A New Modified Extended Tanh-Function Method and Its Application

In this paper, a new modified extended tanh-function method is presented for constructing soliton-like,periodic form solutions of nonlinear evolution equation (NEEs). This method is more powerful than the extended tanhfunction method [Phys. Lett. A277 (2000) 212] and the moditied extended tanh-function method [Phys. Lett. A285(2001) 355]. Abundant new solutions of two physically important NEEs are obtained by using this method and symbolic computation system Maple.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

Comment on: “Solution of the Dirac equation for the Woods–Saxon potential with spin and pseudospin symmetry” [Phys. Lett. A 338 (2005) 90]

The bound-state method employed by Guo and Sheng [J.Y. Guo, Z.-Q. Sheng, Phys. Lett. A 338 (2005) 90] is shown inadequate since only one of their solutions remains compatible, in the spin-symmetric low-mass regime, with the physical boundary conditions. We clarify the problem and construct a new, correct solution in the pseudospin-symmetric regime.     

Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation

Based on the computerized symbolic system Maple and a Riccati equation, a Riccati equation expansion method is presented by a general ansatz. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the Bogoyavlenskii's generalized breaking soliton equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.     

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).     

Generalized statistical complexity measures: Geometrical and analytical properties

We discuss bounds on the values adopted by the generalized statistical complexity measures [M.T. Martin et al., Phys. Lett. A 311 (2003) 126; P.W. Lamberti et al., Physica A 334 (2004) 119] introduced by López Ruiz et al. [Phys. Lett. A 209 (1995) 321] and Shiner et al. [Phys. Rev. E 59 (1999) 1459]. Several new theorems are proved and illustrated with reference to the celebrated logistic map.     

Shape Coexistence for 179Hg in Relativistic Mean-Field Theory

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Generating Generalized Bessel Equations by Virtue of Bose Operator Algebra and Entangled State Representations

With the help of Bose operator identities and entangled state representation and based on our previous work [Phys. Lett. A 325 (2004) 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method.     

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.     

Analysis to Some Solutions Obtained by Modified Extended tanh-Function Method

First, two tanh-coth type solutions of a class of nonlinear wave equation are derived by using a simplified modified extended tanh-function method. Then, further analysis to some tanh-coth type solutions of nonlinear evolution equations are given. The results show that when balance number $ m$ is one or two, the tanh-coth type solutions obtained by the modifiedextended tanh-function method can be obtained by using the hyperbolic-function method.