Two-dimensional lumps in nonlinear dispersive systems

Abstract

Two-dimensional lump solutions which decay to a uniform state in all directions are obtained for the Kadomtsev–Petviashvili and a two-dimensional nonlinear Schrödinger type equation. The amplitude of these solutions is rational in its independent variables. These solutions are constructed by taking a ’’long wave’’ limit of the corresponding N-soliton solutions obtained by direct methods. The solutions describing multiple collisions of lumps are also presented.

Solitons and Infinite Dimensional Lie Algebras

“…On the other hand, we can show that (see (39) In a similar manner we may introduce an arbitrary number of discrete variables and write down a hierarchy of discrete KP equations, which is equivalent to the continuous KP hierarchy. The same procedure is applicable to all the equations discussed in Sections 1-9.…”

Stable multiple vortices in collisionally inhomogeneous attractive Bose-Einstein condensates

“…The discovery of robust coherent nonlinear excitations, in the form of dromions [10] (localized 2D patterns produced by overlapping of 1D ghost solitons) and lumps (weakly localized 2D solitons) [11] in other 2D models has prompted looking for similar nonlinear excitations in BEC as well, following the investigation of more straightforward nonlinear modes -in particular, vortices [12] and Faraday waves [14] -in effectively 2D "pancake-shaped" BECs. Faraday waves are undulating nonlinear excitations generated by time-periodic shaking of the trapping potential, while vortices are created by stirring the condensate with the help of properly designed laser beams, by coherent transfer of orbital angular momentum to the condensate by the two-photon stimulated Raman process [12,13], or by imprinting an appropriate phase pattern onto a trapped condensate [15], see recent survey [16].…”

Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy

“…Then we obtain non-singular rational solutions of the KPI equation. In fact, for N = 1 and n 1 = 1 we get the 1-lump solution [24,2,37] for KPI and for n 1 = 2 the Johnson-Thompson solution [19] recently studied in [5]. For N = 1 we can increase the degree n 1 of the heat polynomial p 1 , as is already suggested [19], and get more involved rational behaviour showing, as is studied in [5], nontrivial interaction of localized lumps.…”