Higher-order vector Peregrine solitons and asymptotic estimates for the multi-component nonlinear Schrödinger equations

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…

Effect of local Peregrine soliton emergence on statistics of random waves in the 1-D focusing Nonlinear Schrödinger equation

The Peregrine soliton is often considered as a prototype of the rogue waves. After recent advances in the semi-classical limit of the 1-D focusing Nonlinear Schrödinger (NLS) equation this conjecture can be seen from another perspective. In the present paper, connecting deterministic and statistical approaches, we numerically demonstrate the effect of the universal local appearance of Peregrine solitons on the evolution of statistical properties of random waves. Evidences of this effect are found in recent experimental studies in the contexts of fiber optics and hydrodynamics. The present appr…

Numerical study of the stability of the Peregrine breather

The Peregrine breather is widely discussed as a model for rogue waves in deep water. We present here a detailed numerical study of perturbations of the Peregrine breather as a solution to the nonlinear Schrödinger (NLS) equations. We first address the modulational instability of the constant modulus solution to NLS. Then we study numerically localized and nonlocalized perturbations of the Peregrine breather in the linear and fully nonlinear setting. It is shown that the solution is unstable against all considered perturbations.

Multi-parametric solutions to the NLS equation

The structure of the solutions to the one dimensional focusing nonlin-ear Schr{ö}dinger equation (NLS) for the order N in terms of quasi rational functions is given here. We first give the proof that the solutions can be expressed as a ratio of two wronskians of order 2N and then two determinants by an exponential depending on t with 2N -- 2 parameters. It also is proved that for the order N , the solutions can be written as the product of an exponential depending on t by a quotient of two polynomials of degree N (N + 1) in x and t. The solutions depend on 2N -- 2 parameters and give when all …

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。

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.