Notes on Multi-linear Variable Separation Approach

The multi-linear variable separation approach method is very useful to solve (2+1)-dimensional integrable systems. In this letter, we extend this method to solve (1+1)-dimensional Boiti system, (2+1)-dimensional Burgers system, (2+1)-dimensional breaking soliton system, and (2+1)-dimensional Maccari system. Some new exact solutions are obtained and the universal formula obtained from many (2+1)-dimensional systems is extended or modified.     

N-Soliton Solutions of (2+1)-Dimensional Non-isospectral AKNS System

The bilinear form of the (2+1)-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a (2+1)-dimensional non-isospectral Schrödinger equation and its N-soliton solutions are constructed.     

Darboux Transformation and New Soliton-like Solutions for (1+1)-Dimensional Higher-Order Broer-Kaup System

In this letter, we construct a kind of new Darboux transformation for the (1+1)-dimensional higher-order Broer-Kaup (HBK) system with the help of a gauge transformation of a spectral problem. By applying this new Darboux transformation, some new soliton-like solutions of the (1+1)-dimensional HBK system are obtained.     

Multi-linear Variable Separation Approach to Solve a (2+1)-Dimensional Generalization of Nonlinear Schrödinger System

By using a Bäcklund transformation and the multi-linear variable separation approach, we find a new general solution of a (2+1)-dimensional generalization of the nonlinear Schrödinger system. The new “universal” formula is defined, and then, rich coherent structures can be found by selecting corresponding functions appropriately.     

Abundant New Explicit Exact Soliton—Like Solutions and Painleve Test for the Generalized Burgers Equation in （2＋1）—Dimensional Space

We obtain Backlund transformation and some new kink-like solitary wave solutions for the generalized Burgers equation in (2 1)-dimensional space,ut 1/2(uδy^-1ux)x-uxx=0,by using the extended homogeneous balance method.As is well known,the introduction of the concept of dromions (the exponentially localized solutions in (2 1)-dimensional space)has triggered renewed interest in (2 1)-dimensional soliton systems.The solutions obtained are used to show that the variable ux admits exponentially localized solutions rather than the physical field u(x,y,t) itself.In addition,it is shown that the equation passes Painleve test.     

Symmetry Reduction and Exact Solutions of the (3+1)-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the (3+1)-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the (3+1)-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the (3+1)-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the (3+1)-dimensional breaking soliton equation.     

New Double-Periodic Soliton Solutions for the (2+1)-Dimensional Breaking Soliton Equation

Under investigation is the (2+1)-dimensional breaking soliton equation. Based on a special ansätz functions and the bilinear form, some entirely new double-periodic soliton solutions for the (2+1)-dimensional breaking soliton equation are presented. With the help of symbolic computation software Mathematica, many important and interesting properties for these obtained solutions are revealed with some figures.     

Multisoliton Solutions of the （3＋1）—Dimensional Nizhnik—Novikov—Veselov Equation

Using the standard truncated Painleve analysis,we can obtain a Backlund transformation of the (3 1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation and get some(3 1)-dimensional single-,two- and three-soliton solutions and some new types of multisoliton solutions of the (3 1)-dimensional NNV system from the Backlund transformation and the trivial vacuum solution.     

Study of （2＋1）-Dimensional Higher-Order Broer-Kaup System

Painleve property and infinite symmetries of the （2＋1）-dimensional higher-order Broer-Kaup （HBK） system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.ups to （2＋1）-dimensional HBK system. Based on our theorem, some new forms of solutions are obtained. We also find infinite number of conservation laws of the （2＋1）-dimensional HBK system.     

Integrability and Solutions of the(2+1)-dimensional Hunter–Saxton Equation

In this paper,the(2+1)-dimensional Hunter-Saxton equation is proposed and studied.It is shown that the(2+1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations.Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation,a non-isospectral Lax-pair of the(2+1)-dimensional Hunter–Saxton equation is derived.In addition,exact singular solutions with a finite number of corners are obtained.Furthermore,the(2+1)-dimensional μ-Hunter–Saxton equation is presented,and its exact peaked traveling wave solutions are derived.     

Multi-linear Variable Separation Approach to Solve a （1＋1）-Dimensional Coupled Integrable Dispersionless System

The multi-linear variable separation approach （MLVSA ） is very useful to solve （2＋1）-dimensional integrable systems. In this letter, we extend this method to solve a （1＋1）-dimensional coupled integrable dispersion-less system. Namely, by using a Backlund transformation and the MLVSA, we find a new general solution and define a new ＂universal formula＂. Then, some new （1＋1）-dimensional coherent structures of this universal formula can be found by selecting corresponding functions appropriately. Specially, in some conditions, bell soliton and kink soliton can transform each other, which are illustrated graphically.     

