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.     

Second-order approximate symmetry classification and optimal system of a class of perturbed nonlinear wave equations

A complete second-order approximate symmetry classification of a class of perturbed nonlinear wave equations with a arbitrary function is performed by means of the method originated from Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is given. Based on the Lie symmetry reduction method, large classes of approximate reductions and invariant solutions of the equations are constructed.     

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations is performed. First, complete infinite-order approximate symmetry classification of the equation is obtained by means of the method originated by Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is derived and used to construct general formulas of approximate symmetry reductions and similarity solutions. Second, we study approximate homotopy symmetry of the equation and construct connections between the two symmetry methods for the first-order and higher-order cases, respectively. The series solutions derived by the two methods are compared.     

The simplest equation method to study perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

The simplest equation method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations.In this paper, the simplest equation method is used to construct exact solutions of nonlinear Schrödinger’s equation and perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity. It is shown that the proposed method is effective and general.     

齐次平衡法若干新的应用

齐次平衡法是求非线性发展方程孤波解的一种有效方法．该文将以KdV方程为例把齐次平衡法向三个方面拓广应用：1）获得非线性发展方程新的具有更为丰富形式的精确解;2）寻找非线性发展方程的Backlund变换、Lax表示;3）求非线性发展方程的对称性约化和相似解．     

Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System

Conditional Lie‐Bäcklund symmetry (CLBS) method is developed to study system of evolution equations. It is shown that reducibility of a system of evolution equations to a system of ordinary differential equations can be fully characterized by the CLBS of the considered system. As an application of the approach, a class of two‐component nonlinear diffusion equations is studied. The governing system and the admitted CLBS can be identified. As a consequence, exact solutions defined on the polynomial, exponential, trigonometric, and mixed invariant subspaces are constructed due to the corresponding symmetry reductions.     

Group Classifications, Symmetry Reductions and Exact Solutions to the Nonlinear Elastic Rod Equations

In this paper, the Lie symmetry analysis are performed on the three nonlinear elastic rod (NER) equations. The complete group classifications of the generalized nonlinear elastic rod equations are obtained. The symmetry reductions and exact solutions to the equations are presented. Furthermore, by means of dynamical system and power series methods, the exact explicit solutions to the equations are investigated. It is shown that the combination of Lie symmetry analysis and dynamical system method is a feasible approach to deal with symmetry reductions and exact solutions to nonlinear PDEs.     

Solutions of Two Nonlinear Evolution Equations Using Lie Symmetry and Simplest Equation Methods

In this paper, we study two nonlinear evolution partial differential equations, namely, a modified Camassa–Holm–Degasperis–Procesi equation and the generalized Korteweg–de Vries equation with two power law nonlinearities. For the first time, the Lie symmetry method along with the simplest equation method is used to construct exact solutions for these two equations.     

The formation of trapped surfaces in spherically-symmetric Einstein–Euler spacetimes with bounded variation

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein–Euler equations of general relativity. We formulate the initial value problem in Eddington–Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solutions to the Einstein equations. This result generalizes a theorem by Christodoulou for regular vacuum spacetimes (but without symmetry restriction). Our method of proof relies on a generalization of the “random choice” method for nonlinear hyperbolic systems and on a detailed analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.     

Symmetry group analysis and similarity solutions of variant nonlinear long-wave equations

Symmetry group properties and similarity solutions of the variant nonlinear long-wave equations in the form of system of nonlinear partial differential equations are analyzed. Lie symmetry group analysis of the variant nonlinear long-wave equations presents that the system has only two-parameter point symmetry group that corresponds to only traveling wave solutions. The symmetry groups yield the general reduced similarity form of the system, which is in the system of nonlinear ordinary differential equations. By using the improved tanh method the similarity solutions are obtained from the reduced system of equations. In addition, some graphical representations of the solitary and periodic solutions are presented.