Exact Two-Soliton Solutions for Discrete mKdV Equation

An exact two-soliton solution of discrete mKdv equation is derived by using the Hirota direct approach. In addition, we plot the soliton solutions to discuss the properties of solitons. It is worth while noting that we obtain the completely elastic interaction between the two solitons.     

Exact Solutions to Triple sine-Gordon Equation

In this paper, a new transformation is introduced to solve triple sine-Gordon equation. It is shown that this intermediate transformation method is powerful to solve complex special type nonlinear evolution equation.     

Solitary Waves of a Perturbed sine—Gordon Equation

Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves,different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained.All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.     

Chaotic Behavior and Instability of Perturbed sine-Gordon Solitons

Solitons of the sine-Gordon system interacting with a disturbed external field are handled by using a direct method. Theoretical analysis reveals that the single soliton perturbed by a periodic field leads to chaotic behavior of the system, and the perturbed double soliton is unstable for most of physically interesting spacetime disturbances.     

N-Soliton-like Solution of Ito Equation

By using the extended Hirota‘s method, the N-soliton-like solution of the Ito equation is obtained. Furthermore, we also investigate the soliton-like solution interaction and find singularity.     

Lump Solution of (2+1)-Dimensional Boussinesq Equation

A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero.The particular lump solutions with specific values of the involved parameters are plotted,as illustrative examples.     

Two Types of New Solutions to KdV Equation

It is common knowledge that the soliton solutions u（x, t） defined by the bell-shape form is required to satisfy the following condition lira u（x, t） = u（±∞, t） = 0. However, we think that the above condition can be modified as lim u（x, t） = u（±∞, t）^x→ = c, where c is a constant, which is called as a stationary height of u（x, t） in the present paper.^x→∞ If u（x, t） is a bell-shape solitary solution, then the stationary height of each solitary wave is just c. Under the constraint c = 0, all the solitary waves coming from the N-bell-shape-sollton solutions of the KdV equation are the same-oriented travelling. A new type of N-soliton solution with the bell shape is obtained in the paper, whose stationary height is an arbitrary constant c. Taking c ≥ 0, the resulting solitary wave is bound to be the same-oriented travelling. Otherwise, the resulting solitary wave may travel at the same orientation, and also at the opposite orientation. In addition, another type of singular rational travelling solution to the KdV equation is worked out.     

A physics-constrained deep residual network for solving the sine-Gordon equation

Despite some empirical successes for solving nonlinear evolution equations using deep learning,there are several unresolved issues.First,it could not uncover the dynamical behaviors of some equations where highly nonlinear source terms are included very well.Second,the gradient exploding and vanishing problems often occur for the traditional feedforward neural networks.In this paper,we propose a new architecture that combines the deep residual neural network with some underlying physical laws.Using the sine-Gordon equation as an example,we show that the numerical result is in good agreement with the exact soliton solution.In addition,a lot of numerical experiments show that the model is robust under small perturbations to a certain extent.     

N-Soliton-like Solution of Ito Equation

By using the extended Hirota‘s method, the N-soliton-like solution of the Ito equation is obtained. Furthermore, we also investigate the soliton-like solution interaction and find singularity.     

Exact Jacobian Elliptic Function Solutions to sine-Gordon Equation

In this paper, four transformations are introduced to solve single sine-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the single sine-Gordon equation.     

The dissipative sine-Gordon equation

Application of the decomposition method to the generalizations of the sine-Gordon equation provide efficient analytic solution without linearization or perturbation.     

Exact Jacobian Elliptic Function Solutions to sine-Gordon Equation

In this paper, four transformations are introduced to solve single sine-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the single sine-Gordon equation.     

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.     

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.     

奇异物质与暗能量作用的sine-Gordon孤子星模型

研究了具有奇异物质和暗能量作用的sine-Gordon孤子星模型,根据场方程计算了物态方程的解和星体质量,发现物质密度和压强与孤子态和星体质量有关.另外,还对星体平衡和暗能量的稳定性质进行了分析和讨论,结果表明孤子星内部以奇异物质与暗能量的混合态形式存在. 关键词： 奇异物质 暗能量 sine-Gordon孤子星