Variable-coefficient auxiliary equation method for exact solutions of non-linear evolution equations

In this paper, a variable-coefficient auxiliary equation method is proposed to seek more general exact solutions of non-linear evolution equations. Being concise and straightforward, this method is applied to the Kawahara equation, Sawada–Kotera equation and (2+1)-dimensional Korteweg–de Vries equations. As a result, many new and more general exact solutions are obtained including Jacobi elliptic, hyperbolic and trigonometric function solutions. It is shown that the proposed method provides a straightforward and effective method for non-linear evolution equations in mathematical physics.     

Abundant travelling wave solutions for nonlinear Kawahara partial differential equation using extended trial equation method

In this paper, we use the extended trial equation method (ETEM) to construct the exact solutions in many different functions such as the trigonometric function, elliptic integral function, rational function, hyperbolic function and Jacobi elliptic functions for nonlinear evolution equation in mathematical physics via the fifth-order modified nonlinear Kawahara equation. In this method, the balance number is not constant as we have shown in other methods, but it is changed by changing the trial equation derivative definition. This method is powerful, reliable, effective and simple for solving more complicated nonlinear partial differential equations in mathematical physics.     

广义BBM方程的无穷序列新解

利用第二种椭圆方程的已知解与解的非线性叠加公式,构造了广义BBM方程的由Jacobi椭圆函数解、双曲函数和三角函数组成的无穷序列新解.     

Symbolic computation of exact solutions for the compound KdV-Sawada–Kotera equation

The generalized F-expansion method is applied to construct the exact solutions of the compound KdV-Sawada–Kotera equation by the aid of the symbolic computation system Maple. Some new exact solutions which include Jacobi elliptic function solutions, soliton solutions and triangular periodic solutions are obtained via this method.     

Exact analytic solitary wave solutions for the RKL model

Interest in finding exact solitary wave solutions of nonlinear evolution equations by means of different methods has grown steadily in recent years. These exact solutions are important to understand the mechanism of the complicated nonlinear physical phenomena. By use of the Jacobi elliptic function method, we find the exact analytic solitary wave solutions for the RKL model with cubic-quintic non-Kerr terms, describing the propagation of extremely short pulses in optical fibers. These new solutions may be useful for describing the propagation of optical pulses in non-Kerr media.     

Numerical solutions of Burgers’ equation with a large Reynolds number

In this article the exact solution of Burgers’ equation represented as an infinite series is transformed into a simpler form involving the elliptic function? ₃(υ, q). To evaluate? ₃(υ, q), we use the Jacobi Imaginary Transformation. It is made clear that the solutions obtained by the proposed approach are numerically stable and precise.     

On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function of four variables. In addition, the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found.     

Hybrid Hamilton–Jacobi–Poisson wall distance function model

Expensive to compute wall distances are used in key turbulence models and also for the modeling of peripheral physics. A potentially economical, robust, readily parallel processed, accuracy improving, differential equation based distance algorithm is described. It is hybrid, partly utilising an approximate Poisson equation. This also allows auxiliary front propagation direction/velocity information to be estimated, effectively giving wall normals. The Poisson normal can be used fully, in an approximate solution of the eikonal equation (the exact differential equation for wall distance). Alternatively, a weighted fraction of this Poisson front direction (effectively, front velocity, in terms of the eikonal equation input) information and that implied by the eikonal equation can be used. Either results in a hybrid Poisson–eikonal wall distance algorithm. To improve compatibility of wall distance functions with turbulence physics a Laplacian is added to the eikonal equation. This gives what is termed a Hamilton–Jacobi equation. This hybrid Poisson–Hamilton–Jacobi approach is found to be robust on poor quality grids. The robustness largely results from the elliptic background presence of the Poisson equation. This elliptic component prevents fronts propagated from solid surfaces, by the hyperbolic eikonal equation element, reflecting off zones of rapidly changing grid density. Where this reflection (due to poor grid quality) is extreme, the transition of front velocity information from the Poisson to Hamilton–Jacobi equation can be done more gradually. Consistent with turbulence modeling physics, under user control, the hybrid equation can overestimate the distance function strongly around convex surfaces and underestimate it around concave. If the former trait is not desired the current approach is amenable to zonalisation. With this, the Poisson element is automatically removed around convex geometry zones.     

