Solving Non-Isospectral mKdV Equation and Sine-Gordon Equation Hierarchies with Self-Consistent Sources via Inverse Scattering Transform

N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources and the hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scattering transform.     

Breather Lattice Solutions to Negative mKdV Equation

In this paper, dependent and independent variable transformations are introduced to solve the negative mKdV equation systematically by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the negative mKdV equation, including breather lattice solution and periodic wave solution.     

A hyperbolic function approach to constructing exact solitary wave solutions of the Hybrid lattice and discrete mKdV lattice

Some new exact solitary wave solutions of the Hybrid lattice and discrete mKdV lattice are obtained by using a hyperbolic function approach. This approach can also be applied to other nonlinear differential-difference equations.     

Some Special Solutions of Three Nonlinear Lattices

Some soliton solutions and periodic solutions of hybrid lattice, discretized mKdV lattice, and modified Volterra lattice have been obtained by introducing a new method. This approach allows us to directly construct some explicit exact solutions for polynomial nonlinear differential-difference equations.     

Some Special Solutions of Three Nonlinear Lattices

Some soliton solutions and periodic solutions of hybrid lattice, discretized mKdV lattice, and modified Volterra lattice have been obtained by introducing a new method. This approach allows us to directly construct some explicit exact solutions for polynomial nonlinear differential-difference equations.     

Exact Solutions Expressible in Rational Formal Hyperbolic and Elliptic Functions for Nonlinear Differential-Difference Equation

A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.     

Generalized differential transform method to differential-difference equation

In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Padé technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.     

Exact Solutions Expressible in Rational Formal Hyperbolic and Elliptic Functions for Nonlinear Differential-Difference Equation

A new approach is presented by means of a new general ansätz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, discretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.     

Coupled Modified Korteweg-de Vries Lattice in (2+1) Dimensions and Soliton Solutions

The coupled semi-discrete modified Korteweg-de Vries equation in (2 1)-dimensions is proposed. It is shown that it can be decomposed into two (1 1)-dimensional differential-difference equations belonging to mKdV lattice hierarchy by considering a discrete isospectral problem. A Darboux transformation is set up for the resulting (2 1)- dimensional lattice soliton equation with the help of gauge transformations of Lax pairs. As an illustration by example,the soliton solutions of the mKdV lattice equation in (2 1)-dimensions are explicitly given.     

利用耦合的Riccati方程组构造微分-差分方程精确解

通过引入耦合的Riccati方程组得到一个构造非线性微分-差分方程精确解的代数方法.作为实例,将该方法应用到了一般格子方程,相对论的Toda格子方程和(2+1)维Toda格子方程.借助符号计算软件Mathematica,获得了这些方程的扭结型孤波解和复数解.该方法也适合求解其他非线性微分-差分方程的精确解. 关键词： 耦合Riccati方程组 格子方程 相对论的Toda格子方程 (2+1)维Toda格子方程     

Higher-Dimensional Integrable Systems Arising from Motions of Curves on S^2（R） and S^3（R）

We show that higher-dimensional integrable systems including the （2＋1）-dimensional generalized sine-Gordon equation and the （2＋1）-dimensional complex mKdV equation are associated with motions of surfaces induced by endowing with an extra space variable to the motions of curves on S^2（R） and S^3（R）.     

Some New Lie Symmetry Groups of Differential-Difference Equations Obtained from a Simple Direct Method

In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A： Math. Gen. 38 （2005） L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and （2＋1）-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.     

Exact Discrete Soliton Solutions and Periodic Solutions to （2＋1）-Dimensional Toda Lattice with a New Algebraic Method

In this paper, with the aid of symbolic computation, we present a uniform method for constructing soliton solutions and periodic solutions to （2＋1）-dimensional Toda lattice equation.     

A connection between the （G′/G）-expansion method and the truncated Painlev ＇e expansion method and its application to the mKdV equation

Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Painlev＇e expansion method by introducing an intermediate expansion method. Then the generalized （G′/G）-（G/G′） expansion method is naturally derived from the standpoint of the nonstandard truncated Painlev＇e expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the （ G′/ G）-expansion method.     

Quantum Optical Squeezing Transform for Generalizing Fractional Fourier Transform

By establishing the relation between the optical scaled fractional Fourier transform （FFT） and quantum mechanical squeezing-rotating operator transform, we employ the bipartite entangled state representation of two-mode squeezing operator to extend the scaled FFT to more general cases, such as scaled complex FFT and entangled scaled FFT. The additiyity and eigenmodes are presented in quantum version. The relation between the scaled FFT and squeezing-rotating Wigner operator is studied.     

非线性离散微分方程的双曲函数法求解

本文推广了双曲函数方法用于求解非线性离散系统。求解离散的(2+1)维Toda系统和离散的mKdV系统，成功地得到了离散钟型孤立子、离散冲击波型孤立子及一些新的精确行波解。     

Realization of quantum Fourier transform over ZN

Since the difficulty in preparing the equal superposition state of amplitude is 1/√N, we construct a quantile transform of quantum Fourier transform （QFT） over ZN based on the elementary transforms, such as Hadamard transform and Pauli transform. The QFT over Z_N can then be realized by the quantile transform, and used to further design its quantum circuit and analyze the requirements for the quantum register and quantum gates. However, the transform needs considerable quantum computational resources and it is difficult to construct a high-dimensional quantum register. Hence, we investigate the design of t-bit quantile transform, and introduce the definition of t-bit semiclassical QFT over Z_N. According to probability amplitude, we prove that the transform can be used to realize QFT over ZN and further design its quantum circuit. For this transform, the requirements for the quantum register, the one-qubit gate, and two-qubit gate reduce obviously when compared with those for the QFT over Z_N.     

Application of Exp-Function Method to Discrete Nonlinear SchrSdinger Lattice Equation with Symbolic Computation

In this paper, we present an extended Exp-function method to differential-difference equation（s）. With the help of symbolic computation, we solve discrete nonlinear Schrodinger lattice as an example, and obtain a series of general solutions in forms of Exp-function.     

N-Soliton Solutions of Modified KdV Equation under Bargmann Constraint

By means of Hirota method, N-soliton solutions of the modified KdV equation under the Bargmann constraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair of the modified KdV equation.     

New Families of Exact Excitations to (2+1)-Dimensional Toda Lattice System via an Extended Projective Approach

In this paper, the extended projective approach, which was recently presented and successfully used in some continuous nonlinear physical systems, is generalized to nonlinear partial differential-difference systems (DDEs). As a concrete example, new families of exact solutions to the (2+1)-dimensional Toda lattice system are obtained by the extended projective approach.