New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations

In this paper, we construct many new types of Jacobi elliptic function solutions of nonlinear evolution equations using the so-called new extended auxiliary equation method. The effectiveness of this method is demonstrated by applications to three higher order nonlinear evolution equations, namely, the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, the higher order dispersive nonlinear Schrödinger equation and the generalized nonlinear Schrödinger equation. The solitary wave solutions and periodic solutions are obtained from the Jacobi elliptic function solutions. Comparing our new results and the well-known results are given.     

A deep learning method for solving third-order nonlinear evolution equations

It has still been difficult to solve nonlinear evolution equations analytically. In this paper, we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly. Specifically, the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters. In particular, numerical experiments on several third-order nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation, modified KdV equation, KdV–Burgers equation and Sharma–Tasso–Olver equation, demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.     

An extended functional transformation method and its application in some evolution equations

In this paper, an extended functional transformation is given to solve some nonlinear evolution equations. This function, in fact,is a solution of the famous KdV equation, so this transformation gives a transformation between KdV equation and other soliton equations. Then many new exact solutions can be given by virtue of the solutions of KdV equation.     

试探方程法及其在非线性发展方程中的应用

提出了一种比较系统的求解非线性发展方程精确解的新方法, 即试探方程法. 以一个带5阶 导数项的非线性发展方程为例, 利用试探方程法化成初等积分形式，再利用三阶多项式的完 全判别系统求解，由此求得的精确解包括有理函数型解, 孤波解, 三角函数型周期解, 多项 式型Jacobi椭圆函数周期解和分式型Jacobi椭圆函数周期解 关键词： 试探方程法 非线性发展方程 孤波解 Jacobi椭圆函数 周期解     

An extension of the modified Sawada—Kotera equation and conservation laws

Based on the modified Sawada-Kotera equation, we introduce a 3 × 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawada-Kotera equation, by which a Lax pair of the modified Sawada-Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawada-Kotera equation, the Kaup-Kupershmidt equation, and the modified Sawada-Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada-Kotera equation are constructed with the aid of their Lax pairs.     

The structure of higher symmetries of nonlinear evolution equations

Within the framework of jet manifolds, we show that the symmetries of nonlinear partial evolution equations in arbitrary dimensions are linear in the leading orders. A necessary condition for the existence of an infinite-dimensional symmetry algebra for a given equation is derived. As an example, the results for a class of nonlinear diffusion equations in (1+2) dimensions are given.     

一类非线性发展方程的复合型双孤子新解

通过下列步骤,构造了一类非线性发展方程的无穷序列复合型双孤子新解: 步骤一, 给出两种函数变换,把一类非线性发展方程化为二阶非线性常微分方程; 步骤二, 再通过函数变换, 二阶非线性常微分方程转化为一阶非线性常微分方程组,并获得了该方程组的首次积分; 步骤三, 利用首次积分与两种椭圆方程的新解与Bäcklund 变换, 构造了一类非线性发展方程的无穷序列复合型双孤子新解.     

圆杆波导中的一个非线性波动方程及准确周期解

在小变形条件下，采用Cox的非线性应力应变关系，计及横向Possion效应，借助Hamilton变分原理导出了非线性弹性圆杆波导中的纵向波动方程. 利用Jacobi椭圆余弦函数展开法，对该方程与截断的非线性波动方程进行求解，得到了两类非线性波动方程的准确周期解，它们可以进一步退化为孤波解. 关键词： 非线性波 Possion效应 Jacobi椭圆余弦函数     

Trial Equation Method to Nonlinear Evolution Equations with Rank Inhomogeneous:Mathematical Discussions and Its Applications

A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation, generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.     

The auxiliary elliptic-like equation and the exp-function method

The auxiliary equation method is very useful for finding the exact solutions of the nonlinear evolution equations. In this paper, a new idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the auxiliary elliptic-like equation are derived using exp-function method, and then the exact solutions of the nonlinear evolution equations are derived with the aid of auxiliary elliptic-like equation. As examples, the RKL models, the high-order nonlinear Schrödinger equation, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system and the generalized ZK-BBM equation are investigated and the exact solutions are presented using this method.     

Weierstrass Elliptic Function Solutions to Nonlinear Evolution Equations

This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation. Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations.     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

Analytic treatment of nonlinear evolution equations using first integral method

In this paper, we show the applicability of the first integral method to combined KdV?CmKdV equation, Pochhammer?CChree equation and coupled nonlinear evolution equations. The power of this manageable method is confirmed by applying it for three selected nonlinear evolution equations. This approach can also be applied to other nonlinear differential equations.     

A Modified Homogeneous Balance Method and Its Applications

A modified homogeneous balance method is proposed by improving some key steps in the homogeneous balance method. Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneous balance method. Generalized Boussinesq equation, KP equation, and mKdV equation are chosen as examples to llustrate our method. This approach is also applicable to a large variety of nonlinear evolution equations.     

The Fredholm determinant method for discrete integrable evolution equations

The Fredholm determinant method defines in an explicit way a mapping from a linear evolution equation to a nonlinear soliton equation. Here, the method is extended to discrete soliton equations like the Toda and Langmuir lattice equations. Though the discrete version looks very similar to the continuous one, the proof is quite different. Explicit solution formulas are given, and the continuous limiting case is considered.     

一类高维耦合的非线性演化方程的简单求解

利用一个简单的变换，一类高维耦合的非线性演化方程可以被约化为一低维的简单方程，将已有的求解法应用于简单方程，十分简捷的获得了原方程大量的精确解. 关键词： 非线性耦合方程 精确解 tanh函数方法     

Beam-core evolution equation and space-charge limit

The nonlinear beam-core evolution equation approach is proposed as a powerful tool to estimate the acceptable beam current in a given circular accelerator. The approach is justified by the macroparticle simulation over a wide beam current parameter region. Space-charge effects on the beam-core evolution are discussed by Poincaré mapping on the beam-core phase space (σ, dσ/ds). The instability seen in the beam-core evolution is rigorously analyzed as an eigenvalue problem in the coupled linear system derived from the linearized beam-core evolution equations. A threshold current resulting in the instability is given by both the nonlinear beam-core evolution equation approach and the coupled linear system approach. As an example, a fast cycling induction synchrotron is evaluated in which the space-charge effects are significant because it is injector-free.     

New periodic wave solutions via Exp-function method

The Exp-function method with the aid of symbolic computational system is used to obtain the generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics, namely, nonlinear partial differential (BBMB) equation, generalized RLW equation and generalized shallow water wave equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

构造非线性发展方程无穷序列类孤子精确解的一种方法

辅助方程法已构造了非线性发展方程的有限多个新精确解. 本文为了构造非线性发展方程的无穷序列类孤子精确解, 分析总结了辅助方程法的构造性和机械化性特点. 在此基础上,给出了一种辅助方程的新解与Riccati方程之间的拟Bäcklund变换. 选择了非线性发展方程的两种形式解,借助符号计算系统 Mathematica,用改进的(2+1) 维色散水波系统为应用实例,构造了该方程的无穷序列类孤子新精确解. 这些解包括无穷序列光滑类孤子解, 紧孤立子解和尖峰类孤立子解.