The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term

By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries （KdV） equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.

The extended auxiliary the KdV equation with equation method for variable coefficients

This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.

EXPLICIT SOLUTIONS TO THE COUPLED KdV EQUATIONS WITH VARIABLE COEFFICIENTS

By means of sn-function expansion method and cn-function expansion method, several kinds of explicit solutions to the coupled KdV equations with variable coefficients are obtained, which include three sets of periodic wave-like solutions. These solutions degenerate to solitary wave-like solutions at a certain limit. Some new solutions are presented.

Prolongation structure of the variable coefficient KdV equation

The prolongation structure methodologies of Wahlquist-Estabrook [Wahlquist H D and Estabrook F B 1975 J. Math. Phys. 16 1] for nonlinear differential equations are applied to a variable-coefficient KdV equation. Based on the obtained prolongation structure, a Lie algebra with five parameters is constructed. Under certain conditions, a Lie algebra representation and three kinds of Lax pairs for the variable coefficient KdV equation are derived.

EXACT SOLUTIONS FOR GENERAL VARIABLE-COEFFICIENT KdV EQUATION

By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don't exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable coefficient KdV equation is given.

A new method of new exact solutions and solitary wave-like solutions for the generalized variable coefficients Kadomtsev-Petviashvili equation

Using the solution of general Korteweg-de Vries （KdV） equation, the solutions of the generalized variable coefficient Kadomtsev-Petviashvili （KP） equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.

Direct Reduction and Exact Solutions for Generalized Variable Coefficients 2D KdV Equation under Some Integrability Conditions

在同类的平衡(HB ) 之中基于关上的连接方法和 ClarksonKruskal (CK ) 方法，我们学习概括可变系数 2D KdV 方程的类似减小。同时，这在在可变系数之间的一些 integrability 条件下面导致在平常的微分方程形式的直接减小，这被显示出。二个不同盒子被讨论了，为那些平常的微分方程的解决方案的搜索产出在在可变系数之间的一些限制下面的许多准确旅行和在夸张、三角法的函数形式的 solitonic 波浪解决方案。

Exact Solutions for a Nonisospectral and Variable-Coefficient KdV Equation

The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilinear transformation from its Lax pairs and find solutions with the help of the obtained bilinear transformation.

Parallel Physics-Informed Neural Networks Method with Regularization Strategies for the Forward-Inverse Problems of the Variable Coefficient Modified KdV Equation

This paper mainly introduces the parallel physics-informed neural networks(PPINNs)method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-de Vries(VC-MKdV)equation.For the forward problem of the VC-MKdV equation,the authors use the traditional PINN method to obtain satisfactory data-driven soliton solutions and provide a detailed analysis of the impact of network width and depth on solving accuracy and speed.Furthermore,the author finds that the traditional PINN method outperforms the one with locally adaptive activation functions in solving the data-driven forward problems of the VC-MKdV equation.As for the data-driven inverse problem of the VC-MKdV equation,the author introduces a parallel neural networks to separately train the solution function and coefficient function,successfully addressing the function discovery problem of the VC-MKdV equation.To further enhance the network’s generalization ability and noise robustness,the author incorporates two regularization strategies into the PPINNs.An amount of numerical experimental data in this paper demonstrates that the PPINNs method can effectively address the function discovery problem of the VC-MKdV equation,and the inclusion of appropriate regularization strategies in the PPINNs can improves its performance.

Nonsingular Positon Solutions of a Variable-Coefficient Modified KdV Equation

The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, the nonsingular positon solutions of the variable-coefficient modified KdV equation are firstly discovered analytically and graphically.

Variable Coefficient KdV Equation for Amplitude of Nonlinear Solitary Rossby Waves in a Sort of Time-Dependent Zonal Flow

On the basis of the quasi-geostrophic vorticity equation,theoretical research has been down upon the evolution of the amplitude of solitary Rossby waves employing the perturbation method,and come to the conclusion that the evolution of the amplitude satisfies the variable coefficient Korteweg-de Vries(KdV) equation.

Solutions for non-isospectral variable coefficient KdV equation

By bilinear approach we derive N-soliton-like solutions for a variable coefficient KdV equation with some x-dependent coefficients. This equation can be considered as a non-isospectral variable coefficient KdV equation. Solutions in Hirota’s form and Wronskian form are given, respectively.

Wronskian Form of N-Solitonic Solution for a Variable-Coefficient Korteweg-de Vries Equation with Nonuniformities

By the symbolic computation and Hirota method,the bilinear form and an auto-Bcklund transformationfor a variable-coefficient Korteweg-de Vries equation with nonuniformities are given.Then,the N-solitonic solution interms of Wronskian form is obtained and verified.In addition,it is shown that the(N-1)-and N-solitonic solutionssatisfy the auto-Bcklund transformation through the Wronskian technique.

The extended symmetry approach for studying the general Korteweg-de Vries-type equation

The extended symmetry approach is used to study the general Korteweg-de Vries-type （KdV-type） equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation.

变系数KdV-mKdV组合方程的精确解

应用齐次平衡原则和辅助函数法,将变系数KdV-mKdV组合方程转化成变系数常微分方程,利用Maple软件得出变系数KdV-mKdV组合方程的几类精确解,比如有类孤子解、三角函数解、椭圆函数解和幂函数解.

Small-divisor Equation of Higher Order with Large Variable Coefficient and Application to the Coupled KdV Equation

In this paper, we establish an estimate for the solutions of small-divisor equation of higher order with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the coupled KdV equation subject to small Hamiltonian perturbations.

推广的F-展开法及变系数KdV和mKdV的精确解

该文首先推广了新近提出的F-展开法,利用该方法导出了变系数KdV和mKdV方程的类椭圆函数解;并在极限的情况下,得到变系数KdV和mKdV方程变波速和变波长的类孤子解以及其他形式解．

广义变系数五阶KdV和BBM方程的孤立子解(英文)

利用假设孤立波方法,研究了广义变系数五阶KdV方程和BBM方程,得到了广义变系数五阶KdV方程和BBM方程的孤立子解。对于得到的孤立子解,为了保证解的存在性,给出了孤立子解存在的条件。

带强迫项变系数组合kdv-Burgers方程的显式精确解

借助Mathematica软件和两个推广形式的Riccati方程组,求出了带强迫项变系数组合kdv-Burgers方程的一些精确解,包括各种类孤立波解、类周期解和变速孤立波解。

变系数组合KdV方程的新的孤立波解

在辅助方程法的基础上,给出辅助方程和函数变换相结合的一种方法,并借助符号计算系统Mathematica,获得了变系数组合KdV方程的新的孤立波解和三角函数解。这种方法在寻找其它变系数非线性发展方程的新的孤立波解和三角函数解方面具有普遍意义。