In this paper we use a similarity transformation connecting some families of Nonlinear Schrödinger equations with time-varying coefficients with the autonomous cubic nonlinear Schrödinger equation. This transformation allows one to apply all known results for that equation to the original non-autonomous case with the additional dynamics introduced by the transformation itself. In particular, us...

This thesis focuses on the development and the implementation of domain decomposition methods for the linear and non-linear, one dimensional and two dimensional Schrödinger equations. In the first part, we focus on the Schwarz waveform relaxation method (SWR) for the one dimensional Schrödinger equation. In the case the potential is linear and time-independent, we propose a new algorithm that i...

We solve the (2+1)-dimensional Davey-Stewartson (DS) equation, a multidimensional analog of the nonlinear Schrödinger equation. A rather general solution for the variables separation with two arbitrary functions is first obtained by applying a special Bäcklund transformation and introducing the seed solutions. And then some new special types of two-dimensional coherent structures are obtained. ...

We study a damped stochastic non-linear Schrödinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markovian transition semi-group toward a unique invariant probability measure. This kind of method was originally developped to prove exponential mixing for strongly dissipative equations such as the Nav...

D−dimensional central and complex potentials of a Coulomb plus quartic-polynomial form are considered in a PT −symmetrized radial Schrödinger equation. Arbitrarily large finite multiplets of bound states are shown obtainable in an elementary form. Relations between their energies and couplings are determined by a finite-dimensional secular equation. The Bender’s and Boettcher’s one-dimensional ...

Given a particular solution of a one-dimensional stationary Schrödinger equation this equation of second order can be reduced to a first order linear ordinary differential equation. This is done with the aid of an auxiliary Riccati differential equation. In the present work we show that the same fact is true in a multidimensional situation also. For simplicity we consider the case of two or thr...

We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrödinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker-Planck type equation with the wave vector performing a Brownian motion on the (d − 1)-dimensional sphere of constant energy...

We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schrödinger equation in a random δ-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and usi...

Essentially, the Darboux proposition is based on the covariance properties of ordinary and partial differential equations with respect to a gauge transformation in the special case of second order differential equations of the SturmLiouville type. In this work, the one-dimensional Schrödinger equation with a position-dependent mass (SEPDM) is transformed into a Schrödinger-like equation with a ...

Using three different representations of the bicomplex numbers T ∼= ClC(1, 0) ∼= ClC(0, 1), which is a commutative ring with zero divisors defined by T = {w0 + w1i1 + w2i2 + w3j | w0, w1, w2, w3 ∈ R} where i1 = −1, i 2 2 = −1, j 2 = 1 and i1i2 = j = i2i1, we construct three classes of bicomplex pseudoanalytic functions. In particular, we obtain some specific systems of Vekua equations of two co...