General Solution and Fractal Localized Structures for the （2＋1）—Dimensional Generalized Ablowitz—Kaup—Newell—Segur System

Using the standard truncated Painleve expansions,we derive a quite general solution of the (2 1)-dimensional generalized Ablowitz-Kaup-Newell-Segur system.Except for the usual localized solutions,such as dromions,lumps,ring soliton solutions,etc,some special localized excitations with fractal behaviour i.e.the fractal dromion and fractal lump excitations,are obtained by some types of lower-dimensional fractal patterns.     

Fractal localized structures related to Jacobian elliptic functions in the higher-order Broer－Kaup system

This work reveals a novel phenomenon—that the localized coherent structures of a (2﹢1)﹣dimensional physical model possesses fractal behaviours. To clarify the interesting phenomenon, we take the (2﹢1)﹣dimensional higher-order Broer－Kaup system as a concrete example. Starting from a B?cklund transformation, we obtain a linear equation, and then a general solution of the system is derived. From this some special localized excitations with fractal behaviours are obtained by introducing some types of lower-dimensional fractal patterns that related to Jacobian elliptic functions.     

New localized excitations in a (2+1)-dimensional Broer—Kaup system

Starting with the extended homogeneous balance method and a variable separation approach, a general variable separation solution of the Broer—Kaup system is derived. In addition to the usual localized coherent soliton excitations like dromions, lumps, rings, breathers, instantons, oscillating soliton excitations, peakon and fractal localized solutions, some new types of localized excitations, such as compacton and folded excitations, are obtained by introducing appropriate lower-dimensional piecewise smooth functions and multiple-valued functions, and some interesting novel features of these structures are revealed.     

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.     

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.     

Hyperelliptic Function Solutions of Three Genus for KP Equation Using Direct Method

In this paper, we will use a simple and direct method to obtain some particular solutions of （2＋1）- dimensional and （3＋ 1）-dimensional KP equation expressed in terms of the Kleinian hyperelliptic functions for a given curve y^2 = f（x） whose genus is three. We observe that this method generalizes the auxiliary method, and can obtain the hyperelliptic functions solutions.     

Similarity Reductions of Nonisospectral KP Equation by a Direct Method

On bases of the direct method developed by Clarkson and Kruskal [J. Math. Phys. 27 （1989） 2201], the （2＋1）-dimensional nonisospectral Kadomtsev-Petviashvili （KP） equation has been reduced to three types of （1 ＋1）- dimensional partial differential equations. We focus on solving the third type of reduction and dividing them into three subcases, from which we obtain rich solutions including some arbitrary functions.     

Symmetry Reduction of the(2+1)-Dimensional Modified Dispersive Water-Wave System

Using the standard truncated Painlev expansion, the residual symmetry of the(2+1)-dimensional modified dispersive water-wave system is localized in the properly prolonged system with the Lie point symmetry vector. Some different transformation invariances are derived by utilizing the obtained symmetries. The symmetries of the system are also derived through the Clarkson-Kruskal direct method, and several types of explicit reduction solutions relate to the trigonometric or the hyperbolic functions are obtained. Finally, some special solitons are depicted from one of the solutions.     

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.     

Two Classes of New Exact Solutions to （2＋1）-Dimensional Breaking Soliton Equation

The singular manifold method is used to obtain two general solutions to a （2＋1）-dimensional breaking soliton equation, each of which contains two arbitrary functions. Then the new periodic wave solutions in terms of the Jacobi elliptic functions are generated from the general solutions. The long wave limit yields the new types of dromion and solitary structures.     

New periodic wave solutions,localized excitations and their interaction for （2＋l）-dimensional Burgers equation

Based on the B/icklund method and the multilinear variable separation approach （MLVSA）, this paper finds a general solution including two arbitrary functions for the （2＋1）-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for （2＋l）-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions （which contains two arbitrary functions）. Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave （called semi-localized） patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations （two-dromion solution）.     

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.     

Exact Solutions for String Cosmology

Some more general cosmological solutions of Bianchi types Ⅱ，Ⅷ，and IX for a cloud string are presented.The physical implications of the solutions are briefly discussed.Our solutions include some of results previously given in the literature as special cases.     

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.     

Comment on ＂Symbolic Computation and q-Deformed Function Solutions of the （2＋1）-Dimensional Breaking Soliton Equation＂

In a recent article [Commun. Theor. Phys. （Beijing, China） 47 （2007） 270], Cao et al. gave some nontrivial solutions of a Riccati equation by using symbolic and algebra computation. They took these solutions, which are in the form of q-deformed hyperbolic and triangular functions as new solutions. In this comment, we will show that these solutions are just the special cases of some known solutions of the Riccati equation and thus they are not new solutions.     

New exact excitations and soliton fission and fusion for the （2＋1）-dimensional Broer-Kaup-Kupershmidt system

With the help of an extended mapping approach, a series of new types of exact excitations with two arbitrary functions of the (2 1)-dimensional Broer-Kaup-Kupershmidt (BKK) system is derived. Based on the derived solitary wave excitation, some specific soliton fission and fusion solutions of the higher-dimensional BKK system are also obtained.     

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.     

New Exact Solutions and Evolution of Solitons in Background Waves for （2＋1）-Dimensional Modified Dispersive Water-Wave System

With the help of the conditional similarity reduction method, some new exact solutions of the （2＋1）- dimensional modified dispersive water-wave system （MDWW） are obtained. Based on the derived solution, we investigate the evolution of solitons in the background waves.     

Exotic Localized Coherent Structures of New（2＋1）－Dimensional Soliton Equation

The variable separation approach is used to obtain localized coherent structures of the new(2 1)-dimensional nonlinear partial differential equation.Applying the Baecklund transformation and introducing the arbitrary functions of the seed solutions,the abundance of the localized structures of this model are derived.Some special types of solutions solitoff,dromions,dromion lattice,breathers and instantons are discussed by selecting the arbitrary functions appropriately .The breathers may breath in their amplititudes,shapes,distances among the peaks and even the number of the peaks.     

Effects of Fractal Size Distributions on Velocity Distributions and Correlations of a Polydisperse Granular Gas

By the Monte Carlo method, the effect of dispersion of disc size distribution on the velocity distributions and correlations of a polydisperse granular gas with fractal size distribution is investigated in the same inelasticity. The dispersion can be described by a fractal dimension D, and the smooth hard discs are engaged in a two- dimensional horizontal rectangular box, colliding inelastically with each other and driven by a homogeneous heat bath. In the steady state, the tails of the velocity distribution functions rise more significantly above a Gaussian as D increases, but the non-Gaussian velocity distribution functions do not demonstrate any apparent universal form for any value of D. The spatial velocity correlations are apparently stronger with the increase of D. The perpendicular correlations are about half the parallel correlations, and the two correlations are a power-law decay function of dimensionless distance and are of a long range. Moreover, the parallel velocity correlations of postcollisional state at contact are more than twice as large as the precollisional correlations, and both of them show almost linear behaviour of the fractal dimension D.