【作者】 陈悦；

【导师】 王俊；

【作者基本信息】 江苏大学 ， 控制科学与工程， 2021， 博士

【摘要】 非线性偏微分方程可用于描述自然界中的各种非线性波动现象,它在流体力学、振动与控制、信号与图像处理、计算机通信、光学等领域有着广泛的应用。然而,到目前为止并没有一个统一的方法能够获得非线性偏微分方程的全部解。因此,发现新的求解非线性偏微分方程有效的方法成为人们研究的热门课题。本文针对多种求非线性偏微分方程精确解的方法进行了比较,基于Jacobi椭圆函数法提出了一种新的求解非线性偏微分方程有效的方法—改进的Khater法。将新方法与七种经典的求解法(包括(G’/G)-展开法、拓展tanh-函数法、Kudrvashov法、改进的Kudrvashov法、改进的tan(φ/2)-扩展方法、新颖的(G’/G)-展开方法、改进的(G’/G)-扩展方法)进行了比研,说明了新方法的有效性和优越性。并运用改进的Khater法和扩展简单方程法、F-展开法、Jacobi椭圆函数法研究了非线性长短波相互作用系统、具有断相位对称性的(3+1)维三次-五次复 Ginzburg-Landau(CQCGL)动力学系统、耦合 Boiti-Leon-Pempinelli(BLP)系统、(3+1)维 Kadomtsev-Petviashvili(KP)系统、分数阶 Kudryashov-Sinelshchikov(KS)波动系统和分数阶非线性Drinfeld-Sokolov-Wilson(DSW)系统,获得了相应系统的明孤子、暗孤子、呼吸型孤子、相互作用的多孤子、扭结波、反扭结波和三角函数解。利用标准的线性稳定分析法研究了相关系统的调制不稳定性,并推导出了色散关系和增益函数。本文针对复杂的、耦合的、含分数阶导数的非线性波动系统进行了研究,获得了相关系统的许多新解,为这些系统的应用提供了理论基础。更多还原

【Abstract】 Nonlinear partial differential equations can be used to describe various nonlinear wave phenomena in nature.Moreover,they are many applications in other fields,such as fluid mechanic,vibration and control,signal and image processing,computer communication,optics,etc.However,since there is no unified method to obtain all the solutions of nonlinear partial differential equations,it becomes hot topic to discover the new effective methods for solving nonlinear partial differential equations.In this thesis,a variety of methods for finding exact solutions of nonlinear partial differential equations are compared.Based on the Jacobi elliptic functions method,a new and effective method for solving nonlinear partial differential equations—the modified Khater method is proposed.Comparing with eight classic methods(including（G’/G)-expansion method,extended tanhfunction method,Kudryashov method,improved Kudryashov method,improved tan(φ/2)expansion method,novel(G’/G)-expansion method,improved(G’/G)-expansion method),it shows the effectiveness and superiority of the new method.The improved Khater method,the improved extended simple equation,F-expansion method and Jacobi elliptical function method are used to study the nonlinear long-short wave interaction system,the(3+1)-dimensional cubicquintic complex Ginzburg-Landau dynamical equation with broken phase symmetry,the coupled Boiti-Leon-Pempinelli system,the(3+1)-dimensional Kadomtsev-Petviashvili equation,the fractional Kudryashov-Sinelshchikov wave equation and the fractional nonlinear DrinfeldSokolov-Wilson system.Then a bright soliton,dark soliton,breathing soliton,interacting multiple solitons,kink wave soliton,anti-kink wave soliton and trigonometric function solution are obtained.The modulation instability of the related systems is studied by using the standard linear stability analysis method.Furthermore,it derives the dispersion relation and gain function.In this thesis,the solutions,numerical simulation and stability analysis of the complex nonlinear wave system,coupled nonlinear wave system,nonlinear wave system with fractional derivatives are investigated,and many new solutions of related systems have been obtained,which provides a theoretical support for the application of these physical systems.更多还原