Exact Vacuum Solutions to the Einstein Equation

In this paper,the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation.This procedure of resolution is based on a canonical form of the metric.According to this procedure,the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations, which are much convenient for the resolution.     

Solitons and Other Solutions for the Generalized KdV–mKdV Equation with Higher-order Nonlinear Terms

The generalized sub-ODEmethod, the rational (G' ⁄ G)-expansionmethod, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations (PDEs). Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV (GKdV-mKdV) equation with higher-order nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.     

Exact Periodic Solutions to Generalized BBM Equation and Relevant Conclusions

In this paper, we consider generalized BBM equation with nonlinear terms of high order.In the case of p=1/2, p=1 and p=2, the exact periodic solutions to G-BBM equation are obtained by means of proper transformation, which degrades the order of nonlinear terms. And we prove that if p ≠ 1/2,p ≠ 1 or p ≠ 2, G-BBM equation does not exist this kind of periodic solution.     

Abundant Exact Explicit Solutions to a Modified cKdV Equation

In this paper, we construct abundant exact explicit solutions to a modi?ed cKdV equation by employing the three forms of (ω/g)-expansion method, i.e., (g0/g2)-expansion method, (g0/g)-expansion method and (g0)expansion method. The solutions obtained are under di?erent constraint conditions and are in the form of hyperbolic, trigonometric and rational functions, respectively, including kink (antikink) wave solutions, singular wave solutions and periodic singular wave solutions which have potential applications in physical science and engineering.     

New Multiple Soliton Solutions of High-Dimensional Nonlinear Klein-Gordon Equation

The generalized nonlinear K-G equation can be written as The well known sine-Gordon(F(φ)=sinφ),φ~4(F(φ)=1/2(φ~3—φ))are just the special cases.After the method proposed by Gibbon et.al(Phys.Lett.A65(1978)380),we introduce the base equation     

Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Ampère Equation

By means of the classical symmetry method,a hyperbolic Monge-Ampère equation is investigated.The symmetry group is studied and its corresponding group invariant solutions are constructed.Based on the a...     

Exact Solutions and Integrability of the Duffing – Van der Pol Equation

The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painlevé test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations.     

Solitons and Other Solutions for the Nonlinear Convection–Diffusion–Reaction Equation with Power-Law Nonlinearity by the Extended Simplest Equation Method

Computational Mathematics and Modeling - Based on the extended simplest equation method, we construct solitons and other solutions for the nonlinear convection-diffusion-reaction equation with...     

1＋2维Caudrey—Dodd—Gibbon—Kotera—Sawada方程的非线性叠加公式

本文给出了1+2维Caudrey-Dodd-Gibbon-Kotera-Sawada方程(CDGKS)的一个B(?)cklund变换(BT)并导出了一个新颖的解的非线性叠加公式,借助于所得到的BT和非线性叠加公式,我们生成了1+2维CDGKS方程的一些新的特解。     

Lie Symmetry Analysis and Exact Solutions for?the?Extended mKdV Equation

By using Lie symmetry analysis and the method of dynamical systems for the extended mKdV equation, the all exact solutions based on the Lie group method are given. Especially, the bifurcations and traveling wave solutions are obtained. To guarantee the existence of the above solutions, all parameter conditions are determined. Furthermore, the exact analytic solutions are considered by using the power series method. Such solutions for the equation are important in both applications and the theory of nonlinear science.     

Backlund Transformation and Exact Solution to Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the homogeneous balance principle, we derive a Backlund transformation (BT) to (3+1)-dimensionaI Kadomtsev-Petviashvili (K-P) equation with variable coefficients if the variable coefficients are linearly dependent. Based on the BT, the exact solution of the (3+1)-dimensional K-P equation is given. By the same method, we derive a BT and the solution to (2+1)-dimensional K-P equation. The variable coefficients can change the amplitude of solitary wave, but cannot change the form of solitary wave.     

Decay of Solutions to Damped Korteweg–de Vries Type Equation

In the present paper we establish results concerning the decay of the energy related to the damped Korteweg–de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when a initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H ^k -level, k∈ℕ. In addition, the decay of the energy regarding a initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect.     

Singular Solutions to Monge-Ampère Equation

We construct merely Lipschitz and $C^{1,α}$ with rational $α ∈ (0, 1 − 2/n]$ viscosity solutions to the Monge-Ampère equation with constant right hand side.     

Exact Solitary-wave Solutions and Periodic Wave Solutions for Generalized Modified Boussinesq Equation and the Effect of Wave Velocity on Wave Shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.     

Exact Solutions for Self-dual Yang–Mills Equations

The (constrained) canonical reduction of four dimensional self-dual Yang–Millstheory to 2, (2+1) dimensional sine-Gordon theory and 2 dimensional Liouvilles theory areconsidered. The Bäcklund transformations (BTs) areimplemented to obtain new classes of exact solutions for the reduced 2 dimensional sine-Gordonand Liouville models. Another transformation is developed and used to obtain exact solution forthe 2+1 and the original 3+1 sine-Gordon models.     

The Thomas–Fermi Problem and Solutions of the Emden–Fowler Equation

Computational Mathematics and Mathematical Physics - A two-point boundary value problem is considered for the Emden–Fowler equation, which is a singular nonlinear ordinary differential...