Soliton Motion in (1+1)-Dimensions

By using the variable separation approach, which is based on the corresponding Bäcklund transformation, new exact solutions of a (1+1)-dimensional nonlinear evolution equation are obtained. Abundant new soliton motions of the potential field can be found by selecting appropriate functions.     

New Family of Exact Solutions and Chaotic Soltions of Generalized Breor-Kaup System in （2＋1）-Dimensions via an Extended Mapping Approach

Starting from an extended mapping approach, a new type of variable separation solution with arbitrary functions of generalized （2＋1）-dimensional Broer-Kaup system （GBK） system is derived. Then based on the derived solitary wave solution, we obtain some specific chaotic solitons to the （2＋1）-dimensional GBK system.     

(2+1)维修正Veselov-Novikov系统的新型折叠子及其弹性碰撞

借助Mathematica软件，在Bcklund变换的基础上采用多线性变量分离(MLVS)方法，得到了(2+1)维修正Veselov-Novikov系统的一个含低维任意函数的新的精确解.选取合适的多值函数，构造出新型的折叠子，对其进行了分类并且研究了各种类型的二折叠子之间的完全弹性碰撞.另外还给出了折叠子与隐形折叠子的相互作用.最后把MLVS方法推广到一个新的(1+1)维非线性系统. 关键词： 修正Veselov-Novikov系统 折叠子 弹性碰撞 变量分离     

Folded Localized Excitations in a Generalized (2+1)-Dimensional Perturbed Nonlinear SchrÖdinger System

Starting from a special Bäcklund transform and a variable separation approach, a quite general variable separation solution of the generalized (2+1)-dimensional perturbed nonlinear Schrödinger system is obtained. In addition to the single-valued localized coherent soliton excitations like dromions, breathers, instantons, peakons, and previously revealed chaotic localized solution, a new type of multi-valued (folded) localized excitation is derived by introducing some appropriate lower-dimensional multiple valued functions.     

(3+1)维非线性Burgers系统的新的分离变量解及其局域激发结构与分形结构

将扩展的Riccati方程映射法推广到了(3+1)维非线性Burgers系统，得到了系统的分离变量解；由于在解中含有一个关于自变量(x,y,z,t)的任意函数，通过对这个任意函数的适当选取，并借助于数学软件Mathematica进行数值模拟，得到了系统的新而丰富的局域激发结构和分形结构.结果表明，扩展的Riccati方程映射法在求解高维非线性系统时，仍然是一种行之有效的方法，并且可以得到比(2+1)维非线性系统更为丰富的局域激发结构. 关键词： 扩展的Riccati方程映射法 (3+1)维非线性Burgers方程 局域激发结构 分形结构     

New Bilinear Bäcklund Transformation and Higher Order Rogue Waves with Controllable Center of a Generalized (3+1)-Dimensional Nonlinear Wave Equation

In this paper, we first obtain a bilinear form with small perturbation u_0 for a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Based on that, a new bilinear B?cklund transformation which consists of four bilinear equations and involves seven arbitrary parameters is constructed. After that, by applying a new symbolic computation method, we construct the higher order rogue waves with controllable center to the generalized(3+1)-dimensional nonlinear wave equation. The rogue waves present new structure, which contain two free parametersα and β. The dynamic properties of the higher order rogue waves are demonstrated graphically. The graphs tell that the parameters α and β can control the center of the rogue waves.     

An Extended Subequation Rational Expansion Method and Solutions of (2+1)-Dimensional Cubic Nonlinear Schrödinger Equation

An extended subequation rational expansion method is presented and used to construct some exact analytical solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation. From our results, many known solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of new non-travelling wave and coefficient function's soliton-like solutions, and elliptic solutions are demonstrated by some plots.     

A new (2+1)-dimensional supersymmetric Boussinesq equation and its Lie symmetry study

According to the conjecture based on some known facts of integrable models, a new (2+1)-dimensional supersymmetric integrable bilinear system is proposed. The model is not only the extension of the known (2+1)-dimensional negative Kadomtsev--Petviashvili equation but also the extension of the known (1+1)-dimensional supersymmetric Boussinesq equation. The infinite dimensional Kac--Moody--Virasoro symmetries and the related symmetry reductions of the model are obtained. Furthermore, the traveling wave solutions including soliton solutions are explicitly presented.     

Variable-Coefficient Hyperbola Function Method and Its Application to （2＋1）-Dimensional Variable-Coefficient Broer-Kaup System

Based on a new intermediate transformation, a variable-coeFficient hyperbola function method is proposed.Being concise and straightforward, it is applied to the (2 1)-dimensional variable-coeFficient Broer-Kaup system. As a result, several new families of exact soliton-like solutions are obtained, besides the travelling wave. When imposing some conditions on them, the new exact solitary wave solutions of the (2 1)-dimensional Broer Kaup system are given. The method can be applied to other variable-coeFficient nonlinear evolution equations in mathematical physics.