A conservative numerical method for the fractional nonlinear Schrodinger equation in two dimensions

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

Existence of positive solutions with peaks on a Clifford torus for a fractional nonlinear Schrodinger equation

We consider the fractional nonlinear Schr?dinger equation in this paper. Applying the finite reduction method, we prove that the equation has positive solutions with peaks on a Clifford torus under some suitable conditions.

A LINEARLY-IMPLICIT ENERGY-PRESERVING ALGORITHM FOR THE TWO-DIMENSIONAL SPACE-FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION BASED ON THE SAV APPROACH

The main objective of this paper is to present an efficient structure-preserving scheme,which is based on the idea of the scalar auxiliary variable approach,for solving the twodimensional space-fractional nonlinear Schrodinger equation.First,we reformulate the equation as an canonical Hamiltonian system,and obtain a new equivalent system via introducing a scalar variable.Then,we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction.After that,applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version.As expected,the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step.Finally,numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrodinger equation

The fractional quadric-cubic coupled nonlinear Schrodinger equation is concerned,and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method.The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated.Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index,respectively.The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied.Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.

Assorted soliton structures of solutions for fractional nonlinear Schrodinger types evolution equations

Fractional order nonlinear evolution equations have emerged in recent times as being very important model for depicting the interior behavior of nonlinear phenomena that exist in the real world.In particular,Schrodinger-type fractional nonlinear evolution equations constitute an aspect of the field of quantum mechanics.In this study,the(2+1)-dimensional time-fractional nonlinear Schrodinger equation and(1+1)-dimensional time-space fractional nonlinear Schrodinger equation are revealed as having different and novel wave structures.This is shown by constructing appropriate analytic wave solutions.A success-ful implementation of the advised rational(1/φ'(ξ))-expansion method generates new outcomes of the considered equations,by comparing them with those already noted in the literature.On the basis of the conformable fractional derivative,a composite wave variable conversion has been used to adapt the suggested equations into the differential equations with a single independent variable before applying the scheme.Finally,the well-furnished outcomes are plotted in different 3D and 2D profiles for the purpose of illustrating various physical characteristics of wave structures.The employed technique is competent,productive and concise enough,making it feasible for future studies.