Stability of Stationary Solutions of the Schrödinger-langevin Equation

Existence and Stability of Solitons for the Nonlinear Schrödinger Equation on Hyperbolic Space

We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.

Lévy flights and Lévy-Schrödinger semigroups

We analyze two different confining mechanisms for Lévy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Lévy-Schrödinger semigroups which induce so-called topological Lévy processes (Lévy flights with locally modified jump rates in the master equation). Given a stationary probability func...

Stability of exact solutions of the defocusing nonlinear Schrödinger equation with periodic potential in two dimensions

The cubic nonlinear Schrödinger equation with repulsive nonlinearity and elliptic function potential in two-dimensions models a repulsive dilute gas Bose–Einstein condensate in a lattice potential. A family of exact stationary solutions is presented and its stability is examined using analytical and numerical methods. All stable trivial-phase solutions are off-set from the zero level. Our resul...

Explicit multiple singular periodic solutions and singular soliton solutions to KdV equation

 Based on some stationary periodic solutions and stationary soliton solutions, one studies the general solution for the relative lax system, and a number of exact solutions to the Korteweg-de Vries (KdV) equation are first constructed by the known Darboux transformation, these solutions include double and triple singular periodic solutions as well as singular soliton solutions whose amplitude d...

A Map Approach to Stationary Solutions of the Discrete Nonlinear Schrödinger Equation