In this paper, we develop and analyze a new multiscale discontinuous Galerkin (DG) method for one-dimensional stationary Schrödinger equations with open boundary conditions which have highly oscillating solutions. Our method uses a smaller finite element space than the WKB local DG method proposed in Wang and Shu (J Comput Phys 218:295– 323, 2006) while achieving the same order of accuracy with...

In this paper, we construct time quasi-periodic solutions for the nonlinear lattice Schrödinger equation iq̇n + (qn+1 + qn−1) + tanπ(nα̃+ x)qn + |qn|qn = 0, n ∈ Z, where α̃ satisfies a certain Diophantine condition and x ∈ R/Z. We prove that for sufficiently small, the equation admits a family of small-amplitude time quasi-periodic solutions for “most” of x belonging to R/Z.

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...

In this work, we explore a generalization of the Dirac and Klein–Gordon (KG) oscillators, provided with deformed linear momentum inspired in nonextensive statistics, that gives place to Morse potential relativistic contexts by first principles. (1 + 1)-dimensional case, oscillators are mapped into quantum potential. Using Pekeris approximation, (3 study thermodynamics S-waves states (l = 0) H 2...

A canonical Hamiltonian formulation of the nonlinear Schrödinger equation has been derived in this paper. This formulation governs the dynamics of pulse propagation in a one-dimensional, periodic Kerr medium when the frequency content of the pulse is sufficiently narrow relative to a carrier frequency, and sufficiently far removed from a photonic band gap of the medium. Our Hamiltonian is numer...

The dynamics of models described by a one-dimensional discrete nonlinear Schrödinger equation is studied. The nonlinearity in these models appears due to the coupling of the electronic motion to optical oscillators which are treated in adiabatic approximation. First, various sizes of nonlinear cluster embedded in an infinite linear chain are considered. The initial excitation is applied either ...

The nonlinear Schrödinger equation is of tremendous interest in both theory and applications. Various regimes of pulse propagation in optical fibers are modeled by some form of the nonlinear Schrödinger equation. In this paper we introduce sequential and parallel numerical methods for numerical simulations of the 1+ 2 dimensional nonlinear Schrödinger type equations. The parallel methods are im...