Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy

In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. Prom the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation.

(2+2)-Dimensional Discrete Soliton Equations and Integrable Coupling System

在这份报纸，我们延长一(2+2 ) 维连续零个弯曲方程到(2+2 ) 维分离零个弯曲方程，然后一新(2+2 ) 维的立方的 Volterra 格子层次被获得。而且，联合系统的 integrable (2+2 ) 维的立方的 Volterra 格子层次和概括 Toda 格子 soliton 方程被使用一个谎言介绍代数学的系统 sl (4 ) 。

The multicomponent (2+1)-dimensional Glachette–Johnson (GJ) equation hierarchy and its super-integrable coupling system

This paper presents a set of multicomponent matrix Lie algebra, which is used to construct a new loop algebra A^-M. By using the Tu scheme, a Liouville integrable multicomponent equation hierarchy is generated, which possesses the Hamiltonian structure. As its reduction cases, the multicomponent （2＋1）-dimensional Glachette-Johnson （G J） hierarchy is given. Finally, the super-integrable coupling system of multicomponent （2＋1）-dimensional GJ hierarchy is established through enlarging the spectral problem.

Infinite Conservation Laws for Nonlinear Integrable Couplings of Toda Hierarchy

We construct nonlinear integrable couplings of discrete soliton hierarchy,then the infinite conservation laws for the nonlinear integrable couplings of the lattice hierarchy are established.For explicit application of the method proposed,the infinite conservation laws of nonlinear integrable couplings of the Toda lattice hierarchy are presented.The obtained integrable couplings of the Toda lattice equations and conservation laws can be used to describe the possible formation mechanisms for hydrodynamics,solid state physics and plasma physics,respectively.

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.

N-Fold Darboux Transformation for the Nonlocal Nonlinear Schrodinger(NNLS)Equation with the Self-Induced PT-Symmetric Potential

The Darboux transformation (DT) method is studied in a lot of local equations, but there are few of work to solve nonlocal equations by DT. In this letter, we solve the nonlocal nonlinear Schrödinger equation (NNLSE) with the self-induced PT-symmetric potential by DT. Then the N-fold DT of NNLSE is derived with the help of the gauge transformation between the Lax pairs. Then we derive some novel exact solutions including the bright soliton, breather wave soliton. In particularly, the dynamic features of one-soliton, two-soliton, three-soliton solutions and the elastic interactions between the two solitons are displayed.

The Multi-Function Jaulent-Miodek Equation Hierarchy with Self-Consistent Sources

与前後一致的来源一起的 soliton 方程层次的多功能被构造。然后，与前後一致的来源一起的 Jaulent-Miodek (JM ) 方程层次被导出。而且，与前後一致的来源一起的多功能 JM 方程层次被使用高度维的宽松的对介绍。[从作者抽象]

A New Method to Construct Integrable Coupling System for Burgers Equation Hierarchy by Kronecker Product

Kronecker 产品能被使用构造在这篇论文联合 soliton 方程层次的系统的新 integrable，这被显示出。直接应用到汉堡包光谱问题导致 integrable 联合的一个新奇 soliton 方程层次系统。它显示 Kronecker 产品是一个有效、直接的方法构造 integrable 政变石楠。

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.

Some space-time fractional bright–dark solitons and propagation manipulations for a fractional Gross–Pitaevskii equation with an external potential

Some nonautonomous bright–dark solitons(NBDSs)and nonautonomous controllable behaviors in the conformable space-time fractional Gross–Pitaevskii(FGP)equation with some external potentials are derived.We consider the relations between the space-time FGP equation and the fractional nonlinear Schr?dinger equation and analyze the properties of the obtained equation with group velocity dispersion and spatiotemporal dispersion.Then,some constraint conditions of the valid soliton solutions are given.Furthermore,we consider the effect ofαandβin NBDSs of the space-time FGP equation.Some fractional spatial–temporal controlling wave prolong phenomena are considered,and some different propagation dynamics are generated via the different parametersαandβ.We study novel shape bright soliton solution,novel‘h’-shape dark soliton and some interactions of nonautonomous bright–dark solitons.The reported results of some novel interactions are considered,which can explain some models of the electrical and optical fields.

A conservative local discontinuous Galerkin method for the solution of nonlinear Schrdinger equation in two dimensions

In this study, we present a conservative local discontinuous Galerkin(LDG) method for numerically solving the two-dimensional nonlinear Schrdinger(NLS) equation. The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux. The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central, alternative and upwind-based flux. We will propose two kinds of time discretization methods for the semi-discrete formulation. One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation. The other one is Krylov implicit integration factor(IIF) method which demands much less computational effort. Various numerical experiments are presented to demonstrate the conservation law of mass and energy, the optimal rates of convergence, and the blow-up phenomenon.

Mixed Local-Nonlocal Vector Schrdinger Equations and Their Breather Solutions

To the best of our knowledge, all nonlinearities in the known nonlinear integrable systems are either local or nonlocal. A natural problem is whether there exist some nonlinear integrable systems with both local and nonlocal nonlinearities, and how to solve this kinds of spectral nonlinear integrable systems with both local and nonlocal nonlinearities. Recently, some novel mixed local-nonlocal vector Schrdinger equations are presented, which are different from the single local and nonlocal coupled Schrdinger equation. We investigate the Darboux transformation of mixed local-nonlocal vector Schrdinger equations with a spectral problem. Starting from a special Lax pairs, the mixed localnonlocal vector Schrdinger equations are constructed. We obtain the one-and two-and N-soliton solution formulas of the mixed local-nonlocal vector Schrdinger equations with N-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-solitons are exhibited, the overtaking elastic interactions among the two-breather solitons are considered. We find that unlike the local and nonlocal cases, the mixed local-nonlocal vector Schrdinger equations have some novel results.The results in this paper might be helpful for understanding some physical phenomena described in plasmas.

Order shrinkage and selection for the INGARCH(p,q)model

The integer-valued generalized autoregressive conditional heteroskedastic(INGARCH)model is often utilized to describe data in biostatistics,such as the number of people infected with dengue fever,daily epileptic seizure counts of an epileptic patient and the number of cases of campylobacterosis infections,etc.Since the structure of such data is generally high-order and sparse,studies about order shrinkage and selection for the model attract many attentions.In this paper,we propose a penalized conditional maximum likelihood(PCML)method to solve this problem.The PCML method can effectively select significant orders and estimate the parameters,simultaneously.Some simulations and a real data analysis are carried out to illustrate the usefulness of our method.

The Discrete Approximation Problem for a Special Case of Hermite-Type Polynomial Interpolation

Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize the result of P.Gniadek.As an application,the authors consider the discrete approximation problem for a special case when the interpolation condition contains all partial derivatives of order less than n and one nth order differential polynomial.In addition,for the case of n≥3,the authors use the concept of Cartesian tensors to give a sufficient condition to find a sequence of discrete points,such that the Lagrange interpolation problems at these points converge to the given Hermite-type interpolant.

A MULTIVARIATE MULTIQUADRIC QUASI-INTERPOLATION WITH QUADRIC REPRODUCTION

In this paper, by using multivariate divided differences to approximate the partial derivative and superposition, we extend the multivariate quasi-interpolation scheme based on dimension-splitting technique which can reproduce linear polynomials to the scheme quadric polynomials. Furthermore, we give the approximation error of the modified scheme. Our multivariate multiquadric quasi-interpolation scheme only requires information of lo- cation points but not that of the derivatives of approximated function. Finally, numerical experiments demonstrate that the approximation rate of our scheme is significantly im- proved which is consistent with the theoretical results.

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.