scholarly journals Long time dynamics for the one dimensional non linear Schrödinger equation

2013 ◽  
Vol 63 (6) ◽  
pp. 2137-2198 ◽  
Author(s):  
Nicolas Burq ◽  
Laurent Thomann ◽  
Nikolay Tzvetkov
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
One Dimensional ◽  
Time Dynamics ◽  
Non Linear ◽  
Long Time ◽  
The One ◽  
Long Time Dynamics

scholarly journals Multiple Soliton Interactions on the Surface of Deep Water

Fluids ◽  
2020 ◽  
Vol 5 (2) ◽  
pp. 65 ◽  
Author(s):  
Dmitry Kachulin ◽  
Alexander Dyachenko ◽  
Sergey Dremov
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Time Dynamics ◽  
Nonlinear Schrödinger ◽  
Long Time ◽  
Long Time Dynamics ◽  
Number Of Particles

The paper presents the long-time dynamics with multiple collisions of breathers in the super compact Zakharov equation for unidirectional deep water waves. Solutions in the form of breathers were found numerically by the Petviashvili method. In the terms of envelope and the assumption of the narrow spectral width the super compact equation turns into the well known exact integrable model—nonlinear Schrödinger equation, and the breather solution in this case turns into envelope soliton. The results of numerical simulations show that two main scenarios of long-time dynamics occur during numerous collisions of breathers. In the first case, one of the breathers regularly takes a number of particles from the other one at each collision and in the second one a structure resembling the bi-soliton solution of nonlinear Schrödinger equation arises during the collision. Despite these scenarios, it is shown that after numerous collisions the only one breather having initially a larger number of particles remains.

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

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