The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line

2019 ◽  
Vol 26 (3) ◽  
pp. 483-508
Author(s):  
Boling Guo ◽  
Nan Liu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hilbert Problem ◽  
Nonlinear Schrodinger Equation ◽  
Half Line ◽  
Time Dynamics ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Long Time ◽  
Riemann Hilbert Problem ◽  
Long Time Dynamics

scholarly journals The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 170 ◽  
Author(s):  
Tongshuai Liu ◽  
Huanhe Dong
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hilbert Problem ◽  
Initial Boundary ◽  
Lax Pair ◽  
Nonlinear Schrodinger Equation ◽  
Half Line ◽  
Prolongation Structure ◽  
Riemann Hilbert Problem

In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential q ( x , t ) can be represented by the solution of this Riemann-Hilbert problem.

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.

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