The Jacobi elliptic function method and its application for two component BKP hierarchy equations

The periodic wave solutions for the two component BKP hierarchy are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

The Dirichlet problem for a 3D elliptic equation with two singular coefficients

In the present work, we investigate the Dirichlet problem for a three-dimensional (3D) elliptic equation with two singular coefficients. We find four fundamental solutions of the equation, containing hypergeometric functions of Appell. Then using an “a-b-c” method, the uniqueness for the solution of the Dirichlet problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition formulas, formulas of differentiation and some adjacent relations for Appell’s hypergeometric functions were used in order to find the explicit solution for the formulated problem.     

具强迫项的变系数Burgers方程的特殊孤波结构

利用推广的双曲函数展开法,得到了具强迫项的变系数Burgers方程的几组带有任意函数和任意常数的精确解.根据得到的解,分析了各种可能的孤波结构,发现了运动学特征不同于通常扭结孤立波的特殊扭结孤立波.     

Domain wall arrays, fronts, and bright and dark solitons in a generalized derivative nonlinear Schrödinger equation

We obtain some exact solutions of a generalized derivative nonlinear Schrödinger equation, including domain wall arrays (periodic solutions in terms of elliptic functions), fronts, and bright and dark solitons. In certain parameter domains, fundamental bright and dark solitons are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. Moreover, we also find the chirping reversal phenomena of fronts, and bright and dark solitons, and discuss two different ways to produce the chirping reversal.     

Soliton solutions of BLMP equation by Lie symmetry approach

Shallow water wave equations are usually described by Korteweg–de Vries (KdV)-type equations. In this paper, we have used Lie transformation group theory to solve (2+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation. We have obtained some exact solutions of BLMP equation in the explicit form through similarity reduction. All the reported results are expressed in closed form and analysed physically through their evolution profiles. The physical analysis reveals that the nature of solutions is parabolic, quasi-periodic, multisoliton and asymptotic.     

Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation

In this paper, we investigate a Cauchy problem for the semi-linear elliptic equation. This problem is well known to be severely ill-posed and regularization methods are required. We use a modified quasi-boundary value method to deal with it, and a convergence estimate for the regularized solution is obtained under an a priori bound assumption for the exact solution. Finally, some numerical results show that our given method works well.     

非线性演化方程的新Jacobi椭圆函数解

基于sinh-Gordon方程的椭圆函数解,构造新的试探解来扩展sinh-Gordon方程展开法.利用该方法研究了KdV-mKdV方程,双sine-Gordon方程和BBM方程,获得了这些方程的新Jacobi椭圆函数解.该方法也能用来求解其他数学物理中的非线性演化方程.     

广义Burgers方程的对称分类及其约化

利用李群方法对广义Burgers方程ut+f(x,t)(ux-uxx)=0的对称分类及其约化作具体讨论,其中f是关于自变量x,u的光滑函数,得到了f(x,t)的八种分类对称及相应的约化方程.该结果对于广义Burgers方程精确解的研究有重要意义.     

Auxiliary equation method for time-fractional differential equations with conformable derivative

In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.     

(2+1)维KD方程的解及分岔行为

通过行波变换,将非线性偏微分方程化为常微分方程,利用辅助常微分方程的解来构造偏微分方程的精确解,获得了(2+1)维Konopelchenko-Dubrovsky方程的孤波解和周期解.然后直接研究变换以后的常微分方程,揭示该方程控制的动力系统的鞍结分岔行为,画出了系统的分岔图.     

Symmetries and exact solutions for a 2 + 1-dimensional shallow water wave equation

Classical and nonclassical reductions of a 2 + 1-dimensional shallow water wave equation are classified. Using these reductions, we derive some exact solutions, including solutions expressed as the nonlinear superposition of solutions of a generalised variable-coefficient Korteweg-de Vries equation. Many of the reductions obtained involve arbitrary functions and so the associated families of solutions have a rich variety of qualitative behaviours. This suggests that solving the initial value problem for the 2 + 1-dimensional shallow water equation under discussion could pose some fundamental difficulties.The nonlinear overdetermined systems of partial differential equations whose solutions yield the reductions were analysed and solved using the MAPLE package diffgrob2, which we describe briefly.     

位移法浅水波方程的解及其特性

不同于传统流体力学,在Lagrange坐标下推导浅水波方程.若将水平位移作为基本变量,则推导出的浅水波数学模型可描述为固体力学的非线性大位移问题.运用不可压缩条件,通过变分原理推导出位移法浅水波方程,给出椭圆函数形式的行波解,并分析孤波解产生的条件.该基础研究建立了在分析结构力学中分析浅水波问题的理论基础,有利于进一步开展水动力学的研究.