Periodic Semifolded Solitary Waves for （2＋1）-Dimensional Variable Coefficient Broer-Kaup System

Applying the extended mapping method via Riccati equation, many exact variable separation solutions for the （2＆1 ）-dimensional variable coefficient Broer-Kaup equation are obtained. Introducing multiple valued function and Jacobi elliptic function in the seed solution, special types of periodic semifolded solitary waves are derived. In the long wave limit these periodic semifolded solitary wave excitations may degenerate into single semifolded localized soliton structures. The interactions of the periodic semifolded solitary waves and their degenerated single semifolded soliton structures are investigated graphically and found to be completely elastic.     

Nonpropagating Solitary Waves in (2+1)-Dimensional Generalized Dispersive Long Wave Systems

An extended mapping approach is used to obtain a new type of variable separation excitation, with three arbitrary functions, of the 2+1-dimensional generalized dispersive long wave equation (DLWE). By selecting appropriate functions, the richness of nonpropagating solitons, such as nonpropagating dromion, nonpropagating ring, nonpropagating lump, and nonpropagating foldon, etc., is displayed for the ($2+1$)-dimensional generalized dispersive long wave equation (DLWE) in this paper. Meanwhile, we conclude that the solution v₁ and v₂ are essentially equivalent to the “universal” formula. PACS numbers: 02.30.Jr, 02.30.Ik, 05.45.Yv     

Oscillating Solitons for (2+1)-Dimensional Nonlinear Models

Using extended homogeneous balance method and variable separation hypothesis,we found new variableseparation solutions with three arbitrary functions of the (2 1)-dimensional dispersive long-wave equations.Based on derived solutions,we revealed abundant oscillating solitons such as dromion,multi-dromion,solitoff,solitary waves,and so on,by selecting appropriate functions.     

Folded Solitary Waves and Foldons in(2+1) Dimensions

A general type of localized excitations, folded solitary waves and foldons, is defined and studied bothanalytically and graphically. The folded solitary waves and foldons may be “folded“ in quite complicated ways andpossess quite rich structures and abundant interaction properties. The folded phenomenon is quite universal in the realnatural world. The folded solitary waves and foldons are obtained from a quite universal formula and the universalformula is valid for some quite universal (2 1)-dimensional physical models. The “universal“ formula is also extendedto a more general form with many more independent arbitrary functions.     

Standing,Periodic and Solitary Waves in (1+1)-Dimensional Caudry-Dodd-Gibbon-Sawada-Kortera System

In the paper, the variable separation approach, homoclinic test technique and bilinear method are successfully extended to a (1+1)-dimensional Caudry-Dodd-Gibbon-Sawada-Kortera (CDGSK) system, respectively. Based on the derived exact solutions, some significant types of localized excitations such as standing waves, periodic waves, solitary waves are simultaneously derived from the (1+1)-dimensional Caudry-Dodd-Gibbon-Sawada-Kortera system by entrancing appropriate parameters.     

Folded Solitary Waves and Foldons in （2＋1） Dimensions

A general type of local/zed excitations, folded solitary waves and foldons, is defined and studied both analytically and graphically. The folded solitary waves and foldons may be “folded“ in quite complicated ways and possess qnite rich structures and abundant interaction properties. The folded phenomenon is quite universal in the real natural world. The folded solitary waves and foldons are obtained from a quite universal formula and the universal formul is valid for some quite universal (2 1)-dimensional physical mode/s. The “universal“ formula is also extended to a more general form with many more independent arbitrary functions.     

New B(a)cklund Transformation and New Exact Solutions to (2+1)-Dimensional KdV Equation

A new Backlund transformation for (2 1)-dimensional KdV equation is first obtained by using homogeneousbalance method. And making use of the Backlund transformation and choosing a special seed solution, we get specialtypes of solitary wave solutions. Finally a general variable separation solution containing two arbitrary functions isconstructed, from which abundant localized coherent structures of the equation in question can be induced.     

The Solitary Waves for Cubic-Quintic Nonlinear Schrödinger Equation with Variable Coefficients

The cubic-quintic nonlinear Schrödinger equation (CQNLS) plays important parts in the optical fiber and the nuclear hydrodynamics. By using the homogeneous balance principle, the bell type, kink type, algebraic solitary waves, and trigonometric traveling waves for the cubic-quintic nonlinear Schrödinger equation with variable coefficients (vCQNLS) are derived with the aid of a set of subsidiary high-order ordinary differential equations (sub-equations for short). The method used in this paper might help one to derive the exact solutions for the other high-order nonlinear evolution equations, and shows the new application of the homogeneous balance principle.     

Quasi-periodic and Non-periodic Waves in (2+1)-Dimensional Nonlinear Systems

New exact quasi-periodic and non-periodic solutions for the (2 1)-dimensional nonlinear systems are studied by means of the multi-linear variable separation approach (MLVSA) and the Jacobi elliptic functions with the space-time-dependent modulus. Though the result is valid for all the MLVSA solvable models, it is explicitly shown for the long-wave and short-wave interaction model.     

