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

Abstract

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.

Singular Manifold Method for an Equation in 2 + 1 Dimensions

“…Actually, (2.1) generalizes the Ermakov-Pinney equation [13] that, as has been proved in [15], is related by means of a reciprocal transformation to an equation with peakon solutions: the Degasperis-Procesi equation. Furthermore, (2.1) can be considered as a modified version of the generalized Hirota-Satsuma equation presented in [2] and [8] as a model for an incompressible fluid.…”

Soliton solutions of the Hamiltonian DSI and DSIII equations

“…The equation DSI can be obtained [6,7] as the compatibility condition between two Lax operators T 1 and T 2 which commute in the "weak" sense [15] …”

Multidimensional localized solitons

“…Here we want to quote only some references more relevant historically or more close to our specific approach. Precisely [65,66] for the discovery of the one dimensional solitons, [67,68] for the extension of the theory to the nonlinear Schrödinger equation, [69]- [71] for the introduction of the so-called ∂-method, [72]- [74], [54], [12], [75]- [80] for the extension of the Spectral Transform to 2 + 1 dimensions, [76]- [79] in particular for the introduction of the "weak" Lax representation of the integrable equations in 2 + 1 dimensions, [81,82] for the Bäcklund transformations and [83] for the extension of the Bäcklund transformations to 2 + 1 dimensions.…”