Multidimensional localized solitons

Abstract

Recently it has been discovered that some nonlinear evolution equations in 2 + 1 dimensions, which are integrable by the use of the Spectral Transform, admit localized (in the space) soliton solutions. This article briefly reviews some of the main results obtained in the last five years thanks to the renewed interest in soliton theory due to this discovery. The theoretical tools needed to understand the unexpected richness of behaviour of multidimensional localized solitons during their mutual scattering are furnished. Analogies and especially discrepancies with the unidimensional case are stressed.

Solution of the dispersionless Hirota equations

“…The potentials in (4.6) occur in χ 1 . The D-bar data, or departure from analyticity of χ, corresponds to spectral data in a sometimes complicated way and we refer to [1,5,6,9,11,13,16,20,22,23] for more on this. Now one expects some connections between inverse scattering for KdV and the dKdV theory since ǫx = X means e.g.…”

Vertex dynamics in multi-soliton solutions of Kadomtsev–Petviashvili II equation

“…Except for specific choices of the ξ-coefficients and/or the wave numbers, Eq. (5) then generates three four-soliton processes: As t → ±∞, two solutions have two junctions ((M,N) = (4,1) and (4,3), which are related by space-time inversion) and one solution has four ((M,N) = (4,2)).…”

Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4‐d Spaces and Deformations via Ds Hierarchy

“…For these solutions, called dromions, the functions p and q decrease exponentially fast in all directions on the plane ξ η (see, e.g. [53,49,51]). For general dromion solution of equation (A13) one has [54] p 2 = −4 log det 1 − εA ξη (8.16) where the rectangular matrix A is of the form A = β †ᾱ (8.17) and…”