I analyze the one-dimensional, cubic Schrödinger equation, with nonlinearity constructed from the current density, rather than, as is usual, from the charge density. A soliton solution is found, where the soliton moves only in one direction. Relation to higher-dimensional Chern–Simons theory is indicated. The theory is quantized and results for the two-body quantum problem agree at weak couplin...

with the Riesz space-fractional derivative of order 0 < α ≤ 2 in the presence of the linear potential V(x) = βx. The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of α = 1, which are useable f...

It is shown how one can get numerical prediction of quantum mechanical particle behaviour without using the Schrödinger equation. The main steps of this development are the non-differentiability hypothesis, the equations of motion entailed by this hypothesis, and the numerical formulation of a simple one-dimensional problem: the particle in a box.

We study four different approximations for finding the profile of discrete solitons in the one-dimensional Discrete Nonlinear Schrödinger (DNLS) Equation. Three of them are discrete approximations (namely, a variational approach, an approximation to homoclinic orbits and a Green-function approach), and the other one is a quasi-continuum approximation. All the results are compared with numerical...

One-dimensional scattering problem admitting a complex, PT −symmetric shortrange potential V (x) is considered. Using a Runge-Kutta-discretized version of Schrödinger equation we derive the formulae for the reflection and transmission coefficients and emphasize that the only innovation emerges in fact via a complexification of one of the potential-characterizing parameters. PACS 03.65.Ge

For the stationary one-dimensional nonlinear Schrödinger equation (or Gross-Pitaevskii equation) nonlinear resonant transmission through a finite number of equidistant identical barriers is studied using a (semi–) analytical approach. In addition to the occurrence of bistable transmission peaks known from nonlinear resonant transmission through a single quantum well (respectively a double barri...

This article considers the linear 1-d Schrödinger equation in (0, π) perturbed by a vanishing viscosity term depending on a small parameter ε > 0. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls vε as ε goes to zero. It is shown that, for any time T sufficiently large but independent of ε and for each initial datum in H−1(0, ...

We consider time-dependent Schrödinger equations in one dimension with doublewell potential and an external nonlinear perturbation. If the initial state belongs to the eigenspace spanned by the eigenvectors associated to the two lowest eigenvalues, then, in the semiclassical limit, we show that the reduction of the time-dependent equation to a 2-mode equation gives the dominant term of the solu...

Effective mass Schrödinger equation is solved exactly for a given potential. NikiforovUvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. On...

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...