J. Leon

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.

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.

Nonlinear evolutions with singular dispersion laws and forced systems

Abstract We develop a general scheme for the spectral transform theory which allows us first to discuss the properties of forced “integrable” systems related to some nonlinear evolution of the spectral transforms, andsecondto build very general solvable nonlinear evolutions related to new types of Lax pairs involving nonlinear, nonlocal and nonanalytic spectral problems. The link between both results is then made.

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.

A new spectral transform for the Davey-Stewartson I equation

Abstract We propose to modify the original spectral transform for the Davey-Stewartson I equation as defined by Fokas and Ablowitz, when the auxiliary function is different from zero at larger distances. The new spectral transform has the following advantages: (i) the form of its time evolution is the same for all equations in the hierarchy and can be explicitly integrated as in the one-dimensional case, (ii) each soliton corresponds to a couple of discrete eigenvalues of the associated spectral problem as in the one-dimensional case, (iii) this spectral transform coincides with the one suggested by the Backlund gauge transformation theory.

Nonlinear supratransmission as a fundamental instability

The nonlinear supratransmission is the property of a nonlinear system possessing a natural forbidden band gap to transmit energy of a signal with a frequency in the gap by means of generation of nonlinear modes (gap solitons). This process is shown to result from a generic instability of the evanescent wave profile generated in a nonlinear medium by the incident signal.

Nonlinear energy transmission in the gap

Abstract Numerical simulations of the scattering of a linear plane wave incoming onto a nonlinear medium (sine–Gordon) reveals that: (i) nonlinearity allows energy transmission in the forbidden band, (ii) this nonlinear transmission occurs beyond an energy threshold of the incoming wave, (iii) the process begins (at the threshold) with large amplitude breathers, and then energy is generically transmitted both by kink–antikink pairs and breathers.

General evolution of the spectral transform from the ∂-approach

Abstract A general time evolution of the spectral transform is constructed by means of ∂ -analysis. It corresponds either to perturbed nonlinear evolution equations or to generalized equations of coherent pulse propagation. A general theorem characterising the integrable cases is obtained. The method furnishes also a simple and powerful tool to build classes of evolution equations related to singular dispersion relations.

Integrable nonlinear evolutions in 2+1 dimensions with non-analytic dispersion relations

The authors develop a method of obtaining two-dimensional integrable evolutions corresponding to singular dispersion relations. The method is applied to the 2*2 linear first-order eigenvalue problem and the inverse spectral transform scheme is established. The Backlund transformation and a non-linear superposition formula are used to obtain the soliton solutions.

Solution of the boundary value problem for the integrable discrete SRS system on the semi-line

The complete solution of the initial-boundary value problem for the integrable discrete version of the stimulated Raman scattering system on the semi-line is constructed by means of the inverse spectral transform. The spectral data obey a Riccati time evolution equation which allows for soliton generation out of a medium initially at rest. It is also proved that the construction of the solution at any finite distance actually results in solving an algebraic system.