scholarly journals Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data

2018 ◽  
Vol 35 (1) ◽  
pp. 217-265 ◽  
Author(s):  
Jiaqi Liu ◽  
Peter A. Perry ◽  
Catherine Sulem
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Nonlinear Schrödinger ◽  
Long Time ◽  
Derivative Nonlinear Schrödinger Equation

SYMPLECTIC NUMERICAL METHODS IN DYNAMICS OF NONLINEAR WAVES

1999 ◽  
Vol 10 (05) ◽  
pp. 967-980 ◽  
Author(s):  
A. G. SHAGALOV
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Integrable Equation ◽  
Nonlinear Schrodinger Equation ◽  
Symplectic Integrators ◽  
Nonlinear Schrödinger ◽  
Higher Derivative ◽  
Long Time ◽  
Derivative Nonlinear Schrödinger Equation

The symplectic integrator of the Gauss–Legendre type is tested on the nonlinear Schrödinger equation. Preservation of high integrals (up to 10 or more) and quasiperiodic motion have been detected for dynamics on both stable soliton and homoclinic manifolds, which indicate applicability of symplectic integrators for adequate simulation of integrable equation. The tested integrator is applied to the problem of long-time stability of the solitons in higher-derivative nonlinear Schrödinger equation. The slow logarithmic-type depletion of the soliton amplitude with time has been detected.

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