A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schroedinger system

We derive a new method for a coupled nonlinear Schr/Sdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove the proposed method preserves the charge and energy conservation laws exactly. In numerical tests, we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions. Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws. These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.     

A conservative Fourier pseudospectral algorithm for the nonlinear Schrdinger equation

In this paper, we derive a new method for a nonlinear Schrodinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in ｜｜·｜｜2 norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.     

Multisymplectic implicit and explicit methods for Klein-Gordon-Schrdinger equations

We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schro¨dinger equations.We prove that the implicit method satisfies the charge conservation law exactly.Both methods provide accurate solutions in long-time computations and simulate the soliton collision well.The numerical results show the abilities of the two methods in preserving the charge,energy,and momentum conservation laws.     

Multisymplectic implicit and explicit methods for Klein–Gordon–Schrödinger equations

We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schrödinger equations. We prove that the implicit method satisfies the charge conservation law exactly. Both methods provide accurate solutions in long-time computations and simulate the soliton collision well. Numerical results show the abilities of the two methods in preserving charge, energy, and momentum conservation laws.     

A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrodinger Equations with Multiply Components

Considering the coupled nonlinear Schrodinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time computations. Furthermore, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.     

A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrdinger Equations with Multiply Components

Considering the coupled nonlinear Schr¨odinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time computations.Furthermore, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.     

A nonlinear discrete integrable coupling system and its infinite conservation laws

We construct a nonlinear integrable coupling of discrete soliton hierarchy,and establish the infinite conservation laws(CLs) for the nonlinear integrable coupling of the lattice hierarchy.As an explicit application of the method proposed in the paper,the infinite conservation laws of the nonlinear integrable coupling of the Volterra lattice hierarchy are presented.     

A local energy-preserving scheme for Klein Gordon Schrdinger equations

A local energy conservation law is proposed for the Klein–Gordon–Schr ¨odinger equations, which is held in any local time–space region. The local property is independent of the boundary condition and more essential than the global energy conservation law. To develop a numerical method preserving the intrinsic properties as much as possible, we propose a local energy-preserving(LEP) scheme for the equations. The merit of the proposed scheme is that the local energy conservation law can hold exactly in any time–space region. With the periodic boundary conditions, the scheme also possesses the discrete change and global energy conservation laws. A nonlinear analysis shows that the LEP scheme converges to the exact solutions with order O(τ2+ h2). The theoretical properties are verified by numerical experiments.     

A novel hierarchy of differential integral equations and their generalized bi-Hamiltonian structures

With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 x 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.     

Two integrable generalizations of WKI and FL equations:Positive and negative flows,and conservation laws

With the aid of Lenard recursion equations, an integrable hierarchy of nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is proposed, in which the first nontrivial member in the positive flows can be reduced to a new generalization of the Wadati–Konno–Ichikawa(WKI) equation. Further, a new generalization of the Fokas–Lenells(FL) equation is derived from the negative flows. Resorting to these two Lax pairs and Riccati-type equations, the infinite conservation laws of these two corresponding equations are obtained.     

Estimating topological charge of propagating vortex from single-shot non-imaged speckle

Encoding information using the topological charge of vortex beams has been proposed for optical communications. The conservation of the topological charge on propagation and the detection of the topological charge by a receiver are significant in these applications and have been well established in free-space. However, when vortex beams enter a diffuser,the wavefront is distorted, leading to a challenge in the conservation and detection of the topological charge. Here, we present a technique to measure the value of the topological charge of a vortex beam obscured in the randomly scattered light. The results of the numerical simulations and experiments are presented and are in good agreement. In particular, only a single-shot measurement is required to detect the topological charge of vortex beams, indicating that the method is applicable to a dynamic diffuser.     

Symmetries and variational calculation of discrete Hamiltonian systems

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.     

