Symmetries for exact solutions to the nonlinear Schrödinger equation

Periodic energy switching of bright solitons in mixed coupled nonlinear Schr{ö}dinger equations with linear self and cross coupling terms

The bright soliton solutions of the mixed 2-coupled nonlinear Schr{ö}dinger (CNLS) equations with linear self and cross coupling terms have been obtained by identifying a transformation that transforms the corresponding equation to the integrable mixed 2-CNLS equations. The study on the collision dynamics of bright solitons shows that there exists periodic energy switching, due to the coupling terms. This periodic energy switching can be controlled by the new type of shape changing collisions of bright solitons arising in mixed 2-CNLS system, characterized by intensity redistribution, amplitud…

Deep-water waves: On the nonlinear Schr{ö}dinger equation and its solutions

We present a brief discussion on the nonlinear Schr{ö}dinger equation for modeling the propagation of the deep-water wavetrains and a discussion on its doubly-localized breather solutions that can be connected to the sudden formation of extreme waves, also known as rogue waves or freak waves.

Shape changing (intensity redistribution) collisions of solitons in mixed coupled nonlinear Schr{ö}dinger equations

A novel kind of shape changing (intensity redistribution) collision with potential application to signal amplification is identified in the integrable $N$-coupled nonlinear Schr{ö}dinger (CNLS) equations with mixed signs of focusing and defocusing type nonlinearity coefficients. The corresponding soliton solutions for N=2 case are obtained by using Hirota's bilinearization method. The distinguishing feature of the mixed sign CNLS equations is that the soliton solutions can both be singular and regular. Although the general soliton solution admits singularities we present parametric conditions …

Perturbation theory for the modified nonlinear Schr{ö}dinger solitons

The perturbation theory based on the Riemann-Hilbert problem is developed for the modified nonlinear Schr{ö}dinger equation which describes the propagation of femtosecond optical pulses in nonlinear single-mode optical fibers. A detailed analysis of the adiabatic approximation to perturbation-induced evolution of the soliton parameters is given. The linear perturbation and the Raman gain are considered as examples.

Peregrine comb: multiple compression points for Peregrine rogue waves in periodically modulated nonlinear Schr{ö}dinger equations

It is shown that sufficiently large periodic modulations in the coefficients of a nonlinear Schr{ö}dinger equation can drastically impact the spatial shape of the Peregrine soliton solutions: they can develop multiple compression points of the same amplitude, rather than only a single one, as in the spatially homogeneous focusing nonlinear Schr{ö}dinger equation. The additional compression points are generated in pairs forming a comb-like structure. The number of additional pairs depends on the amplitude of the modulation but not on its wavelength, which controls their separation distance. The…

含时变平衡损耗-增益与时空调制非线性相互作用的非线性薛定谔方程

Non-linear Schr$\ddot{o}$dinger equation with time-dependent balanced loss-gain and space-time modulated non-linear interaction

本研究针对含平衡损耗-增益、线性耦合与时空调制非线性作用的一维矢量非线性薛定谔方程，通过非幺正变换与坐标变换映射为可解方程，结合超对称量子力学方法构造了多类势场下的精确局域非线性模与奇异相位解。