New Method for Finding a Series of Exact Solutions to Generalized Breaking Soliton Equation

In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like,period-form solutions of nonlinear evolution equations (NEEs). This method is more powerful than the extended tanhfunction method [Phys. Left. A 277 (2000) 212] and the modified extended tanh-function method [Phys. Left. A 285 (2001) 355]. Abundant new families of the exact solutions of Bogoyavlenskii‘s generalized breaking soliton equation are obtained by using this method and symbolic computation system Maple.     

On a New Modified Extended Tanh-Function Method

In this paper, a new modified extended tanh-function method is presented for constructing multiple soliton-like, periodic form and rational solutions of nonlinear evolution equations (NLEEs). This method is more powerful thanthe extended tanh-function method [Phys. Lett. A277 (2000) 212] and the modified extended tanh-function method[Phys. Lett. A299 (2002) 179] Abundant new solutions of two physically important NLEEs are obtained by using thismethod and symbolic computation system Maple.     

A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations

With the aid of the symbolic computation, we improve Xie's algorithm [F. Xie, Z.Y. Yan, H. Zhang, Phys. Lett. A 285 (2001) 76], and present a new extended method. Based on the new general ansatz (3), we successfully solve a compound KdV-MKdV equation, and obtain some special solutions which contain soliton solutions, and periodic solutions. The method can also be applied to other nonlinear partial differential equations.     

Extended Complex tanh-Function Method and Exact Solutions to (2+1)-Dimensional Hirota Equation

In this paper,a new extended complex tanh-function method is presented for constructing traveling wave,non-traveling wave,and coefficient functions' soliton-like solutions of nonlinear equations.This method is nore powerful than the complex tanh-function method [Chaos,Solitons and Fractals 20 (2004) 1037].Abundant new solutions of (2 1)-dimensional Hirota equation are obtained by using this method and symbolic computation system Maple.     

New Exact Solutions of the Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces

In this paper, by improving some procedure of extended tanh-function method, some new exact solutions to the integrable Broer-Kaup equations in (2+1)-dimensional spaces are obtained, which include soliton-like solutions, solitary wave solutions, trigonometric function solutions, and rational solutions.     

Comment on ``Application of the (G'/G)-Expansion Method for Nonlinear Evolution Equations' [Phys. Lett. A 372 (2008) 3400]

In this paper, some travelling wave solutions involving parameters of the Modified Zakharov-Kuznetsov equation [Phys. Lett. A 372 (2008) 3400] are investigated. We will how that these solutions are not new travelling wave solutions.     

New Exact Solutions of the (2+1)-Dimensional Asymmetric Nizhnik-Novikov-Veselov System

Using symbolic and algebra computation, the extended tanh-function method (ETM) based on mapping method is further extended. The new variable separation solutions of the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov (ANNV) system are derived.     

Explicit Exact Solutions for the (2+1)-Dimensional Higher-Order Broer-Kaup System

Using the modified extended tanh-function method, explicit and exact traveling wave solutions for the (2+1)-dimensional higher-order Broer-Kaup (HBK) system, comprising new soliton-like and period-form solutions, are obtained.     

New Exact Solutions to （2＋1）-Dimensional Dispersive Long Wave Equations

Based upon a further extended tanh method [Phys. Lett. A307 (2003) 269; Chaos, Solitons and Fractals 17 (2003) 669] and the symbolic computation system, Maple, we consider the (2 1)-dimensional dispersive long waveequations. We obtain many new solutions of the equation. These solutions contain solitomlike solutions, periodic form solutions, and some rational solutions.     

New Soliton-like Solutions and Multi-soliton Structures for Broer-Kaup System with Variable Coefficients

By using the further extended tanh method [Phys. Lett. A 307 （2003） 269; Chaos, Solitons ＆ Fractals 17 （2003） 669] to the Broer-Kaup system with variable coefficients, abundant new soliton-like solutions and multl-solitonlike solutions are derived. Based on the derived multi-soliton-like solutions which contain arbitrary functions, some interesting multi-soliton structures are revealed.     

改进的tanh函数方法与广义变系数KdV和MKdV方程新的精确解

利用改进的tanh函数方法将广义变系数KdV方程和MKdV方程化为一阶变系数非线性常微分方 程组-通过求解这个变系数非线性常微分方程组，获得了广义变系数KdV方程和MKdV方程新的 精确类孤子解、有理形式函数解和三角函数解- 关键词： 改进的tanh函数方法 类孤子解 有理形式函数解 三角函数解     

Dynamics of solitons in Bose-Einstein condensate with time-dependent atomic scattering length

The evolution of solitons in Bose--Einstein condensates (BECs) with time-dependent atomic scattering length in an expulsive parabolic potential is studied. Based on the extended hyperbolic function method, we successfully obtain the bright and dark soliton solutions. In addition, some new soliton solutions in this model are found. The results in this paper include some in the literature ({\em Phys. Rev. Lett.} {\bf 94} (2005) 050402 and {\em Chin. Phys. Lett.} {\bf 22} (2005) 1855).     

