Solitary wave solutions for nonlinear fractional Schrödinger equation in Gaussian nonlocal media

This article is devoted to the study of nonlinear fractional Schrödinger equation with a Gaussian nonlocal response. We firstly prove the existence of solitary wave solutions by using the variational method and Mountain Pass Theorem. Numerical simulations are presented to verify the findings of the existence theorem. And we also investigate the impacts of Gaussian nonlocal response and fractional-order derivatives on the solitary waves, which enable us to perform control experiments for the development of rogue waves in quantum mechanics and optics.     

Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

Abstract

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field.     

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrödinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrödinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.     

On fractional Schrödinger equation in α-dimensional fractional space

The Schrödinger equation is solved in $\alpha$-dimensional fractional space with a Coulomb potential proportional to $\frac{1}{r^{\beta ?2}}$, $2\le \beta \le 4$. The wave functions are studied in terms of spatial dimensionality $\alpha$ and $\beta$ and the results for $\beta =3$ are compared with those obtained in the literature.     

Variable sinh-Gaussian solitons in nonlocal nonlinear Schrödinger equation

Based on the nonlocal nonlinear Schrödinger equation that governs phenomenologically the propagation of laser beams in nonlocal nonlinear media, we theoretically investigate the propagation of sinh-Gaussian beams (ShGBs). Mathematical expressions are derived to describe the beam propagation, the intensity distribution, the beam width, and the beam curvature radius of ShGBs. It is found that the propagation behavior of ShGBs is variable and closely related to the parameter of sinh function (PShF). If the PShF is small, the transverse pattern of ShGBs keeps invariant during propagation for a proper input power, which can be regarded as solitons. If the PShF is large, it varies periodically, which is similar to the evolution of temporal higher-order solitons in nonlinear optical fiber. Numerical simulations are carried out to illustrate the typical propagation characteristics.     

Conformal structure-preserving schemes for damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise

Since many physical phenomena are often influenced by dispersive medium, energy compensation and random perturbation, exploring the dynamic behaviors of the damped-driven stochastic system has becoming a hot topic in mathematical physics in recent years. In this paper, inspired by the stochastic conformal structure, we investigate the geometric numerical integrators for the damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise. To preserve the conformal structures of the system, by using symplectic Euler method, implicit midpoint method and Fourier pseudospectral method, we propose three kinds of stochastic conformal schemes satisfying corresponding discrete stochastic multiconformal-symplectic conservation laws and discrete global/local charge conservation laws. Numerical experiments illustrate the structure-preserving properties of the proposed schemes, as well as favorable results over traditional nonconformal schemes, which are consistent with our theoretical analysis.     

Martingale solutions of stochastic fractional nonlinear Schrödinger equation on a bounded interval*

This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrödinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrödinger equation on a bounded interval.     

Global existence for the nonlinear fractional Schrödinger equation with fractional dissipation

We consider the initial value problem for the fractional nonlinear Schrödinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation.     

A semi-discrete scheme for the stochastic nonlinear Schrödinger equation

Summary. We study the convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrödinger equation. The nonlinear term is a power law and the noise is multiplicative with a Stratonovich product. Our scheme is implicit in the deterministic part of the equation as is usual for conservative equations. We also use an implicit discretization of the noise which is better suited to Stratonovich products. We consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate. However, in the semi discrete case, no Ito formula is available and we have to use a discrete form of this tool. Also, in the course of the proof we need to introduce a cut-off of the diffusion coefficient, which allows to treat the nonlinearity. Then, we prove convergence by a compactness argument. Due to the presence of noise and to the implicit discretization of the noise, this is rather complicated and technical. We finally obtain convergence of the discrete solutions in various topologies. Mathematics Subject Classification (2000):35Q55, 60H15, 65M06, 65M12     

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger equation

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one-soliton solutions and two-soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.     

Inverse scattering transform for a nonlocal derivative nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We give a detailed discussion of a nonlocal derivative nonlinear Schrödinger (NL-DNLS) equation with zero boundary conditions at infinity in terms of the...     

Time dependent nonlinear Schrödinger equations

We prove the existence of global classical solutions of the Cauchy-problem for nonlinear Schrödinger equations u+iAu+f(|u|²)u or u+Au+if(|u|²)u respectively. We need various growth conditions on the nonlinearity f and some restrictions on the admissible space dimension n.     

Pure soliton solutions of the nonlocal Kundu–nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We systematically present an inverse scattering transform for a nonlocal reverse-space higher-order nonlinear Schrödinger equation with nonzero boundary...     

Interaction theory of mirror-symmetry soliton pairs in nonlocal nonlinear Schrödinger equation

In this work, we theoretically investigate the evolution of the soliton pairs in strongly nonlocal nonlinear media, which is modeled by the nonlocal nonlinear Schrödinger equation. Taking two pairs of solitons as an example, which initial incident directions have a mirror symmetry, a set of mathematical expressions are derived to describe the soliton pairs’ propagation, the soliton spacing, the area of the optical field. The results demonstrate that the motion state of the soliton pairs is mirror-symmetry. Numerical simulations are carried out to illustrate the quintessential propagation properties.     

Exotic localized waves in the shifted nonlocal multicomponent nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We theoretically calculate general higher-order soliton solutions of the space-shifted parity–time-symmetric nonlocal multicomponent nonlinear...     

Multiplicity results of nonlinear fractional magnetic Schrödinger equation with steep potential

In this article, we prove the existence and multiplicity of non-trivial solutions for an indefinite fractional elliptic equation with magnetic field and concave–convex nonlinearities. Our multiplicity results are based on studying the decomposition of the Nehari manifold.