time-dependent schrödinger equation for a 1-d model of hydrogen molecular ion h2+ in intenselaser field linearly polarized along the molecular axis is solved. ionization rates are calculated for differentinitial states. the evolution of electronic wavefunction at fixed inter-nuclear separations are simulated andanalyzed. the results obtained for the ground state of this 1-d model of h2+ show ap...

In the present work, we improve a numerical method, developed to solve the Gross-Pitaevkii nonlinear Schrödinger equation. A particular scaling is used in the equation, which permits us to evaluate the wave-function normalization after the numerical solution. We have a two-point boundary value problem, where the second point is taken at infinity. The differential equation is solved using the sh...

Concepts of supersymmetry and parasupersymmetry known for the one-dimensional stationary Schrödinger equation are generalized to the time-dependent equation. Our approach is based on differential transformation operators for the non-stationary Schrödinger equation called Darboux transformation operators and on chains of such operators. As an illustration new exactly solvable time-dependent pote...

It is shown how to compute the instability rates for double-periodic solutions cubic NLS (nonlinear Schrödinger) equation by using Lax linear equations. The wave function modulus of periodic both in space and time coordinates; such generalize standing waves which have time-independent space-periodic modulus. Similar other equation, are spectrally unstable this related bands spectrum outside ima...

In this paper an efficient and accurate method for simulating the propagation of a localized solution of the Schrödinger equation near the semiclassical limit is presented. We are interested computing arbitrarily accurate solutions when the non dimensional Plank’s constant, ε, is small, but not negligible. The method is based on a time dependent transformation of the independent variables, clos...

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, ...

of the periodic nonlinear Schrödinger equation, where u(x, t) is a complex valued function in the class of smooth period one functions. In this work, we explain in what sense the generic level set of the constants of motion for the periodic nonlinear Schrödinger equation is an infinite dimensional torus, why the solution of the Hamiltonian equation is almost periodic in time, and describe how n...

The second order (in time) Schrödinger equation is proposed. The additional term (in comparison to Schrödinger equation) describes the interaction of particles with vacuum filled with virtual particle – antiparticle pairs (zitterbewegung).

We consider the semiclassical limit for the nonlinear Schrödinger equation. We introduce a phase/amplitude representation given by a system similar to the hydrodynamical formulation, whose novelty consists of including some asymptotically vanishing viscosity. We prove that the system is always locally well-posed in a class of Sobolev spaces, and globally well-posed for a fixed positive Planck c...