In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut − uxx + |u|2pu= 0, p ∈N, with periodic boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to 2-dimensional invariant tori of an associated infinite-dimensional dynamical system. The proof is based on infinite-dimensional KAM theory, partial no...

This paper deals with one-dimensional (1D) nonlinear Schrödinger equation with a multiplicative potential, subject to Dirichlet boundary conditions. It is proved that for each prescribed integer b > 1, the equation admits smallamplitude quasi-periodic solutions, whose b-dimensional frequencies are small dilation of a given Diophantine vector. The proof is based on a modified infinitedimensional...

We prove the persistence of finite dimensional invariant tori associated with the defocusing nonlinear Schrödinger equation under small Hamiltonian perturbations. The invariant tori are not necessarily small.

We constract an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator.

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...

in this paper, the numerical solution methods of one- particale, one – dimensional time- independentschrodinger equation are presented that allows one to obtain accurate bound state eigen values andeigen functions for an arbitrary potential energy function v(x). these methods included the fem(finite element method), cooly, numerov and others. here we considered the numerov method inmore details...

In this paper, we establish bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds with boundary. Using these estimates, we can infer the local well-posedness of cubic nonlinear Schrödinger equation in H for every s > 2 3 on such manifolds.

A model describing wave propagation in optically modulated waveguide arrays is proposed. In the weakly guided regime, a two-dimensional semidiscrete nonlinear Schrödinger equation with the addition of a bulk diffraction term and an external "optical trap" is derived from first principles, i.e., Maxwell equations. When the nonlinearity is of the defocusing type, a family of unstaggered localized...

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...

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. ...