Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

Infinite series symmetry reduction solutions to the modified KdV--Burgers equation

From the point of view of approximate symmetry, the modified Korteweg--de Vries--Burgers (mKdV--Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV--Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV--Burgers equation satisfies the Painlevé II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.     

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving...     

Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations

We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations. As concrete examples of its application, we apply this method to the (2 1)-dimensional modified Broer-Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.     

Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations

We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations, As concrete examples of its application, we apply this method to the （2＋1）-dimensional modified Broer- Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.     

远场模型方程的同伦近似对称约化

以同伦近似对称法为理论依据研究了远场模型方程, 通过归纳各阶相似约化解和各阶相似约化方程的通式构造相应的同伦级数解. 各阶相似约化方程均为线性变系数常微分方程, 并且可以从零阶开始依次求解. 同伦模型中的辅助参数影响同伦级数解的收敛性. 关键词： 同伦近似对称法 远场模型方程 同伦级数解     

New Similarity Reductions and Compacton Solutions for Boussinesq-Like Equations with Fully Nonlinear Dispersion

In this paper, similarity rcductions of Boussinesq-like equations with nonlinear dispersion (simply called B(m, n) equations) utt = (un)xx (um) which is a generalized model of Boussinesq equation uts = (u2)xx u and modified Bousinesq equation utt = (u3)xx uxxxx, are considered by using the direct reduction method. As a result,several new types of similarity reductions are found. Based on the reduction equations and some simple transformations,we obtain the solitary wave solutions and compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they re-emerge with the same coherent shape) of B(1, n) equations and B(m, m)equations, respectively.``     

New Similarity Reduction Solutions for the(2+1)-Dimensional Nizhnik-Novikov-Veselov Equation

In this paper,some new formal similarity reduction solutions for the(2+1)-dimensional Nizhnik-Novikov-Veselov equation are derived.Firstly,we derive the similarity reduction of the NNV equation with the optimal system of the admitted one-dimensional subalgebras.Secondly,by analyzing the reduced equation,three types of similarity solutions are derived,such as multi-soliton like solutions,variable separations solutions,and KdV type solutions.     

New Similarity Reductions and Compacton Solutions for Boussinesq-Like Equations with Fully Nonlinear Dispersion

In this paper, similarity reductions of Boussinesq-like equations with nonlinear dispersion (simply called B(m,n) equations) u_tt=(uⁿ)_xx+(u^m)_xxxx, which is a generalized model of Boussinesq equation u_tt=(u²)_xx+u_xxxx and modified Bousinesq equation u_tt=(u³)_xx+u_xxxx, are considered by using the direct reduction method. As a result, several new types of similarity reductions are found. Based on the reduction equations and some simple transformations, we obtain the solitary wave solutions and compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they re-emerge with the same coherent shape) of B(1,n) equations and B(m,m) equations, respectively.     

New Exact Solutions of Broer-Kaup Equations

By a known transformation, (2 1)-dimensional Brioer Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions are performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.     

New Exact Solutions of Broer-Kaup Equations

By a known transformation, (2 1)-dimensional Brioer-Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions axe performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.     

Approximate Homotopy Symmetry Reduction Method: Infinite Series Reductions to Kawahara Equation

The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.     

Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method

<正>This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method.Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders,showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method.The homotopy series solutions to the generalized Kawahara equation are consequently derived.     

具有阻尼项的非线性波动方程的相似约化

利用Clarkson和Kruskal引入的直接约化法,给出了具有阻尼项的非线性波动方程u_tt-2bu_xxt+αu_xxxx=β(uⁿ_x)x(α>0,β≠0,n≥2)三种类型的相似约化.从这些约化方程的Painlevé分析表明该方程在Ablowitz的猜测意义下是不可积的.此外还获得了该方程(n=2)的4种精确类孤波解. 关键词： 波动方程 相似约化 Painlevé分析 精确解     

Truncated series solutions to the(2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method

In this paper, the(2+1)-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional(2 D) similarity reduction equations by using the approximate symmetry method. A step-by-step procedure is used to acquire Jacobi elliptic function solutions to these similarity equations, which generate the truncated series solutions to the original perturbed Boussinesq equation. Aside from some singular area, the series solutions are convergent when the perturbation parameter is diminished.     

An extension of the direct method and similarity reductions of a generalized Burgers equation with an arbitrary derivative function

In this paper,we extend the well-known direct method proposed by Clarkson and Kruskal for finding similarity reductions of partial differential equations.It follows that some new similarity reductions of the generalized Burgers equation,such as travelling wave reduction,logarithmic reduction,power reduction,rational fractional reduction,etc,are derived,in which some of these cannot be obtained solely by using the direct method.The similarity reductions obtained are interpreted by the nonclassical symmetry Lie group.     

Approximate Symmetries and Infinite Series Symmetry Reduction Solutions to Perturbed Kuramoto-Sivashinsky Equation

Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here.     

Similarity structure of wave-collapse

Similarity transformations of the cubic Schrödinger equation (CSE) are investigated. The transformations are used to remove the explicit time variation in the CSE and reduce it to differential equations in the spatial variables only. Two different methods for similarity reduction are employed and the significance of similarity in the evolution of a collapsing wave packet is investigated. Numerical solutions in radial symmetry demonstrate that the similarity behaviour is local in space and time, and that some similarity solutions must be classified as improper solutions. The nature of the collapsing singularity is reexamined.     

Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach

Bosonization approach is applied in solving the most general N=1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a (sKdV-a) equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosonic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are simply obtained for all values of a. Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.     

Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach

Bosonization approach is applied in solving the most general N=1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a (sKdV-a) equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosonic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are simply obtained for all values of a. Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.