A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-BACKLUND SYMMETRY METHOD

A new method based on Lie-Backlund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained. This method is a generalization of Burde＇s Lie point symmetry technique.     

A series of new exact solutions of nonlinear evolution equations

With the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Traveling wave solutions of the generalized nonlinear evolution equations

Solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed. Some of the obtained solutions seem to be new.     

Solving evolution equations using a new iterative method

In this article, we apply the new iterative method proposed by Daftardar‐Gejji and Jafari (J Math Anal Appl 316, (2006), 753–763) for solving various linear and nonlinear evolution equations. The results obtained are compared with the results by existing methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Commutator identities on associative algebras and the integrability of nonlinear evolution equations

We show that commutator identities on associative algebras generate solutions of the linearized versions of integrable equations. In addition, we introduce a special dressing procedure in a class of integral operators that allows deriving both the nonlinear integrable equation itself and its Lax pair from such a commutator identity. The problem of constructing new integrable nonlinear evolution equations thus reduces to the problem of constructing commutator identities on associative algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 477–491, March, 2008.     

非线性发展方程新的周期解

In this paper,some new periodic solutions of nonlinear evolution equations and corresponding travelling wave solutions are obtained by using the double function method and Jacobi elliptic functions.     

应用指数函数展开法求解非线性发展方程

利用指数函数展开法,研究BBM方程与KG方程,在一个特定的变换下,借助Maple软件的符号运算功能,获得BBM方程与KG方程指数函数型新的孤立波解与周期解.这种方法用于求解非线性发展方程是简单而有效的.     

四阶非线性发展方程的精确解和广义条件对称

讨论了允许二阶广义条件对称的四阶非线性发展方程.通过广义条件对称方法得到了其对称约化和精确解.     

Exp-function method for traveling wave solutions of nonlinear evolution equations

In this paper, we apply the exp-function method to construct generalized solitary and periodic solutions of nonlinear evolution equations. The proposed technique is tested on the modified Zakharov-Kuznetsov (ZK) and Zakharov-Kuznetsov-Modified-Equal-Width (ZK-MEW) equations. These equations play a very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical results clearly indicate the reliability and efficiency of the proposed exp-function method.     

Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations

A straightforward algorithm for the symbolic computation of generalized (higher‐order) symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the generalized symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi‐discrete lattice equations. With our Integrability Package, generalized symmetries are obtained for several well‐known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of generalized symmetries exists. The existence of a sequence of such symmetries is a predictor for integrability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Superconvergence analysis of the finite element method for nonlinear hyperbolic equations with nonlinear boundary condition

This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 projection and integral identity technique. Meanwhile, the global superconvergence is obtained based on the interpolated postprocessing techniques.     

SOLITON SOLUTIONS OF A CLASS OF GENERALIZED NONLINEAR SCHRODINGER EQUATIONS

Soliton solutions of a class of generalized nonlinear evolution equations are discussed ana-lytically and numerically. This is done by using a travelling wave method to formulate one-soliton solution and the finite difference method to the numerical solutions and the interactions betweenthe solitons for the generalized nonlinear Sehrodinger equations. the characteristic behavior of thenonlinearity admintted in the system has been investigated and the soliton states of the system in thelimit when a→Oand a→∞ have been studled. The results presented show that the soliton phe-rtomenon is charaeteristics associated with the nonlinearities of the dynamical systems.     

Explicit series solutions to nonlinear evolution equations: The sine-cosine method

In this paper, a sine-cosine method is used to construct many periodic and solitary wave solutions to two nonlinear evolution systems: the coupled quadratic nonlinear equations and the coupled Klein-Gordon-Schrödinger equations. Under different parameter conditions, explicit formulas for some new periodic and solitary wave solutions are successfully obtained. The proposed solutions are found to be important for the explanation of some practical physical problems.     

Global attractor and decay estimates of solutions to a class of nonlinear evolution equations

In this work, we prove the existence of global attractor for the nonlinear evolution equation u_tt?Δu?Δu_t?Δu_tt + g(x, u)=f(x) in X=(H²(Ω)∩H(Ω)) × (H²(Ω)∩H(Ω)). This improves a previous result of Xie and Zhong in (J. Math. Anal. Appl. 2007; 336 :54–69.) concerning the existence of global attractor in H(Ω) × H(Ω) for a similar equation. Further, the asymptotic behavior and the decay property of global solution are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

A direct method of construction of two-zone solutions of nonlinear equations

We suggest an approach that allows one to effectively construct two-zone solutions, including real, of some nonlinear equations without applying the technique of algebraic curves. Thestarting point in the construction is a special addition theorem for theta-functions of two variables. The method is illustrated by the Kadomtsev-Petviashvili, KdV, and sine-Gordon equations. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 401–406, September, 1999.     

A new filled function method for nonlinear equations

In this paper, a new global optimization approach based on the filled function method is proposed for solving box-constrained systems of nonlinear equations. We first convert the nonlinear system into an equivalent global optimization problem, and then propose a new filled function method to solve the converted global optimization problem. Several numerical examples are presented and solved by using different local minimization methods, which illustrate the efficiency of the present approach.     

The Gauss-Newton method for finding singular solutions to systems of nonlinear equations

An approach to the computation of singular solutions to systems of nonlinear equations is proposed. It consists in the construction of an (overdetermined) defining system to which the Gauss-Newton method is applied. This approach leads to completely implementable local algorithms without nondeterministic elements. Under fairly weak conditions, these algorithms have locally superlinear convergence.     

A new trust region method for nonlinear equations

In this paper, a new trust region method for the system of nonlinear equations is presented in which the determining of the trust region radius incorporates the information of its natural residual. The global convergence is obtained under mild conditions. Unlike traditional trust region method, the superlinear convergence of the method is proven under the local error bound condition. This condition is weaker than the nondegeneracy assumption which is necessary for superlinear convergence of traditional trust region method. We also propose an approximate algorithm for the trust region subproblem. Preliminary numerical experiments are reported. Acknowledgements.The authors are indebted to our supervisor, Professor Y.-X. Yuan, for his excellent guidance and Jorge J. Moré for his subroutine. And we would like to thank the referees for their valuable suggestions and comments.     

A new fourth‐order numerical algorithm for a class of three‐dimensional nonlinear evolution equations

In this article, a new compact alternating direction implicit finite difference scheme is derived for solving a class of 3‐D nonlinear evolution equations. By the discrete energy method, it is shown that the new difference scheme has good stability and can attain second‐order accuracy in time and fourth‐order accuracy in space with respect to the discrete H¹ ‐norm. A Richardson extrapolation algorithm is applied to achieve fourth‐order accuracy in temporal dimension. Numerical experiments illustrate the accuracy and efficiency of the extrapolation algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A new approach for solving highly nonlinear partial differential equations by successive differentiation method

In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin–Bona–Mahony equation, Burger's equation, Fornberg–Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd.