scholarly journals Collision-induced amplitude dynamics of pulses in linear waveguides with the generic nonlinear loss

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Quan M. Nguyen
Keyword(s):  
Numerical Simulations ◽  
Perturbation Technique ◽  
Propagation Model ◽  
Schrodinger Equations ◽  
Nonlinear Coupling ◽  
Schrödinger Equations ◽  
Nonlinear Loss ◽  
Theoretical Predictions ◽  
Analytic Expression ◽  
Pulse Interaction

Abstract We study the effects of the generic weak nonlinear loss on fast two-pulse interactions in linear waveguides. The colliding pulses are described by a system of coupled Schrödinger equations with a purely nonlinear coupling in the presence of the weak (2m + 1)-order of nonlinear loss, for any m ≥ 1. We derive the analytic expression for the collision-induced amplitude shift in a fast two-pulse interaction. The analytic calculations are based on a generalization of the perturbation technique for calculating the effects of weak perturbations on fast collisions between solitons of the nonlinear Schrödinger equation. The theoretical predictions are confirmed by the numerical simulations with the full propagation model of coupled Schrödinger equations.

Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials

2006 ◽  
Vol 136 (5) ◽  
pp. 889-907 ◽  
Author(s):  
A. Ambrosetti ◽  
D. Ruiz
Keyword(s):  
Perturbation Technique ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Radial Solutions ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Image Position ◽  
Vanishing Potentials

We prove the existence of radial solutions of concentrating on a sphere for potentials which might be zero and might decay to zero at infinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov–Schmidt reduction.

scholarly journals Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

Symmetry ◽  
2017 ◽  
Vol 9 (8) ◽  
pp. 165 ◽  
Author(s):  
Miloslav Znojil ◽  
František Růžička ◽  
Konstantin Zloshchastiev
Keyword(s):  
Pt Symmetry ◽  
Schrodinger Equations ◽  
Nonlinear Coupling ◽  
Schrödinger Equations

scholarly journals The n -component nonlinear Schrödinger equations: dark–bright mixed N - and high-order solitons and breathers, and dynamics

2018 ◽  
Vol 474 (2215) ◽  
pp. 20170688 ◽  
Author(s):  
Guoqiang Zhang ◽  
Zhenya Yan
Keyword(s):  
Numerical Simulations ◽  
Explicit Formula ◽  
Soliton Solutions ◽  
Transformation Method ◽  
High Order ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

The general n -component nonlinear Schrödinger equations are systematically investigated with the aid of the Darboux transformation method and its extension. Firstly, we explore the condition of the existence for dark–bright mixed soliton solutions and derive an explicit formula of dark–bright mixed multi-soliton solutions in terms of the determinant. Secondly, we present the formula of dark–bright mixed high-order semi-rational solitons, and predict their general N th-order wave structures. Thirdly, we investigate the wing-shaped structures of breather. Finally, we perform the numerical simulations for some representative solitons to study their dynamical behaviours.

scholarly journals Energy-preserving methods for nonlinear Schrödinger equations

2020 ◽  
Author(s):  
Christophe Besse ◽  
Stéphane Descombes ◽  
Guillaume Dujardin ◽  
Ingrid Lacroix-Violet
Keyword(s):  
Numerical Simulations ◽  
Relaxation Method ◽  
Discrete Analogue ◽  
Nonlinear Schrödinger Equations ◽  
Physical Models ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Fully Implicit ◽  
Nonlinear Schrodinger Equations

Abstract This paper is concerned with the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analogue of it. The Crank–Nicolson method is a well-known method of order $2$ but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in Besse (1998, Analyse numérique des systèmes de Davey-Stewartson. Ph.D. Thesis, Université Bordeaux) for the cubic nonlinear Schrödinger equation. This method is also an energy-preserving method and numerical simulations have shown that its order is $2$. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows one to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.

scholarly journals Non-relativistic and relativistic scattering by short-range potentials

2011 ◽  
Vol 369 (1939) ◽  
pp. 1228-1244 ◽  
Author(s):  
H. Arnbak ◽  
P. L. Christiansen ◽  
Yu. B. Gaididei
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Short Range ◽  
Perturbation Technique ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Relativistic Limit ◽  
Scattering Coefficients ◽  
Relativistic Scattering

Relativistic and non-relativistic scattering by short-range potentials is investigated for selected problems. Scattering by the δ ′ potential in the Schrödinger equation and δ potentials in the Dirac equation must be solved by regularization, efficiently carried out by a perturbation technique involving a stretched variable. Asymmetric regularizations yield non-unique scattering coefficients. Resonant penetration through the potentials is found. Approximative Schrödinger equations in the non-relativistic limit are discussed in detail.

scholarly journals Collision-induced amplitude dynamics of fast 2D solitons in the presence of the generic nonlinear loss

2021 ◽  
Author(s):  
Toan Huynh ◽  
Nguyen Minh Quan
Keyword(s):  
Nonlinear Schrödinger Equations ◽  
Theoretical Expression ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Loss ◽  
Nonlinear Schrodinger Equations ◽  
Coupled Nonlinear Schrodinger Equations ◽  
Fast Collision

Abstract We study the amplitude dynamics of two-dimensional (2D) solitons in a fast collision described by the coupled nonlinear Schrödinger equations with a saturable nonlinearity and weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two one-dimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude dynamics in a fast collision of two 2D solitons. Our perturbative approach is based on two major steps. The first step is the standard adiabatic perturbation for the calculations on the energy balance of perturbed solitons and the second step, which is the crucial one, is for the analysis of the collision-induced change in the envelope of the perturbed 2D soliton. Furthermore, we also present the dependence of the collision-induced amplitude shift on the angle of the two 2D colliding-solitons. In addition, we show that the current perturbative technique can be simply applied to study the collision-induced amplitude shift in a fast collision of two perturbed 1D solitons. Our analytic calculations are confirmed by numerical simulations with the corresponding coupled nonlinear Schrödinger equations in the presence of the cubic loss and in the presence of the quintic loss.

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