1,787 research outputs found
Growth of Sobolev norms for the quintic NLS on
We study the quintic Non Linear Schr\"odinger equation on a two dimensional
torus and exhibit orbits whose Sobolev norms grow with time. The main point is
to reduce to a sufficiently simple toy model, similar in many ways to the one
used in the case of the cubic NLS. This requires an accurate combinatorial
analysis.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with
arXiv:0808.1742 by other author
Newtonian Self-Gravitation in the Neutral Meson System
We derive the effect of the Schr\"odinger--Newton equation, which can be
considered as a non-relativistic limit of classical gravity, for a composite
quantum system in the regime of high energies. Such meson-antimeson systems
exhibit very unique properties, e.g. distinct masses due to strong and
electroweak interactions. We find conceptually different physical scenarios due
to lacking of a clear physical guiding principle which mass is the relevant one
and due to the fact that it is not clear how the flavor wave-function relates
to the spatial wave-function. There seems to be no principal contradiction.
However, a nonlinear extension of the Schr\"odinger equation in this manner
strongly depends on the relation between the flavor wave-function and spatial
wave-function and its particular shape. In opposition to the Continuous
Spontaneous Localization collapse models we find a change in the oscillating
behavior and not in the damping of the flavor oscillation.Comment: 10 pages, no figure
Multi-Symplectic Simulation on Soliton-Collision for Nonlinear Perturbed Schrödinger Equation
Seeking solitary wave solutions and revealing their interactional characteristics for nonlinear evolution equations help us lot to comprehend the motion laws of the microparticles. As a local nonlinear dynamic behavior, the soliton-collision is difficult to be reproduced numerically. In this paper, the soliton-collision process in the nonlinear perturbed Schrödinger equation is simulated employing the multi-symplectic method. The multi-symplectic formulations are derived including the multi-symplectic form and three local conservation laws of the nonlinear perturbed Schrödinger equation. Employing the implicit midpoint rule, we construct a multi-symplectic scheme, which is equivalent to the Preissmann box scheme, for the nonlinear perturbed Schrödinger equation. The elegant structure-preserving properties of the multi-symplectic scheme are illustrated by the tiny maximum absolute residual of the discrete multi-symplectic structure at each time step in the numerical simulations. The effects of the perturbation strength on the soliton-collision in the nonlinear perturbed Schrödinger equation are reported in the numerical results in detail
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