Darboux Transformation and N-soliton Solution for Extended Form of Modified Kadomtsev-Petviashvili Equation with Variable-Coefficient

The extended form of modified Kadomtsev-Petviashvili equation with variable-coefficient is investigated in the framework of Painlevé analysis. The Lax pairs are obtained by analysing two Painlevé branches of this equation. Starting with the Lax pair, the N-times Darboux transformation is constructed and the N-soliton solution formula is given, which contains 2n free parameters and two arbitrary functions. Furthermore, with different combinations of the parameters, several types of soliton solutions are calculated from the first order to the third order. The regularity conditions are discussed in order to avoid the singularity of the solutions. Moreover, we construct the generalized Darboux transformation matrix by considering a special limiting process and find a rational-type solution for this equation.     

Soliton and Rogue Wave Solution of the New Nonautonomous Nonlinear Schrödinger Equation

In this paper, a new type (or the second type) of transformation which is used to map the variable coefficient nonlinear Schrödinger (VCNLS) equation to the usual nonlinear Schrödinger (NLS) equation is given. As a special case, a new kind of nonautonomous NLS equation with a t-dependent potential is
introduced. Further, by using the new transformation and making full use of the known soliton and rogue wave solutions of the usual NLS equation, the corresponding kinds of solutions of a special model of the new nonautonomous NLS equation are discussed respectively. Additionally, through using the new transformation, a new expression, i.e., the non-rational formula, of the rogue wave of a special VCNLS equation is given analytically.
The main differences between the two types of transformation mentioned above are listed by three items.     

非线性波方程尖峰孤子解的一种简便求法及其应用

根据尖峰孤子解的特点,提出了一种待定系数法求非线性波方程尖峰孤子解的思路和方法,并利用该方法求解了5个非线性波方程,即CH（Camassa-Holm)方程、五阶KdV-like 方程、广义Ostrovsky方程、组合KdV-mKdV方程和Klein-Gordon方程,比较简便地得到了这些方程的尖峰孤子解.文献中关于CH方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.简要说明了非线性波方程存在尖峰孤子解所须满足的特定条件.该方法也适用于求其他非线性波方程的尖峰孤子解. 关键词： 非线性波方程 尖峰孤子解 待定系数法     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Holm equation is obtained. One-loop soliton solution of the Degasperis-Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

The Soliton Solution of （3＋1）—Dimensional Kadomtsev—Petviashvili Equations

A simple algebraic transformation relation of a special type of solution between the (3 1)-dimensional Kadomtsev-petviashvili(KP) equation and the cubic nonlinear Klein-Gordon equation (NKG) is established.Using known solutions of the NKG equation,we can obtain many soliton solutions and periodic solution of the (3 1)-dimensional KP equation.     

(1+1)维广义的浅水波方程的变量分离解和孤子激发模式

研究(1+1)维广义的浅水波方程的变量分离解和孤子激发模式. 该方程包括两种完全可积(IST可积)的特殊情况，分别为AKNS方程和Hirota-Satsuma方程. 首先把基于Bcklund变换的变量分离(BT-VS)方法推广到该方程，得到了含有低维任意函数的变量分离解. 对于可积的情况，含有一个空间任意函数和一个时间任意函数，而对于不可积的情况，仅含有一个时间任意函数，其空间函数需要满足附加条件. 另外，对于得到的(1+1)维普适公式，选取合适的函数，构造了丰富的孤子激发模式，包括单孤子，正-反孤子，孤子膨胀，类呼吸子，类瞬子等等. 最后，对BT-VS方法作一些讨论. 关键词： 浅水波方程 Bcklund变换 变量分离 孤子     

N-soliton solutions of an integrable equation studied by Qiao

In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 (2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.     

广义随机KdV方程新的精确类孤子解

利用厄米(Hermite)变换求出了广义随机KdV方程新的类孤子解.这种方法的基本思想是通过厄米变换把Wick类型的广义随机KdV变成广义变系数KdV方程，利用特殊的截断展开方法求出 方程的解，然后通过厄米的逆变换求出方程的随机解. 关键词： 随机KdV方程 随机孤子解 白色噪音 截断展开方法 厄米变换     

Integrability of auto-Bäcklund transformations and solutions of a torqued ABS equation

An auto-Bäcklund transformation for the quad equation Q1₁ is considered as a discrete equation, called H2^a, which is a so called torqued version of H2. The equations H2^a and Q1₁ compose a consistent cube, from which an auto-Bäcklund transformation and a Lax pair for H2^a are obtained. More generally it is shown that auto-Bäcklund transformations admit auto-Bäcklund transformations. Using the auto-Bäcklund transformation for H2^a we derive a seed solution and a one-soliton solution. From this solution it is seen that H2^a is a semi-autonomous lattice equation, as the spacing parameter q depends on m but it disappears from the plane wave factor.     

