On the collapse of trial solutions for a damped-driven nonlinear Schrödinger equation

@article{Assing2014OnTC,
  title={On the collapse of trial solutions for a damped-driven nonlinear Schr{\"o}dinger equation},
  author={Sigurd Assing and Astrid Hilbert},
  journal={Nonlinearity},
  year={2014},
  volume={31},
  pages={4955 - 4978}
}
We consider the focusing 2D nonlinear Schrödinger equation, perturbed by a damping term, and driven by multiplicative noise. We show that a physically motivated trial solution does not collapse for any admissible initial condition although the exponent of the nonlinearity is critical. Our method is based on the construction of a global solution to a singular stochastic Hamiltonian system used to connect trial solution and Schrödinger equation. 

References

SHOWING 1-10 OF 24 REFERENCES

Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and white noise.

It is found that a sufficiently large noise variance may cause an initially localized distribution to spread instead of contracting, and that the critical variance necessary to cause dispersion will for small damping be the same as for the undamped system.

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

We study Langevin dynamics of N particles on ℝd interacting through a singular repulsive potential, e.g., the well‐known Lennard‐Jones type, and show that the system converges to the unique invariant

Geometric Ergodicity of Two--dimensional Hamiltonian systems with a Lennard--Jones--like Repulsive Potential

In this paper we establish the ergodicity of Langevin dynamics for simple two-particle system involving a Lennard-Jones type potential. To the best of our knowledge, this is the first such result for

ASYMPTOTIC AND LIMITING PROFILES OF BLOWUP SOLUTIONS OF THE NONLINEAR SCHRODINGER EQUATION WITH CRITICAL POWER

This paper is a sequel to previous ones 38, 39, 41. We continue the study of the blowup problem for the nonlinear Schrodinger equation with critical power nonlinearity (NSC). We introduce a new idea

Nonlinear Markov Processes and Kinetic Equations

A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from

A Hypocoercivity Related Ergodicity Method for Singularly Distorted Non-Symmetric Diffusions

In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial

On unique ergodicity for degenerate diffusions

We investigate the invariant probabilities of a possible degenerate diffusion process on a manifold. Using the support theorems of stroock, Varadhan and Kunita, the possible candidates for supports

Nonlinear Schrödinger equations and sharp interpolation estimates

AbstractA sharp sufficient condition for global existence is obtained for the nonlinear Schrödinger equation $$\begin{array}{*{20}c} {(NLS)} & {2i\phi _t + \Delta \phi + \left| \phi \right|^{2\sigma