The projective Riccati equation expansion method and variable separation solutions for the nonlinear physical differential equation in physics

Using the projective Riccati equation expansion (PREE) method, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for two nonlinear physical models are obtained. Based on one of the variable separation solutions and by choosing appropriate functions, new types of interactions between the multi-valued and single-valued solitons, such as a peakon-like semi-foldon and a peakon, a compacton-like semi-foldon and a compacton, are investigated.     

Annihilation Solitons and Chaotic Solitons for the （2＋1）-Dimensional Breaking Soliton System

By means of an improved mapping method and a variable separation method, a series of variable separation solutions including solitary wave solutions, periodic wave solutions and rational function solutions） to the （2＋1）-dimensional breaking soliton system is derived. Based on the derived solitary wave excitation, we obtain some special annihilation solitons and chaotic solitons in this short note.     

New Families of Rational Form Variable Separation Solutions to （2＋1）-Dimensional Dispersive Long Wave Equations

With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.     

New Exact Solutions and Localized Structures for （3＋1）-Dimensional Burgers System

With an extended mapping approach and a linear variable separation method, new families of variable separation solutions （including solitary wave solutions, periodic wave solutions, and rational function solutions） with arbitrary functions for （3＋1）-dimensionai Burgers system is derived. Based on the derived excitations, we obtain some novel localized coherent structures and study the interactions between solitons.     

Quasi-Periodic Structures Based on Symmetrical Lucas Function of （2＋1）-Dimensional Modified Dispersive Water-Wave System

By introducing the Lucas-Riccati method and a linear variable separation method, new variable separation solutions with arbitrary functions are derived for a （2＋1）-dimensional modified dispersive water-wave system. The main idea of this method is to express the solutions of this system as polynomials in the solution of the Riecati equation that the symmetrical Lucas functions satisfy. From the variable separation sohition and by selecting appropriate functions, some novel Jacobian elliptic wave structure with variable modulus and their interactions with dromions and peakons are investigated.     

Folded localized excitations in the （2＋1）-dimensional modified dispersive water-wave system

By using a mapping approach and a linear variable separation approach, a new family of solitary wave solutions with arbitrary functions for the (2+1)-dimensional modified dispersive water-wave system (MDWW) is derived. Based on the derived solutions and using some multi-valued functions, we obtain some novel folded localized excitations of the system.     

Variable separation solutions and new solitary wave structures to the (1+1)-dimensional equations of long-wave－short-wave resonant interaction

A variable separation approach is proposed and extended to the (1+1)-dimensional physical system. The variable separation solutions of (1+1)-dimensional equations of long-wave－short-wave resonant interaction are obtained. Some special type of solutions such as soliton solution, non-propagating solitary wave solution, propagating solitary wave solution, oscillating solitary wave solution are found by selecting the arbitrary function appropriately.     

Complex Wave Excitations in Generalized Broer-Kaup System

Starting from an improved projective method and a linear variable separation approach, new families of variable separation solutions （including solltary wave solutlons, periodic wave solutions and rational function solutions） with arbitrary functions [or the （2＋ 1）-dimensional general/zed Broer-Kaup （GBK） system are derived. Usually, in terms of solitary wave solutions and/or rational function solutions, one can find abundant important localized excitations. However, based on the derived periodic wave solution in this paper, we reveal some complex wave excitations in the （2＋1）-dimensional GBK system, which describe solitons moving on a periodic wave background. Some interesting evolutional properties for these solitary waves propagating on the periodic wave bactground are also briefly discussed.     

Soliton excitations and chaotic patterns for the (2+1)-dimensional Boiti-Leon-Pempinelli system

By improved projective equation approach and a linear variable separation approach, a new family of exact solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli (BLP) system is derived. Based on the derived solitary wave solution, some dromion and solitoff excitations and chaotic behaviours are investigated.     

Chaotic solutions of (2+1)-dimensional Broek Kaup equation with variable coefficients

In this paper,an improved projective approach is used to obtain the variable separation solutions with two arbitrary functions of the (2+1)-dimensional Broek-Kaup equation with variable coefficients (VCBK). Based on the derived solitary wave solution and using a known chaotic system,some novel chaotic solutions are investigated.     

扩展的(2+1)维浅水波方程的尖峰孤子解及其相互作用

利用改进的 Riccati方程映射法和变量分离法, 得到了扩展的(2+1)维浅水波方程的变量分离解(包括孤波解, 周期波解和有理函数解). 根据得到的孤波解, 构造出了方程的几种不同形状的尖峰孤子结构, 研究了孤子的相互作用.     

(3+1)维Burgers系统的瞬内嵌孤子和瞬锥形孤子

利用投射方程法和变量分离法,得到了(3+1)维Burgers系统的变量分离解(包括孤波解、周期波解和有理函数解).根据孤波解和有理函数解,构造出Burgers系统新颖的局域结构,例如瞬内嵌孤子和瞬锥形孤子.     

Localized Excitations of (2+1)-Dimensional Korteweg-de Vries System Derived from a Periodic Wave Solution

With the aid of an improved projective approach and a linear variable separation method,new types of variable separation solutions (including solitary wave solutions,periodic wave solutions,and rational function solutions)with arbitrary functions for (2 1)-dimensional Korteweg-de Vries system are derived.Usually,in terms of solitary wave solutions and rational function solutions,one can find some important localized excitations.However,based on the derived periodic wave solution in this paper,we find that some novel and significant localized coherent excitations such as dromions,peakons,stochastic fractal patterns,regular fractal patterns,chaotic line soliton patterns as well as chaotic patterns exist in the KdV system as considering appropriate boundary conditions and/or initial qualifications.     

(2+1)维色散长波方程的新精确解及其复合波激发

利用Riccati方程展开法和变量分离法,得到了(2+1)维色散长波方程的变量分离解.根据得到的孤波解,构造出该方程新颖的复合波局域结构,研究了复合波随时间的演化.     

Complex wave excitations general （2＋1）-dimensional and chaotic patterns for a Korteweg-de Vries system

Starting from an improved mapping approach and a linear variable separation approach, a new family of exact solutions （including solitary wave solutions, periodic wave solutions and rational function solutions） with arbitrary functions for a general （2＋1）-dimensional Korteweg de solutions, we obtain some novel dromion-lattice solitons, system Vries system （GKdV） is derived. According to the derived complex wave excitations and chaotic patterns for the GKdV