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Dynamics of Solitons of the Generalized Nonlinear Schrödinger Equation in an Inhomogeneous and Nonstationary Medium: Evolution and Interaction
Geomagnetism and Aeronomy ( IF 0.6 ) Pub Date : 2021-05-10 , DOI: 10.1134/s0016793221020031
V. Yu. Belashov , O. A. Kharshiladze , E. S. Belashova

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.



中文翻译:

非均匀非平稳介质中广义非线性薛定ding方程孤子的动力学:演化与相互作用

摘要

广义非线性薛定ding(NLS)方程的类孤子解的相互作用的稳定性和动力学,描述了调制的非线性波和脉冲的包络线的动力学(包括波崩和波的自聚焦现象)。已经研究了等离子体(包括空间一)以及非线性光学系统中的分布环境的不均匀性和非平稳性。该方程还用于其他物理领域,例如超导理论和低温物理学,深粘性流体表面的小振幅引力波等。应注意,所研究的方程并不完全可积,其解析解决方案通常是未知的(也许,用于孤立波类型的平滑解)。但是,Belashov–Karpman系统的其他方程(广义的Kadomtsev–Petviashvili方程和带有非线性项导数的三维NLS方程)较早开发的方法,使得可以分析这些方程的可能解的稳定性方程并进行孤子相互作用动力学的数值研究。该方法已在研究中实现。通过分析获得了二维和三维类孤子解的稳定性的充分条件,以及各种形状的脉冲的稳定和不稳定(随着通气孔的形成)的演化以及二维和二维相互作用的情况。三脉冲结构,导致形成稳定和不稳定的溶液,进行了数值研究。该结果可用于电离层和磁层等离子物理学的许多应用,以及许多其他物理学领域。

更新日期:2021-05-10
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