroborate in an independent way the results of the paper, Monte Carlo methods were used. These numerical estimations are in full agreement with the analytical predictions.

On a Volterra system

A 2 + 1 Volterra system is considered, which in the continuum limit reduces to the well known Kadomtsev - Petviashvili equation and the 1 + 1 reduction becomes the Volterra system. The inverse scattering transform is developed, and special solutions are obtained, including lump type solutions as well as solutions which depend on arbitrary functions.

On an algorithmic construction of lump solutions in a 2+1 integrable equation

The singular manifold method is used to generate lump solutions of a generalized integrable nonlinear Schrodinger equation in 2 + 1 dimensions. We present several essentially different types of lump solutions. The connection between this method and the Ablowitz–Villarroel scheme is also analysed.

On the Hamiltonian formalism for the Davey-Stewartson system

The Hamiltonian formalism and the action-angle variables for a generalized version of the Davey-Stewartson system is developed. Special cases include the usual Davey-Stewartson II system and the delta limit of the Davey-Stewartson I equations.

Contact symmetries and integrable non-linear dynamical systems

The authors present a systematic method of classifying and constructing invariants for Lagrangians containing arbitrary polynomial non-linear potentials. It is based on the assumption that these Lagrangians are invariant under contact groups of transformations. For a finite number of degrees of freedom they can prove integrability for a large class of polynomial potentials. The method can be extended in several directions.

On the inverse scattering transform of the 2+1 Toda equation

Abstract The method of solution to the 2 + 1 dimensional Toda equation is described in some detail. This equation reduces directly to the well know Toda lattice in 1 + 1 dimension and, by an appropriate asymptotic reduction, to the Kadomtsev-Petviashvili equation in a continuous limit. The solution exhibits a number of interesting aspects depending on certain choices of signs, of which there are four, in the equation. For two choices of sign the equation is well posed and linearly stable/unstable. For the other choices of sign the equation is linearly illposed. In these cases we can relate the solution to a boundary value problem and give a formal construction of the solution. For one choice of signs in the illposed case an analogue of the Sommerfeld radiation condition is developed in order to identify a unique solution. In general the method of solution of the “Toda molecule” equation requires an implementation of the dbar technique to cases where the associated eigenfunctions possess both smooth regions of nonholomorphicity and a discontinuity across a curve, which in this problem is the unit circle. Special lump type solutions and solutions depending on suitable arbitrary functions are presented.

Initial Value Problems and Solutions of the Kadomtsev-Petviashvili Equation

Initial value problems and solutions associated with the Kadomtsev-Petviashvili equation are analyzed. The discussion includes the inverse scattering transform for suitably decaying data, solutions decaying off a background line, multi-pole lump soliton solutions and solutions which are slowly decaying. Existence and uniqueness of the associated eigenfunctions are discussed in terms of natural functional norms.

ON THE DISCRETE SPECTRUM OF SYSTEMS IN THE PLANE AND THE DAVEY-STEWARTSON II EQUATION ∗

The discrete spectrum of first order systems in the plane and localized solutions of the Davey--Stewartson II equation are studied via the inverse scattering transform. Localized nonsingular algebraically decaying potentials are found which correspond to a discrete spectrum whose related eigenfunctions have, in general, multiple poles and are associated to kernels with dimension