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.     

Embedded-Soliton and Complex Wave Excitations of （3＋1）-Dimensional Burgers System

Starting from the extended mapping approach and a linear variable separation method, we find new families of variable separation solutions with some arbitrary functions for the （3＋1）-dimensionM Burgers system. Then based on the derived exact solutions, some novel and interesting localized coherent excitations such as embedded-solitons, taper-like soliton, complex wave excitations in the periodic wave background are revealed by introducing appropriate boundary conditions and/or initial qualifications. The evolutional properties of the complex wave excitations are briefly investigated.     

Chaos and Fractals in a （2＋1）—Dimensional Soliton System

Considering that there are abundant coherent solitent soliton excitations in high dimensions,we reveal a novel phenomenon that the localized excitations possess chaotic and fractal behaviour in some(2 1)-dimensional soliton systems.To clarify the interesting phenomenon,we take the generalized(2 1)-dimensional Nizhnik-Novikov-Vesselov system as a concrete example,A quite general variable separation solutions of this system is derived via a variable separation approach first.then some new excitations like chaos and fractals are derived by introducing some types of lower-dimensional chaotic and fractal patterns.     

A Novel Class of Coherent Localized Structures for the Maccari System

By means of the Baecklund transformation, a quite general variable separation solution of the (2 1)-dimensional Maccari systems is derived. In addition to some types of the usual localized excitations such as dromion, lumps, ring soliton and oscillated dromion, breathers solution, fractal-dromion, fractal-lump and chaotic soliton structures can be easily constructed by selecting the arbitrary functions appropriately, a new novel class of coherent localized structures like peakon solution and compacton solution of this new system are found by selecting apfropriate functions.     

Localized Coherent Structures with Chaotic and Fractal Behaviors in a （2＋l）－Dimensional Modified Dispersive Water－Wave System

In this work, we reveal a novel phenomenon that the localized coherent structures of some (2 1)-dimensional physical models possess chaotic and fractal behaviors. To clarify these interesting phenomena, we take the (2 l)-dimensional modified dispersive water-wave system as a concrete example. Starting from a variable separation approach,a general variable separation solution of this system is derived. Besides the stable located coherent soliton excitations like dromions, lumps, rings, peakons, and oscillating soliton excitations, some new excitations with chaotic and fractal behaviors are derived by introducing some types of lower dimensional chaotic and fractal patterns.     

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.     

Instantaneous solitons and fractal solitons for a (2+1)-dimensional nonlinear system

By an improved projective equation approach and a linear variable separation approach, a new family of exact solutions of the (2+1)-dimensional Broek--Kaup system is derived. Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as instantaneous solitons and fractal solitons are investigated.     

Solitons in a generalized (2+1)-dimensional Ablowitz－Kaup－Newell－Segur system

In the previous Letter (Zheng C L and Zhang J F 2002 Chin. Phys. Lett. 19 1399), a localized excitation of the generalized Ablowitz－Kaup－Newell－Segur (GAKNS) system was obtained via the standard Painlevé truncated expansion and a special variable separation approach. In this work, starting from a new variable separation approach, a more general variable separation excitation of this system is derived. The abundance of the localized coherent soliton excitations like dromions, lumps, rings, peakons and oscillating soliton excitations can be constructed by introducing appropriate lower-dimensional soliton patterns. Meanwhile we discuss two kinds of interactions of solitons. One is the interaction between the travelling peakon type soliton excitations, which is not completely elastic. The other is the interaction between the travelling ring type soliton excitations, which is completely elastic.     

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.     

Localized Excitations in a Sixth-Order （1＋1）-Dimensional Nonlinear Evolution Equation

In this letter, by means of the Lax pair, Darboux transformation, and variable separation approach, a new exact solution of a sixth-order （1＋1）-dimensional nonlinear evolution equation, which includes some arbitrary functions, is obtained. Abundant new localized excitations can be found by selecting appropriate functions and they are illustrated both analytically and graphically.     

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

Starting from a special Baecklund transform and a variable separation approach, a quite general variable separation solution of the generalized ( 2 1 )-dimensional perturbed nonlinear Schroedinger system is obtained. In addition to the single-valued localized coherent soliron 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.     

Multi Dromion-Solitoff and Fractal Excitations for （2＋1）-Dimensional Boit i-Leon-Manna-Pempinelli System

In this short note, a new projective equation （Ф = σФ ＋ Ф^2） is used to obtain the variable separation solutions with two arbitrary functions of the （2＋1）-dimensional Boiti-Leon-Manna Pempinelli system （BLMP）. Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as multi dromion-solitoffs and fractal-solitons are investigated.     

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.     

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.     

General Solution and Localized Coherent Soliton Structures of the (2+1)-Dimensional Generalized Davey-Stewarson Equations

In this paper, the variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized Davey-Stewarson equations: iqt 1/2(qxx qyy) (R S)q = O, Rx=-σ/2|q|2y Sy = -σ/2|q|2/x.Applying a special Backlund transformation and introducing arbitrary functions of the seed solutions, an abundance of the localized structures of this model is derived. By selecting the arbitrary functions appropriately, some special typesof localized excitations such as dromions, dromion lattice, breathers, and instantons are constructed.     

General Solution and Localized COherent Soliton Structures of the （2＋1）—Dimensional Generalized Davey—Stewarson Equations

In this paper,the variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized Davey-Stewarson equations:iqt 1/2(qxx=qyy) (R S)q=0,Rx=-σ/2|q|y^2,Sy=-σ/2|q|2/x.Applying a special Baecklund transformation and introducing arbitrary functions of the seed solutions.and abundance of the localized structures of this model is derived,By selceting the arbitrary functions appropriately,some special types of localized excitations such as dromions,dromion lattice,breathers,and instantons are constructed.     

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.     

Semifolded Localized Structures in Three-Dimensional Soliton Systems

By means ofa Painlev6 Backlund transformation and a multi-linear variable separation approach, abundant localized coherent excitations of the three-dimensional Broer Kaup Kupershmidt system with variable coeft~cients are derived. There are possible phase shifts for the interactions of the three-dimensional novel localized structures discussed in this paper.     

Complex solutions and novel complex wave localized excitations for the(2+1)-dimensional Boiti–Leon–Pempinelli system

With the help of the symbolic computation system Maple, the Riccati equation mapping approach and a linear variable separation approach, a new family of complex solutions for the （2＋ 1）-dimensional Boiti-Leon-Pempinelli system （BLP） is derived. Based on the derived solitary wave solution, some novel complex wave localized excitations are obtained.     

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.