scholarly journals Long-Time Dynamics of the Perturbed Schrödinger Equation on Negatively Curved Surfaces

2016 ◽  
Vol 17 (8) ◽  
pp. 1955-1999 ◽  
Author(s):  
Gabriel Rivière
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Curved Surfaces ◽  
Time Dynamics ◽  
Negatively Curved ◽  
Long Time ◽  
Long Time Dynamics

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.

scholarly journals PERTURBATION OF THE SEMICLASSICAL SCHRÖDINGER EQUATION ON NEGATIVELY CURVED SURFACES

2015 ◽  
Vol 16 (4) ◽  
pp. 787-835 ◽  
Author(s):  
Suresh Eswarathasan ◽  
Gabriel Rivière
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Cotangent Bundle ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Quantum Evolution ◽  
Curved Surfaces ◽  
Small Perturbations ◽  
Negatively Curved ◽  
Ehrenfest Time

We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit, and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

Long-time dynamics of a class of nonlocal extensible beams with time delay

2021 ◽  
pp. 108128652110194
Author(s):  
Fengjuan Meng ◽  
Cuncai Liu ◽  
Chang Zhang
Keyword(s):  
Time Delay ◽  
Stability Property ◽  
Beam Equation ◽  
Exponential Attractors ◽  
Time Dynamics ◽  
Bounded Smooth Domain ◽  
Finite Dimensional ◽  
Long Time ◽  
Bounded Smooth ◽  
Long Time Dynamics

This work is devoted to the following nonlocal extensible beam equation with time delay: [Formula: see text] on a bounded smooth domain [Formula: see text]. The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.

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