Dynamics of Solitons of the Generalized Nonlinear Schrödinger Equation in an Inhomogeneous and Nonstationary Medium: Evolution and Interaction

Abstract

The stability and dynamics of the interaction of soliton-like solutions of the generalized nonlinear Schrödinger (NLS) equation describing the dynamics of the envelope of modulated nonlinear waves and pulses (including the phenomenon of wave collapse and the self-focusing of wave beams) in plasma (including space one), as well as in nonlinear optical systems, have been studied with allowance for the inhomogeneity and nonstationarity of the distribution environment. The equation is also used in other areas of physics, such as the theory of superconductivity and low-temperature physics, small-amplitude gravitational waves on the surface of a deep inviscid fluid, etc. It should be noted that the studied equation is not completely integrable, and its analytical solutions are generally unknown (except, perhaps, for smooth solutions of the solitary wave type). However, approaches that were developed earlier for other equations (the generalized Kadomtsev–Petviashvili equation and the three-dimensional NLS equation with the derivative of the nonlinear term) of the Belashov–Karpman system makes it possible to analyze the stability of possible solutions of these equations and to conduct a numerical study of the dynamics of soliton interaction. This approach is implemented in the study. Sufficient conditions for the stability of two- and three-dimensional soliton-like solutions are obtained analytically, and the cases of stable and unstable (with the formation of breathers) evolution of pulses of various shapes, as well as the interaction of two- and three-pulse structures, which leads to the formation of stable and unstable solutions, were studied numerically. The results can be useful in numerous applications for the physics of ionospheric and magnetospheric plasma and in many other areas of physics.