Superconvergence analysis of Crank‐Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation

The purpose of this article is to apply nonconforming finite element(FE) to solve a generalized nonlinear Schrödinger equation. First, a new important property of nonconforming FE (see ( 2.3 ) of Lemma 2 below) is proved by use of BHX lemma and the integral identities techniques. Second, a linearized Crank‐Nicolson fully discrete scheme is constructed and the superclose error estimate of order for original variable u in broken H¹‐norm is also derived by using the properties of element and the splitting argument for nonlinear terms, while previous works always only obtain convergent error estimates with this element. Furthermore, the global superconvergence is arrived at by the interpolated postprocessing technique. Finally, two numerical experiments are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.     

Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.     

Derivation of Korteweg‐de Vries flow equations from fourth order nonlinear Schrödinger equation

We perform a multiple scale analysis on the fourth order nonlinear Schrödinger equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg‐de Vries flow equations in the bi‐Hamiltonian form with the corresponding Hamiltonian functions. Copyright © 2013 John Wiley & Sons, Ltd.     

Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

Local exact controllability of the one‐dimensional NLS (subject to zero‐boundary conditions) with distributed control is shown to hold in a H¹‐neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Exact soliton solution for the fourth‐order nonlinear Schrödinger equation with generalized cubic‐quintic nonlinearity

In this paper, we investigate the fourth‐order nonlinear Schrödinger equation with parameterized nonlinearity that is generalized from regular cubic‐quintic formulation in optics and ultracold physics scenario. We find the exact solution of the fourth‐order generalized cubic‐quintic nonlinear Schrödinger equation through modified F‐expansion method, identifying the particular bright soliton behavior under certain external experimental setting, with the system's particular nonlinear features demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.     

Exact solutions of nonlinear Schrödinger equation by using symbolic computation

The (G′/G,1/G)‐expansion method and (1/G′)‐expansion method are interesting approaches to find new and more general exact solutions to the nonlinear evolution equations. In this paper, these methods are applied to construct new exact travelling wave solutions of nonlinear Schrödinger equation. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. It is shown that the proposed methods provide a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Copyright © 2015 John Wiley & Sons, Ltd.     

Efficient energy‐preserving scheme of the three‐coupled nonlinear Schrödinger equation

An energy‐preserving scheme is proposed for the three‐coupled nonlinear Schrödinger (T‐CNLS) equation. The T‐CNLS equation is rewritten into the classical Hamiltonian form. Then the spatial variable is discretized by using high‐order compact method to convert it into a finite‐dimensional Hamiltonian system. Next, a second‐order averaged vector field (AVF) method is employed in time which results in an energy‐preserving scheme. Some theoretical results such as convergence are investigated. In addition, it provides some numerical examples to illustrate the robustness and reliability of the theoretical results. It also explores the role of the parameters in the model and initial condition on the wave propagation.     

Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential

This paper is concerned with the nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose–Einstein condensate under the magnetic trap. By combining the best constant of Gagliardo–Nirenberg's inequality with the characteristic of this equation, we derive out a global existence condition for the supercritical equation which coincides with the critical case.     

On a discrete‐time collocation method for the nonlinear Schrödinger equation with wave operator

We consider a discrete‐time orthogonal spline collocation scheme for solving Schrödinger equation with wave operator. The scheme is proposed recently by Wang et al. (J Comput Appl Math 235 (2011), 1993–2005) and is showed to have high‐order convergence rate when a parameter θ in the scheme is not less than $\frac{1}{4}$. In this article, we show that the result can be extended to include $\theta\in(0,\frac{1}{4})$ under an assumption. Numerical example is given to justify the theoretical result. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Finite‐dimensional global attractor for a semi‐discrete fractional nonlinear Schrödinger equation

We consider a semi‐discrete in time Crank–Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation u _t ?i (?Δ)^α u +i |u |²u +γ u =f for considered in the the whole space . We prove that such semi‐discrete equation provides a discrete infinite‐dimensional dynamical system in that possesses a global attractor in . We show also that if the external force is in a suitable weighted Lebesgue space, then this global attractor has a finite fractal dimension. Copyright © 2017 John Wiley & Sons, Ltd.     

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

In this work, we discuss the coupled modified nonlinear Schrödinger (CMNLS) equation, which describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. By use of the Fokas approach, the initial‐boundary value problem for the CMNLS equation related to a 3×3 matrix Lax pair on the half‐line is to be analyzed. Assuming that the solution {u(x,t),v(x,t)} of CMNLS equation exists, we will prove that it can be expressed in terms of the unique solution of a 3×3 matrix Riemann‐Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions s(λ) and S(λ) are not independent of each other but meet a global relationship.     

An inverse scattering problem for the Schrödinger equation in a semiclassical process

We study a direct and an inverse scattering problem for a pair of Hamiltonians (H(h),H₀(h)) on , where H₀(h)=−h²Δ and H(h)=H₀(h)+V, V is a short-range potential and h is the semiclassical parameter. First, we show that if two potentials are equal in the classical allowed region for a fixed non-trapping energy, the associated scattering matrices coincide up to O(h^∞) in . Then, for potentials with a regular behaviour at infinity, we study the inverse scattering problem. We show that in dimension n3, the knowledge of the scattering operators S(h), , up to O(h^∞) in , and which are localized near a fixed energy λ>0, determine the potential V at infinity.     

Infinitely many solutions for a resonant sublinear Schrödinger equation

In this paper, we consider a nonlinear sublinear Schrödinger equation at resonance in . By using bounded domain approximation technique, we prove that the problem has infinitely many solutions. Copyright © 2013 John Wiley & Sons, Ltd.