F. Pempinelli

Scattering of localized solitons in the plane

Abstract Localized (exponentially decaying in all directions) soliton solutions of the evolution equations related to the Zakharov-Shabat spectral problem in the plane are explicitly given. They can move along any direction in the plane and the only effect of their interaction is a shift in their position both in the x- and y-directions, independently of their relative initial position in the plane.

On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions

A generalisation in 2+1 dimensions of the Korteweg-de Vries equation is related to the spectral problem ( delta x2- delta y2-p(x,y)) phi (x,y;k)=0. It can contain arbitrary functions of x+y or x-y and time. The Cauchy problem, associated with initial data decaying sufficiently rapidly at infinity, is linearised by an extension of the spectral transform technique to two spatial dimensions. The spectral data are explicitly defined in terms of the initial data and the inverse problem is formulated as a non-local Riemann-Hilbert boundary-value problem. The presence of arbitrary functions of x+y and x-y in the evolution equation implies that the time evolution of the spectral data is linear but non-local. Discrete spectral data are forbidden and, consequently, localised soliton solutions are not allowed.

Spectral transform for a two spatial dimension extension of the dispersive long wave equation

A two-dimensional nonlinear evolution equation is solved in the inverse spectral transform scheme. It coincides, when reduced to one spatial dimension, with the dispersive long wave equation. The Backlund transformation, soliton solution and superposition formula are obtained. The spectral transform is explicitly defined and the corresponding linear evolution of the spectral data is given. The inverse spectral problem is formulated as a non-local Riemann-Hilbert boundary value problem and solved.

Integrable two-dimensional generalisation of the sine- and sinh-Gordon equations

A two-dimensional generalisation of the sine- and the sinh-Gordon equations, which one refers to as the shine-Gordon equations, is obtained and solved through the inverse spectral transform (IST) method. The Backlund transformation and nonlinear superposition formula are constructed and explicit wave solitons are given. It is shown also that a slightly different procedure furnishes an IST-solvable extension in 2+1 dimensions of the dispersive long-wave equation.

On a spectral transform of a KDV-like equation related to the Schrodinger operator in the plane

A generalisation in 2+1 dimensions of the Korteweg-de Vries (KDV) equation is related to the Schrodinger operator in the plane. Spectral data which only depend on one complex spectral parameter are introduced and the inverse problem is formulated as a delta problem. In opposition to what happens in the usual scattering scheme, the final formula for reconstructing the potential does not depend fictitiously on any extra parameter. The time evolution of the spectral data corresponding to a potential that evolves according to the KDV-like equation mentioned above is also obtained.

Multidimensional solitons and their spectral transforms

The soliton solution to the hierarchy of two‐dimensional nonlinear evolution equations related to the Zakharov–Shabat spectral problem (including the Davey–Stewartson equation) are derived and studied. The solitons are localized two‐dimensional structures traveling on straight lines at constant velocities. Their spectral transform is not uniquely defined and this point is discussed by giving two explicit different spectral transforms of the one‐soliton solution and also by giving the general dependence of the spectral transform on the definition of the basic Jost‐like solutions.

Towards an inverse scattering theory for non-decaying potentials of the heat equation

The resolvent approach is applied to the spectral analysis of the heat equation with non-decaying potentials. The special case of potentials with spectral data obtained by a rational similarity transformation of the spectral data of a generic decaying potential is considered. It is shown that these potentials describe N solitons superimposed by Backlund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a -problem explicitly in terms of the corresponding objects associated with the original potential. Regularity conditions of the potential in the cases N = 1 and 2 are investigated in detail. The singularities of the resolvent for the case N = 1 are studied, opening the way to a correct definition of the spectral data for a generically perturbed soliton.

Properties of solutions of the Kadomtsev–Petviashvili I equation

The Kadomtsev–Petviashvili I (KPI) equation is considered as a useful laboratory for experimenting with new theoretical tools able to handle the specific features of integrable models in 2+1 dimensions. The linearized version of the KPI equation is first considered by solving the initial value problem for different classes of initial data. Properties of the solutions in different cases are analyzed in details. The obtained results are used as a guideline for studying the properties of the solution u(t,x,y) of the Kadomtsev–Petviashvili I (KPI) equation with given initial data u(0,x,y) belonging to the Schwartz space. The spectral theory associated to KPI is studied in the space of the Fourier transform of the solutions. The variables p={p1,p2} of the Fourier space are shown to be the most convenient spectral variables to use for spectral data. Spectral data are shown to decay rapidly at large p but to be discontinuous at p=0. Direct and inverse problems are solved with special attention to the behavior of...

Spectral transform and orthogonality relations for the Kadomtsev-Petviashvili I equation☆

Abstract We define a new spectral transform r(k, l) of the potential u in the time dependent Schrodinger equation (associated to the KPI equation). Orthogonality relations for the sectionally holomorphic eigenfunctions of the Schrodinger equation are used to express the spectral transform f ( k, l ) previously introduced by Manakov and Fokas and Ablowitz in terms of r ( k, l ). The main advantage of the new spectral transform r ( k, l ) is that its definition does not require to introduce an additional nonanalytic eigenfunction N . Characterization equations for r ( k, l ) are also obtained.

Raman solitons in transient SRS

We report on the observation of Raman solitons on numerical simulations of transient stimulated Raman scattering with small group-velocity dispersion. The theory proceeds with the inverse scattering transform (IST) for initial boundary-value problems and it is shown that the explicit theoretical solution obtained by IST for a semi-infinite medium fits strikingly well the numerical solution for a finite medium. We are able to explain this in terms of the rapid decrease of the medium dynamical variable (the potential of the scattering theory). The spectral transform reflection coefficient can be computed directly from the values of the input and output fields and this allows one to see the generation of the Raman solitons from the numerical solution. We confirm the presence of these nonlinear modes in the medium dynamical variable by the use of a discrete spectral analysis.