Degenerate Solutions of the Nonlinear Self-Dual Network Equation

The N-fold Darboux transformation(DT) T_n~([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1, 2..., N)and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j →λ_1 and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically,which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t~2).     

Soliton Fission and Fusion in (2+1)-Dimensional Boiti-Leon-Pempinelli System

By means of a special Painlevé-Bäcklund transformation and a multilinear variable separation approach, an exact solution with arbitrary functions of the (2+1)-dimensional Boiti-Leon-Pempinelli system (BLP) is derived. Based on the derived variable separation solution, we obtain some special soliton fission and fusion solutions for the higher dimensional BLP system.     

Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.     

截断展开方法和广义变系数KdV方程新的精确类孤子解

利用特殊的截断展开方法求出了广义变系数KdV方程新的类孤子解.这种方法的基本思想是假定形式解具有截断展开形式,以致可把广义变系数KdV方程转化为一组待定函数的代数方程组,进而给出待定函数容易积分的常微分方程.利用例子证明了这种方法是十分有效的. 关键词： 截断展开法 变系数 KdV方程 孤波解     

Solitons on a Periodic Wave Background of the Modified KdV-Sine-Gordon Equation

The Bäcklund transformation(BT) of the mKdV-sG equation is constructed by introducing a new transformation. Infinitely many nonlocal symmetries are obtained in terms of its BT. The soliton-periodic wave interaction solutions are explicitly derived by the classical Lie-group reduction method. Particularly, some special concrete soliton and periodic wave interaction solutions and their behaviours are discussed both in analytical and graphical ways.     

Mixed Local-Nonlocal Vector Schrödinger 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 Schrödinger equations are presented, which are different from the single local and nonlocal coupled Schrödinger equation. We investigate the Darboux transformation of mixed local-nonlocal vector Schrödinger equations with a spectral problem. Starting from a special Lax pairs, the mixed localnonlocal vector Schrödinger equations are constructed. We obtain the one- and two- and N-soliton solution formulas of the mixed local-nonlocal vector Schrödinger 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 Schrödinger equations have some novel results. The results in this paper might be helpful for understanding some physical phenomena described in plasmas.     

光纤中变系数非线性Schr(o)dinger方程的孤子解及其应用

基于推广的立方非线性Klein—Gordon方程对一般形式的变系数非线性Schrodinger方程进行研究，讨论了无啁啾情形的孤子解，发现了包括亮、暗孤子解和类孤子解在内的一些新的精确解．同时对基本孤子的色散控制方法进行了简单讨论．作为特例，常系数非线性Schrodinger方程和两类特殊的变系数非线性Schrodinger方程的结果和已知的形式一致．此外，还研究了一个周期增益或损耗的光纤系统，得到了有意义的结果．     

Soliton molecules and mixed solutions of the(2+1)-dimensional bidirectional Sawada-Kotera equation

In this paper, we investigate some new interesting solution structures of the(2+1)-dimensional bidirectional Sawada–Kotera(bSK) equation. We obtain soliton molecules by introducing velocity resonance. On the basis of soliton molecules, asymmetric solitons are obtained by changing the distance between two solitons of molecules. Based on the N-soliton solutions,several novel types of mixed solutions are generated, which include the mixed breather-soliton molecule solution by the module resonance of the wave number and partial velocity resonance,the mixed lump-soliton molecule solution obtained by partial velocity resonance and partial long wave limits, and the mixed solutions composed of soliton molecules(asymmetric solitons), lump waves, and breather waves. Some plots are presented to clearly illustrate the dynamic features of these solutions.     

Lump and Stripe Soliton Solutions to the Generalized Nizhnik-Novikov-Veselov Equation

With the aid of the truncated Painlevé expansion, a set of rational solutions of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov (GNNV) equation with the quadratic function which contains one lump soliton is derived. By combining this quadratic function and an exponential function, the fusion and fission phenomena occur between one lump soliton and a stripe soliton which are two kinds of typical local excitations. Furthermore, by adding a corresponding inverse exponential function to the above function, we can derive the solution with interaction between one lump soliton and a pair of stripe solitons. The dynamical behaviors of such local solutions are depicted by choosing some appropriate parameters.     

Pfaffian Solutions and Resonant Interaction Properties of a Coupled BKP Lattice

In this paper, we give a coupled lattice equation with the help of Hirota operators, which comes from a special BKP lattice. Two-soliton and three-soliton solutions to the coupled system are constructed. Furthermore,resonant interaction of the two-soliton solution is analyzed in detail. Under some special resonant condition, it is shown that low soliton can propagate faster than high one. Finally, the N-soliton solution is presented in the Pfaffian form.