Related papers: Solving the linear semiclassical Schr\"odinger equ…
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential…
The integrating factor technique is widely used to solve numerically (in particular) the Schr\"odinger equation in the context of spectral methods. Here, we present an improvement of this method exploiting the freedom provided by the gauge…
For the first time, the general nonlinear Schr\"odinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class…
A method is presented to construct exactly solvable nonlinear extensions of the Schr\"odinger equation. The method explores a correspondence which can be established under certain conditions between exactly solvable ordinary Schr\"odinger…
Here we present exact, stationary, parametric solutions to the Schr\"odinger--Poisson system. We confront two images: on one hand, we draw on the homotopy analysis method which leads us to a nonlinear integral scheme. Indeed, this approach…
This paper is concerned with an alternative analytical solution of time-fractional nonlinear Schrodinger equation and nonlinear coupled Schrodinger equation obtained by employing fractional reduced differential transform method. The…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
In this paper, we integrate neural networks and Gaussian wave packets to numerically solve the Schr\"odinger equation with a smooth potential near the semi-classical limit. Our focus is not only on accurately obtaining solutions when the…
We introduce a new family of trial wave-functions based on deep neural networks to solve the many-electron Schr\"odinger equation. The Pauli exclusion principle is dealt with explicitly to ensure that the trial wave-functions are physical.…
An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schr\"odinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we…
Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new…
Since the kinetic and the potential energy term of the real time nonlinear Schr\"odinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence…
We consider the cubic-quintic nonlinear Schr\"odinger equation: \[ i\partial_t u = -\Delta u - |u|^2u + |u|^4u. \] In the first part of the paper, we analyze the one-parameter family of ground-state solitons associated to this equation with…
We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schr\"odinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique…
We establish uniform bounds for the solutions $e^{it\Delta}u$ of the Schr\"{o}dinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak-* limits of sequences of…
We consider the spectral problem for the two-dimensional Schr\"odinger operator for a charged particle in strong uniform magnetic and periodic electric fields. The related classical problem is analyzed first by means of the…
We report on a number of careful numerical experiments motivated by the semiclassical (zero-dispersion, \epsilon\downarrow 0) limit of the focusing nonlinear Schr\"odinger equation. Our experiments are designed to study the evolution of a…
The three-dimensional Schredinger's equation is analyzed with the help of the correspondence principle between classical and quantum-mechanical quantities. Separation is performed after reduction of the original equation to the form of the…
In [8], some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, kinetic equations, and Schr{\"o}dinger type equations with a rotation term. In this work, these exact…
A global solution of the Schr\"odinger equation for explicitly time-dependent Hamiltonians is derived by integrating the non-linear differential equation associated with the time-dependent wave operator. A fast iterative solution method is…