【作者】 陈悦；

【导师】 卢殿臣；

【作者基本信息】 江苏大学 ， 数学， 2018， 硕士

【摘要】 非线性偏微分方程在物理学、化学、生物学、医学等自然学科的应用之广泛,使之成为广大学者深入研究经久不衰的课题.随着工业社会的不断发展,高阶非线性偏微分方程在图像处理及去噪、信号处理、系统控制工程等热门领域得到了很好的应用.因此,探索高阶偏微分方程的理论和解的性质是一项有意义的工作.本文主要介绍了两类高阶非线性偏微分方程:一类是广义五阶Korteweg-de Vries方程(KdV方程),另一类是广义五阶非线性Schr?dinger方程(DNLS方程).文中在应用扩展的直接代数法获得了三类五阶KdV方程的精确的行波解后,将分数阶的影响纳入考量,应用广义指数函数展开法得到了以三角函数、有理函数、双曲函数、指数函数形式给出的时间分数阶五阶KdV方程的精确的行波解,并绘制了广义五阶KdV方程解的不同形状的三维图,这对人们更直观的理解非线性波传播的运动情况有很大帮助.本文通过对五阶非线性Schr?dinger方程的研究,推导出一个拉格朗日函数以及该方程的不变变分原理,根据Darboux变换得到了五阶非线性Schr?dinger方程的扭结型孤立波解、反扭结型孤立波解、正弦孤立波解和钟型孤立波解,并应用调制不稳定性讨论了所得解的稳定性.研究结论为一类高阶偏微分方程的求解提供了一个可行且高效的方法,对解的稳定性研究为现实应用提供了理论基础.更多还原

【Abstract】 Nonlinear partial differential equations are widely used by the natural sciences such as physics,chemistry,biology,medicine,etc.which has made them become long-term and hot point for a lot of scholars to research.With the development of society,the application of high-order nonlinear partial differential equations in the hot fields of Image processing and denoising,signal processing,system control engineering,etc.Therefore,exploring further theory and obtaining solutions of high-order partial differential equations will be meaningful work.This thesis mainly introduces two types of higher-order nonlinear partial differential equations: one is generalized fifth-order Korteweg-de Vries equation KdV,,and another is generalized fifth-order nonlinear Schr?dinger equation DNLS,.In this paper,after applying the extended direct algebra method to obtain the exact traveling wave solutions of the three types of fifth-order KdV equations,the fractional order are taken into account,then the exact traveling wave solutions of time fractional order fifth-order KdV equation in the form of trigonometric functions,rational functions,hyperbolic functions and exponential functions are obtained.The three-dimensional maps of those equations’ solutions are drawn to facilitate people’s intuitive understanding of the nonlinear wave propagation.By studying the fifth-order nonlinear Schr?dinger equation,we derive a Lagrangian and the invariant variational principle for the fifthorder nonlinear Schr?dinger equation.According to the Darboux transformation,four kinds of exact solitary wave solutions for the fifth-order nonlinear Schr?dinger equation are obtained.These are the kink-type solitary wave solution,the anti-kink solitary wave solution,the sinusoidal solitary wave solution,and the bell-type solitary wave solution.At the same time,we also apply modulation instability to discuss the stability of solutions.The research conclusions provide a feasible and efficient method for solving a class of higher-order partial differential equations.The study of the stability of the solution provides a theoretical basis for practical applications.更多还原