Nonpropagating Solitons in (2+1)-Dimensional Dispersive Long-Water Wave System

With the help of an extended mapping approach, a new type of variable separation excitation with three arbitrary functions of the (2+1)-dimensional dispersive long-water wave system (DLW) is derived. Based on the derived variable separation excitation, abundant non-propagating solitons such as dromion, ring, peakon, and compacton etc. are revealed by selecting appropriate functions in this paper.     

New Baecklund Transformation and New Exact Solutions to （2＋1）-Dimensional KdV Equation

A new Baecklund transformation for (2 1)-dimensional KdV equation is first obtained by using homogeneous balance method. And making use of the Baecklund transformation and choosing a special seed solution, we get special types of solitary wave solutions. Finally a general variable separation solution containing two arbitrary functions is constructed, from which abundant localized coherent structures of the equation in question can be induced.     

Grey Self-similar Solitary Waves in Inhomogeneous Nonlinear Media

This paper analyzes spatial grey self-similar solitary waves propagation and collision in graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities. New exact self-similar solutions are found using a novel transformation and their main features are investigated by using direct computer simulations.     

New Explicit Exact Solutions to (2+1)-Dimensional Generalized Broer-Kaup System

A nonlinear transformation and some multi-solition solutions for the (2+1)-dimensional generalized Broer-Kaup (GBK) system is first given by using the homogeneous balance method. Then starting from the nonlinear transformation, we reduce the (2+1)-dimensional GBK system to a simple linear evolution equation. Solving this equation, we can obtain some new explicit exact solutions of the original equations by means of the extended hyperbola function method.     

Optical Solitary Waves in Fourth-Order Dispersive Nonlinear Schr(o)dinger Equation with Self-steepening and Self-frequency Shift

By making use of the generalized sine-Gordon equation expansion method, we find cnoidal periodic wave solutions and fundamental bright and dark optical solitarywave solutions for the fourth-order dispersive and the quintic nonlinear Schrodinger equation with self-steepening, and self-frequency shift. Moreover, we discuss the formation conditions of the bright and dark solitary waves.     

New Compacton-Like and Solitary Pattern-Like Solutions of (2+1)-Dimensional Generalization of Modified KdV Equation

Recently some (1 1)-dimensional nonlinear wave equations with linearly dispersive terms were shown to possess compacton-like and solitary pattern-like solutions. In this paper, with the aid of Maple, new solutions of (2 1)-dimensional generalization of mKd V equation, which is of only linearly dispersive terms, are investigated using three new transformations. As a consequence, it is shown that this (2 1)-dimensional equation also possesses new compacton-like solutions and solitary pattern-like solutions.     

广义(3+1)维非线性Burgers系统孤波级数解

研究了一类广义(3+1)维非线性Burgers系统.首先,利用同伦映射方法构造了相应的映射关系式.其次,利用迭代方法得到了扰动系统的一个孤波非行波的级数解.     

(2+1)-维色散长波方程的折叠孤立波解

本文利用一种该进的映射法和线性变量分离法,得到（2+1）-维色散长波方程大量的,带有两个任意函数的精确解。并在得到的一个周期波精确解的基础上,通过选择恰当的函数,可以观察到（2+1）-维色散长波方程的折叠孤立波的演化行为。     

New Families of Rational Form Solitary Wave Solutions to (2+1)-Dimensional Broer-Kaup-Kupershmidt System

Taking the (2+1)-dimensional Broer-Kaup-Kupershmidt system as a simple example, some families of rational form solitary wave solutions, triangular periodic wave solutions, and rational wave solutions are constructed by using the Riccati equation rational expansion method presented by us. The method can also be applied to solve more nonlinear partial differential equation or equations.     

Variable Separation Approach to Solve (2 + 1)-Dimensional Generalized Burgers System:Solitary Wave and Jacobi Periodic Wave Excitations

By means of the standard truncated Painlev\'{e} expansion and a variable separation approach, a general variable separation solution of the generalized Burgers system is derived. In addition to the usual localized coherent soliton excitations like dromions, lumps, rings, breathers, instantons, oscillating soliton excitations, peakons, foldons, and previously revealed chaotic and fractal localized solutions, some new types of excitations --- compacton and Jacobi periodic wave solutions are obtained by introducing appropriate lower dimensional piecewise smooth functions and Jacobi elliptic functions.     

Some Special Types of Solitary Wave Solutions for (3+1)-Dimensional Jimbo-Miwa Equation

Using the extended homogeneous balance method, we find some special types of single solitary wave solution and new types of the multisoliton solutions of the (3+1)-dimensional Jimbo-Miwa equation.