Asymptotics for large time of solutions to the nonlinear Schrodinger and Hartree equations

1998 ◽  
Vol 120 (2) ◽  
pp. 369-389 ◽  
Author(s):  
Nakao Hayashi ◽  
P. I. (Pavel Ivanovich) Naumkin
Keyword(s):  
Large Time ◽  
Nonlinear Schrödinger ◽  
Asymptotics For Large Time

scholarly journals Large Time-Stepping Spectral Methods for the Semiclassical Limit of the Defocusing Nonlinear Schrödinger Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-27
Author(s):  
Zongqi Liang
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Spectral Methods ◽  
Schrodinger Equation ◽  
Large Time ◽  
Semiclassical Limit ◽  
Nonlinear Schrodinger Equation ◽  
Time Step ◽  
Nonlinear Schrödinger ◽  
Time Stepping

We analyze a class of large time-stepping Fourier spectral methods for the semiclassical limit of the defocusing Nonlinear Schrödinger equation and provide highly stable methods which allow much larger time step than for a standard implicit-explicit approach. An extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Meanwhile, the first-order and second-order semi-implicit schemes are constructed and analyzed. Finally the numerical experiments are performed to demonstrate the effectiveness of the large time-stepping approaches.

scholarly journals Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations

2015 ◽  
Vol 145 (6) ◽  
pp. 1251-1282 ◽  
Author(s):  
Stefan Le Coz ◽  
Dong Li ◽  
Tai-Peng Tsai
Keyword(s):  
Solitary Waves ◽  
Fast Moving ◽  
Existence And Uniqueness ◽  
Large Time ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
New Construction

We study infinite soliton trains solutions of nonlinear Schrödinger equations, i.e. solutions behaving as the sum of infinitely many solitary waves at large time. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighbourhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

Scattering of solutions to the fourth-order nonlinear Schrödinger equation

2016 ◽  
Vol 18 (03) ◽  
pp. 1550035 ◽  
Author(s):  
Nakao Hayashi ◽  
Jesus A. Mendez-Navarro ◽  
Pavel I. Naumkin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Large Time ◽  
Nonlinear Schrodinger Equation ◽  
Asymptotics Of Solutions ◽  
Nonlinear Schrödinger ◽  
Time Asymptotics ◽  
The Cauchy Problem

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation [Formula: see text] where [Formula: see text] [Formula: see text] We introduce the factorization for the free evolution group to prove the large time asymptotics of solutions.

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