New Families of Exact Solutions for (2+1)-Dimensional Broer-Kaup System

Using a further modified extended tanh-function method, rich new families of the exact solutions for the (2+1)-dimensional Broer-Kaup (BK) system, comprising the non-traveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained.     

Exact analytical solution of a nonlinear equation arising in heat transfer

This Letter shows that the nonlinear equation arising in heat transfer recently investigated in papers [D.D. Ganji, Phys. Lett. A 355 (2006) 337; S. Abbasbandy, Phys. Lett. A 360 (2006) 109; Hafez Tari, D.D. Ganji, H. Babazadeh, Phys. Lett. A 363 (2007) 213] and [M.S.H. Chowdhury, I. Hashim, Phys. Lett. A 372 (2008) 1240] is exactly solvable, analyses the equation fully and, furthermore, gives analytic exact solution in implicit form for each value of parameters of equation.     

Log-Poisson statistics and extended self-similarity in driven dissipative systems

The Bak–Chen–Tang forest fire model [Phys. Lett. A 147 (1990) 297] was proposed as a toy model of turbulent systems, where energy (in the form of trees) is injected uniformly and globally, but is dissipated (burns) locally. We review our previous results on the model [Phys. Rev. E 62 (2000) 1613; Phys. Rev. Lett. 86 (2000) 4215] and present our new results on the statistics of the higher-order moments for the spatial distribution of fires. We show numerically that the spatial distribution of dissipation can be described by Log-Poisson statistics which leads to extended self-similarity [Phys. Rev. E. 48 (1993) R29; Phys. Rev. Lett. 73 (1994) 959]. Similar behavior is also found in models based on directed percolation; this suggests that the concept of Log-Poisson statistics of (appropriately normalized) variables can be used to describe scaling not only in turbulence but also in a wide range of driven dissipative systems.     

Some remarks on finite-gap solutions of the Ernst equation

The relationship between the class of algebro-geometrical (finite-gap) solutions of the Ernst equation constructed by Korotkin and Matveev [Theor. Math. Phys. 77 (1989) 1018; St. Petersburg Math. J. 1 (1990) 379] and the solutions recently constructed by Meinel and Neugebauer [Phys. Lett. A 210 (1996) 160] is discussed. A new formula for the general algebro-geometrical solution is obtained.     

A New Generalization of Extended Tanh-Function Method for Solving Nonlinear Evolution Equations

Making use of a new generalized ansatze and a proper transformation, we generalized the extended tanh-function method. Applying the generalized method with the aid of Maple, we consider some nonlinear evolution equations.As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extendedtanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain othernew and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profilesolitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.     

Soliton-like solutions of certain types of nonlinear diffusion-reaction equations with variable coefficient

With a view to exploring new soliton-like solutions of certain types of nonlinear diffusion-reaction (DR) equations with a variable coefficient, we demonstrate the viability of a method which is the combination of both the symbolic computation technique of Gao and Tian [Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158] and auxiliary equation method of Sirendaoreji [Sirendaoreji, Phys. Lett. A 356 (2006) 124] and used recently for the KdV equation. In particular, the DR equations with quadratic and cubic nonlinearities with a time-dependent velocity in the convective flux term are studied and the existence of soliton-like solutions is shown.     

The soliton-like solutions to the （2＋1）-dimensional modified dispersive water-wave system

By a simple transformation, we reduce the (2 1)-dimensional modified dispersive water-wave system to a simple nonlinear partial differential equation. In order to solve this equation by generalized tanh-function method, we only need to solve a simple system of first-order ordinary differential equations, and by doing so we can obtain many new soliton-like solutions which include the solutions obtained by using the conventional tanh-function method.     

Nonlinear dynamics of a zonal flow near marginal stability

The predator-prey model of the coupled spectral dynamics of drift-wave-zonal-flow turbulence is extended to the regime near the instability threshold. The problem reduces to a nonlinear integral equation, which is similar to that for the mode-particle system [H. L. Berk, Phys. Rev. Lett. 76, 1256 (1996)]], but contains two additional terms due to drift-wave self-nonlinearity. The possibility of pulsating solutions of this equation is investigated. The critical parameter for the Hopf bifurcation and the pulsation period are in reasonable agreement with the bursting behavior recently observed in the global gyrokinetic simulations [Z. Lin, Phys. Rev. Lett. 83, 3545 (1999)]].