Integrability and soliton interaction of a resonant nonlinear Schrödinger equation via binary Bell polynomials. (English) Zbl 1270.37046

Li, Min; Xiao, Jing-Hua; Yan, Tian-Zhong; Tian, Bo

Nonlinear Anal., Real World Appl. 14, No. 3, 1669-1679 (2013).

Summary: Under investigation in this paper is a resonant nonlinear Schrödinger equation for the response of a hypothetical resonance medium to an action of a quasimonochromatic wave or the propagation of one-dimensional long magnetoacoustic waves in a cold collisionless plasma subject to a transverse magnetic field. Binary Bell polynomials are employed to derive the bilinear form, Bäcklund transformation (BT) and Lax pair in the \(3\times 3\) matrix form. Two sets of the binary Bell polynomials are considered. Infinite conservation laws are also constructed from the BT in the binary-Bell-polynomial form. Moreover, two-soliton solutions are obtained through the Hirota method. Finally, the regular, intermediate-state and resonant soliton interactions are analyzed under certain conditions.

MSC:

37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35	Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35C08	Soliton solutions
35Q55	NLS equations (nonlinear Schrödinger equations)
11B73	Bell and Stirling numbers
33C47	Other special orthogonal polynomials and functions

Keywords:

resonant nonlinear Schrödinger equation; integrability; resonant interactions; binary Bell polynomials