A new finite difference scheme for a dissipative cubic nonlinear Schr6dinger equation

This paper considers the one-dimensional dissipative cubic nonlinear SchrSdinger equation with zero Dirichlet boundary conditions on a bounded domain. The equation is discretized in time by a linear implicit three-level central difference scheme, which has analogous discrete conservation laws of charge and energy. The convergence with two orders and the stability of the scheme are analysed using a priori estimates. Numerical tests show that the three-level scheme is more efficient.     

Explicit multi-symplectic method for the Zakharov—Kuznetsov equation

We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.     

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

Starting from a new discrete spectral problem, the corresponding hierarchy of nonlinear lattice equations is proposed. It is shown that the lattice soliton hierarchy possesses the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, infinite conservation laws of the hierarchy are presented.     

A new finite difference scheme for a dissipative cubic nonlinear Schrdinger equation

This paper considers the one-dimensional dissipative cubic nonlinear Schrdinger equation with zero Dirichlet boundary conditions on a bounded domain.The equation is discretized in time by a linear implicit three-level central difference scheme,which has analogous discrete conservation laws of charge and energy.The convergence with two orders and the stability of the scheme are analysed using a priori estimates.Numerical tests show that the three-level scheme is more efficient.     

Conservation Laws and Darboux Transformation for Sharma-Tasso-Olver Equation

A hierarchy of new nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is derived.One of the nontrivial equations in this hierarchy is the famous Sharma-Tasso-Olver equation.Then infinitely many conservation laws of this equation are deduced.Darboux transformation for the Sharma-Tasso-Olver equation is constructed with the aid of a gauge transformation.     

Collisions of Two Soatial Solitons in Inhomogeneous Nonlinear Media

Collisions of spatial solitons occurring in the nonlinear Schroeinger equation with harmonic potential are studied, using conservation laws and the split-step Fourier method. We find an analytical solution for the separation distance between the spatial solitons in an inhomogeneous nonlinear medium when the light beam is self-trapped in the transverse dimension. In the self-focusing nonlinear media the spatial solitons can be transmitted stably, and the interaction between spatial solitons is enhanced due to the linear focusing effect （and also diminished for the linear defocusing effect）. In the self-defocusing nonlinear media, in the absence of self-trapping or in the presence of linear self-defocusing, no transmission of stable spatial solitons is possible. However, in such media the linear focusing effect can be exactly compensated, and the spatial solitons can propagate through.     

Turing pattern in the fractional Gierer–Meinhardt model

It is well-known that reaction–diffusion systems are used to describe the pattern formation models. In this paper,we will investigate the pattern formation generated by the fractional reaction–diffusion systems. We first explore the mathematical mechanism of the pattern by applying the linear stability analysis for the fractional Gierer–Meinhardt system.Then, an efficient high-precision numerical scheme is used in the numerical simulation. The proposed method is based on an exponential time differencing Runge–Kutta method in temporal direction and a Fourier spectral method in spatial direction. This method has the advantages of high precision, better stability, and less storage. Numerical simulations show that the system control parameters and fractional order exponent have decisive influence on the generation of patterns. Our numerical results verify our theoretical results.     

An efficient time-integration method for nonlinear dynamic analysis of solids and structures

This paper presents an efficient time-integration method for obtaining reliable solutions to the second-order nonlinear dynamic problems in structural engineering. This method employs both the backward-acceleration differentiation formula and the trapezoidal rule, resulting in a self-starting, single step, second-order accurate algorithm. With the same computational effort as the trapezoidal rule, the proposed method remains stable in large deformation and long time range solutions even when the trapezoidal rule fails. Meanwhile, the proposed method has the following characteristics: (1) it is applicable to linear as well as general nonlinear analyses; (2) it does not involve additional variables (e.g. Lagrange multipliers) and artificial parameters; (3) it is a single-solver algorithm at the discrete time points with symmetric effective stiffness matrix and effective load vectors; and (4) it is easy to implement in an existing computational software. Some numerical results indicate that the proposed method is a powerful tool with some notable features for practical nonlinear dynamic analyses.