We develop a fourth order numerical method for the computation of multivalued physical observables (density, momentum, etc.) in the semiclassical limit of the one dimensional linear Schrödinger equation in the case of discontinuous potentials. We adopt the level set framework developed in [21] which allows one to compute the multivalued physical observables via solving the classical Liouville e...

In this paper, multi-component generalizations to the Camassa–Holm equation, the modified Camassa–Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex Camassa– Holm equation and the multi-component modified Camassa–Holm equation are provided. It is shown that these equations arise from non-streching invariant c...

We consider the 3–dimensional Schrödinger operator with slowly decaying potential whose radial derivatives are short-range. The long-time asymptotics for solution of the corresponding non-stationary equation is established. In this case, the standard WKB-correction should be replaced by the solution of certain evolution equation. Consider the Schrödinger operator H = −∆+ V, x ∈ R (1) In this pa...

A method is presented to calculate the eigenenergies and eigenfunctions of quantum wires. This is a true three-dimensional method based on a direct implementation of the time-dependent Schrödinger equation. It makes no approximations to the Schrödinger equation other than the finite-difference approximation of the space and time derivatives. The accuracy of our method is tested by comparing it ...

A numerical method for solving the time-independent Schrödinger equation of a particle moving freely in a three-dimensional axisymmetric region is developed. The boundary of the region is defined by an arbitrary analytic function. The method uses a coordinate transformation and an expansion in eigenfunctions. The effectiveness is checked and confirmed by applying the method to a particular exam...

A concept of semiclassically concentrated solutions is formulated for the multidimensional nonlinear Schrödinger equation (NLSE) with an external field. These solutions are considered as multidimensional solitary waves. The center of mass of such a solution is shown to move along with the bicharacteristics of the basic symbol of the corresponding linear Schrödinger equation. The leading term of...

We examine the statistical properties of nonlinear random waves that are ruled by the one-dimensional defocusing and integrable nonlinear Schrödinger equation. Using fast detection techniques in an optical fiber experiment, we observe that the probability density function of light fluctuations is characterized by tails that are lower than those predicted by a Gaussian distribution. Moreover, by...