Folded Localized Excitations of the （2＋1）-Dimensional Broer-Kaup Equation

Considering that the multi-valued (folded) localized excitations may appear in many (2 1)-dimensional soliton equations because some arbitrary functions can be included in the exact solutions, we use some special types of muliti-valued functions to construct folded solitrary waves and foldons in the (2 1)-dimensional Broer-Kaup equation.These folded excitations are invesigated both analytically and graphically in an alternative way.     

Multi-valued solitary waves in multidimensional soliton systems

Considering that folded phenomena are rather universal in nature and some arbitrary functions can be included in the exact excitations of many (2+1)-dimensional soliton systems, we use adequate multivalued functions to construct folded solitary structures in multi-dimensions. Based on some interesting variable separation results in the literature, a common formula with arbitrary functions has been derived for suitable physical quantities of some significant (2+1)-dimensional soliton systems like the generalized Ablowitz－Kaup－Newell－Segur (GAKNS) model, the generalized Nizhnik－Novikov－Veselov (GNNV) system and the new (2+1)-dimensional long dispersive wave (NLDW) system. Then a new special type of two-dimensional solitary wave structure, i.e. the folded solitary wave and foldon, is obtained. The novel structure exhibits interesting features not found in the single valued solitary excitations.     

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.     

Periodic Folded Wave Patterns for (2+1)-Dimensional Higher-Order Broer-Kaup Equation

A general solution including three arbitrary functions is obtained for the (2+1)-dimensional higher-order Broer-Kaup equation by means of WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and their degenerated single folded solitary waves are investigated graphically and are found to be completely elastic.     

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.     

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.     

Periodic folded waves for a （2＋1）-dimensional modified dispersive water wave equation

A general solution, including three arbitrary functions, is obtained for a (2+1)-dimensional modified dispersive water-wave (MDWW) equation by means of the WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In the long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and the degenerated single folded solitary waves are investigated graphically and found to be completely elastic.     

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.     

Folded localized excitations of the (2+1)-dimensional (M+N)-component AKNS system

Starting from the standard truncated Painlevé expansion and a multilinear variable separation approach, a quite general variable separation solution of the (2+1)-dimensional (M+N)-component AKNS (Ablowitz–Kaup–Newell–Segur) system is derived. In addition to the single-valued localized coherent soliton excitations like dromions, breathers, instantons, peakons, and a previously revealed chaotic localized solution, a new type of multi-valued (folded) localized excitation is obtained by introducing some appropriate lower-dimensional multiple valued functions. The folded phenomenon is quite universal in the real natural world and possesses quite rich structures and abundant interaction properties.     

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

Starting from a special Backlund transform and a variable separation approach, a quite general variableseparation solution of the generalized (2+ 1)-dimensional perturbed nonlinear Schrodinger system is obtained. In additionto the single-valued localized coherent soliton excitations like dromions, breathers, instantons, peakons, and previouslyrevealed chaotic localized solution, a new type of multi-valued (folded) localized excitation is derived by introducingsome appropriate lower-dimensional multiple valued functions.     

Two BT-VSA Solvable Models

Variable separation approach that is based on Backlund transformation (BT-VSA) is extended to solve the (3 1)-dimensional Jimbo-Miwa equation and the (1 1)-dimensional Drinfel'd-Sokolov-Wilson equation. New ex act solutions, which include some low-dimensional functions, are obtained. One of the low-dimensional function is arbitrary and another must satisfy a Riccati equation. Some new localized excitations can be derived from (2 1)-dimensional localized excitations and for simplification, we omit those in this letter.     

(1+1)维耦合Ito系统的单值和多值局域激发模式

在(1 1)维非线性动力学系统,人们发现不同的局域激发模式分别存在于不同的非线性系统.可是最近的若干研究表明,在高维非线性动力学系统中,如果选取适当的边值条件或初始条件时,人们可以同时找到若干不同的局域激发模式,如:紧致子、峰孤子、呼吸子和折叠子等.本文的主要目的是寻找(1 1)维非线性耦合Ito系统中的不同的局域激发模式.首先,基于一个特殊的Painlev-éBacklund变换和线性变量分离方法,求得了该系统具有若干任意函数的变量分离严格解.然后,根据得到的变量分离严格解,通过选择严格解中的任意函数,引入恰当的单值分段连续函数和多值局域函数,成功找到了耦合Ito系统若干有实际物理意义的单值和多值局域激发模式,如:峰孤子,紧致子和多圈孤子等.     

Two classes of fractal structures for the （2＋1）-dimensional dispersive long wave equation

Using the mapping approach via a Riccati equation, a series of variable separation excitations with three arbitrary functions for the (2+1)-dimensional dispersive long wave (DLW) equation are derived. In addition to the usual localized coherent soliton excitations like dromions, rings, peakons and compactions, etc, some new types of excitations that possess fractal behaviour are obtained by introducing appropriate lower-dimensional localized patterns and Jacobian elliptic functions.     

Exact Solutions of Some (1+1)-Dimensional Nonlinear Evolution Equations

By means of the variable separation method, new exact solutions of some (1+1)-dimensional nonlinear evolution equations are obtained. Abundant localized excitations can be found by selecting corresponding arbitrary functions appropriately. Namely, the new soliton-like localized excitations and instanton-like localized excitations are presented.     

Fractal Dromion, Fractal Lump, and Multiple Peakon Excitations in a New (2+1)-Dimensional Long Dispersive Wave System

By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: λqt + qxx - 2q ∫ (qr)xdy ＝ 0, λrt - rxx + 2r ∫(qr)xdy ＝ 0, is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, farctal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.     

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.     

The Exact Solutions of Some (2 + 1)-Dimensional Integrable Systems

The equivalence of three (2 1)-dimensional soliton equations is proved, and the quite generalsolutionswitha some arbitrary functions of x, t and y respectively are obtained. By selecting the arbitrary functions, many specialtypes of the localized excitations like the solitoff solitons, multi-dromion solutions, lump, and multi-ring soliton solutionsare obtained.     

(2+1)维广义Nizhnik-Novikov-Veselov系统的新严格解和复合波激发

利用拓展的Riccati方程映射法与变量分离法，得到了(2+1)维广义Nizhnik-Novikov-Veselov(GNNV)系统新的含有两个任意函数的相当广义的变量分离严格解.根据其中的周期波解，找到了该系统的复合波，即在周期波背景下的孤立波，并简要讨论了其演化行为. 关键词： GNNV系统 拓展Riccati映射 周期波解 孤立波     

The Exact Solutions of Some (2+1)-Dimensional Integrable Systems

The equivalence of three (2+1)-dimensional soliton equations is proved, and the quite general solutions with some arbitrary functions of x, t and y respectively are obtained. By selecting the arbitrary functions, many special types of the localized excitations like the solitoff solitons, multi-dromion solutions, lump, and multi-ring soliton solutions are obtained.     

Variable Separation Solutions of Generalized Broer-Kaup System via a Projective Method

Using an extended projective method, a new type of variable separation solution with two arbitrary functions of the (2+1)-dimensional generalized Broer-Kaup system (GBK) is derived. Based on the derived variable separation solution, some special localized coherent soliton excitations with or without elastic behaviors such as dromions, peakons, and foldons etc. are revealed by selecting appropriate functions in this paper.