Jerome Leon

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.

Nonlinear resonant scattering and plasma instability: an integrable model

A detailed study of a system of coupled waves is given for which an initial‐boundary value problem is solved by means of the spectral transform theory. This system represents the nonlinear interaction of an electrostatic high‐frequency wave with the ion acoustic wave in a two component homogeneous plasma. As a result it is understood the plasma instability as (i) a continuous secular transfer of energy from the laser beam to the acoustic wave, (ii) the evolution toward the formation of local singularities of the electrostatic wave (collapsing), (iii) a mutual trapping of the acoustic wave and the scattered Langmuir wave.

Bistability in the sine-gordon equation : The ideal switch

The sine-Gordon equation, used as the representative nonlinear wave equation, presents a bistable behavior resulting from nonlinearity and generating hysteresis properties. We show that the process can be understood in a comprehensive analytical formulation and that it is a generic property of nonlinear systems possessing a natural band gap. The approach allows one to discover that the sine-Gordon equation can work as an ideal switch by reaching a transmissive regime with vanishing driving amplitude.

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.

Bifurcations of solitons in multidimensions

The authors consider the W-soliton solution of the Davey-Stewartson equation. It describes N localized bell shaped objects interacting without changing shape and velocity. The only effect of the interaction is a shift in the position and a shift in the phase. However, at some values of the parameters the asymptotic behaviour of the solution bifurcates. The structurally unstable solutions describe coherent localized objects that during the interaction change shape without exchanging energy and coherent localized objects that change shape and exchange energy.

Spectral transform and solitons for generalized coupled Bloch systems

The spectral transform and Backlund transformation for a generalization of the coupled Bloch system arising in nonlinear optics are studied. The equation is related to a singular dispersion relation and is treated as a representative example of a general method for studying such equations. The spectral transform is developed through the ∂ formalism. The Backlund transformation is derived in a very general way and solved to obtain the one‐soliton and breatherlike solutions. The nonlinear superposition formula is also constructed.

The Dirac inverse spectral transform: Kinks and boomerons

The inverse spectral transform (IST) is derived when using the eigenvalue problem for the one-dimensional Dirac operator: (D)=iσ3(d/dx)+i(r 0 0-q), σ3=(01-10), where the potentials q and r have nonzero asymptotic values. The method used is of AKNS type. It is shown that the nonlinear evolution equations (NEE) obtained are of differential type at any order (and not of integro-differential type). Some particular solutions are studied, and it is shown that their special behavior is a direct consequence of the nonzero boundary condition on (D).