Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis

In this article, we obtain local energy and momentum conservation laws for the Klein‐Gordon‐Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy‐ and momentum‐preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time‐space region. With suitable boundary conditions, the schemes will be charge‐ and energy‐/momentum‐preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order . The theoretical properties are verified by numerical experiments. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1329–1351, 2017     

New analysis and application of fractional order Schrödinger equation using with Atangana–Batogna numerical scheme

In this work, an analytical approximation to the solution of Schrodinger equation has been provided. The fractional derivative used in this equation is the Caputo derivative. The existence and uniqueness conditions of solutions for the proposed model are derived based on the power law. While solving the fractional order Schrodinger equation, Atangana–Batogna numerical method is presented for fractional order equation. We obtain an efficient recurrence relation for solving these kinds of equations. To illustrate the usefulness of the numerical scheme, the numerical simulations are presented. The results show that the numerical scheme is very effective and simple.     

A conservative compact difference scheme for the coupled Klein–Gordon–Schrödinger equation

In this article, a conservative compact difference scheme is presented for the periodic initial‐value problem of Klein–Gordon–Schrödinger equation. On the basis of some inequalities about norms and the priori estimates, convergence of the difference solution is proved with order O(h⁴ +τ ²) in maximum norm. Numerical experiments demonstrate the accuracy and efficiency of the compact scheme. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A linearized,decoupled, and energy‐preserving compact finite difference scheme for the coupled nonlinear Schrödinger equations

In this article, a decoupled and linearized compact finite difference scheme is proposed for solving the coupled nonlinear Schrödinger equations. The new scheme is proved to preserve the total mass and energy which are defined by using a recursion relationship. Besides the standard energy method, an induction argument together with an H¹ technique are introduced to establish the optimal point‐wise error estimate of the proposed scheme. Without imposing any constraints on the grid ratios, the convergence order of the numerical solution is proved to be of with mesh size h and time step τ. Numerical results are reported to verify the theoretical analysis, and collision of two solitary waves are also simulated. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 840–867, 2017     

The attractor of the dissipative coupled fractional Schrödinger equations

In the present paper, we consider the dissipative coupled fractional Schrödinger equations. The global well‐posedness by the contraction mapping principle is obtained. We study the long time behavior of solutions for the Cauchy problem. We prove the existence of global attractor. Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient numerical approach to solve Schrödinger equations with space fractional derivative

We design and analyze an efficient numerical approach to solve the coupled Schrödinger equations with space‐fractional derivative. The numerical scheme is based on leap‐frog in time direction and Fourier method in spatial direction. The advantage of the numerical scheme is that only a linear equation needs to be solved for each time step size, and we proved that the energy and mass of space‐fractional coupled Schrödinger equations (SFCSEs) are conserved in the case of full‐discrete scheme. Moreover, we also analyze the error estimate of the numerical scheme, and numerical solutions converge with the order in L² norm. Numerical examples are illustrated to verify the theoretical results.     

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.     

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.     

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.     

On the fractional Schrödinger problem with non‐symmetric potential

We study the semilinear equation where 0 < s < 1, , V(x) is a sufficiently smooth non‐symmetric potential with , and ? > 0 is a small number. Letting U be the radial ground state of (?Δ)^sU + U ? U^p=0 in , we build solutions of the form for points ?_j,j = 1,?,m, using a Lyapunov–Schmidt variational reduction. Copyright © 2014 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.     

Fourth‐order compact and energy conservative scheme for solving nonlinear Klein‐Gordon equation

In this article, a fourth‐order compact and conservative scheme is proposed for solving the nonlinear Klein‐Gordon equation. The equation is discretized using the integral method with variational limit in space and the multidimensional extended Runge‐Kutta‐Nyström (ERKN) method in time. The conservation law of the space semidiscrete energy is proved. The proposed scheme is stable in the discrete maximum norm with respect to the initial value. The optimal convergent rate is obtained at the order of in the discrete ‐norm. Numerical results show that the integral method with variational limit gives an efficient fourth‐order compact scheme and has smaller error, higher convergence order and better energy conservation for solving the nonlinear Klein‐Gordon equation compared with other methods under the same condition. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1283–1304, 2017     

A conservative difference scheme for two‐dimensional nonlinear Schrödinger equation with wave operator

A conservative difference scheme is presented for two‐dimensional nonlinear Schrödinger equation with wave operator. The discrete energy method and an useful technique are used to analyze the difference scheme. It is shown, both theoretically and numerically, that the difference solution is conservative, unconditionally stable and convergent with second order in maximum norm. A numerical experiment indicates that the scheme is very effective. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 862–876, 2016     

On concentration of solution to a Schrödinger logarithmic equation with deepening potential well

In this work, we prove the existence of positive solution for the following class of problems where λ>0 and is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C¹ functional with a convex lower semicontinuous functional, we prove that for each large enough λ>0, there exists a positive solution for the problem, and that, as λ→+∞, such solutions converge to a positive solution of the limit problem defined on the domain Ω=int(V^?1({0})).     

Fractional logarithmic Schrödinger equations

By means of nonsmooth critical point theory, we obtain existence of infinitely many weak solutions of the fractional Schrödinger equation with logarithmic nonlinearity. We also investigate the Hölder regularity of the weak solutions. Copyright © 2015 JohnWiley & Sons, Ltd