On an algorithmic construction of lump solutions in a 2+1 integrable equation

Abstract

The singular manifold method is used to generate lump solutions of a generalized integrable nonlinear Schrödinger equation in 2 + 1 dimensions. We present several essentially different types of lump solutions. The connection between this method and the Ablowitz-Villarroel scheme is also analysed.

Dynamics of Lump Solutions in a 2 + 1 NLS Equation

“…Particular attention is paid to describing the dynamics and scattering properties of some of these configurations. Lump solutions of the kind considered here have also been obtained via these direct methods in [21]. Note also that in [22,23] and [24] it was proven that Equations (1) satisfy Painleve's test; in addition some special solutions, like line solitons, lumps and dromion solutions, were found.…”

“…As we have already pointed out Equation (1) arises as the compatibility of a pair of operators, see [5,21,24]. Under the boundary conditions 1 (BC) lim r →∞ |u|(x, y, t) = 1 a convenient form of the Lax pair, depending on a complex spectral parameter k, is given by the following pair of linear operators L, M:…”

Lump solitons in a higher-order nonlinear equation in2+1dimensions

“…The same method was applied in Ref. [16] to derive rational solitons (lumps) of a different generalization of the NLSE to 2 + 1 dimensions. Notice that the second derivative includes crossed terms u xy instead of some combination of u xx and u yy as appears in many genaralizations of NLS.…”

“…Rogue waves in 1 + 1 dimensions [17] as well as lumps in 2 + 1 dimensions [16] are rational solutions with nontrivial behavior. This suggests that rogue waves can appear as a reduction of variables in the lump solutions.…”

“…Therefore, it can be regarded as an integrable system of equations, which means that it should have a Lax representation. A powerful method to derive the Lax pair is the singular manifold method [19], which has already been applied to the above-mentioned generalizations of the NLSE to 2 + 1 dimensions [15,16].…”