【作者】 周艳；

【导师】 王毅； 李继彬；

【作者基本信息】 中国科学技术大学 ， 基础数学， 2019， 博士

【摘要】 本文主要应用动力系统方法研究若干非线性波方程的精确行波解及其分支问题。这些方程包括了数学物理中有重要应用的Raman孤立子方程,以及若干耦合非线性方程、离子声波模型和高阶非线性方程。本文详细分析了这些非线性方程对应的行波系统的丰富动力学性质,以及其随参数而改变的分支行为,并通过较为复杂的计算获得了系统丰富的精确行波解。针对光学超导材料中的一类Raman孤立子方程,我们利用动力系统及分支理论方法,研究该方程分别在具有Kerr非线性律和抛物非线性律情形下的精确行波解及其分支。对非线性方程具有形如q(x,t)=φ(x-vt)exp(i(-kx+ωt))的解,其中φ(ξ)为对应的奇异非线性平面动力系统的解函数,我们根据分支理论分析该平面动力系统,从而对具有Kerr非线性律的情形得到23种不同参数条件下的系统相图分支和92种不同形式的精确行波解,这些行波解包括孤立波解、周期波解、扭波和反扭波解、周期尖波解、孤立尖波解以及各种破缺波解等。而对于具有抛物非线性律情形,由于四次非线性项出现,使其精确行波解及其分支问题研究难度大为增加,我们根据分支理论对系统做更精细的刻画,获得了28个具有代表性的相图,进而得到了相应的Raman孤立子系统的62个不同形式的行波解,这些解包括孤立波解、周期波解、扭波和反扭波解、周期尖波解、孤立尖波解、伪尖波解和破缺波解以及其精确的参数表达式。其后,我们相继研究了若干耦合非线性方程、离子声波模型以及高阶非线性方程。对于耦合非线性方程组,经过计算我们发现其相应的行波系统属于第一类奇异行波系统且含有9个参数,利用分支理论和奇异行波系统理论,我们证明了存在合适的参数组使得此系统有扭波和反扭波解、周期波解、周期尖波解、破缺波解及各种不同的孤立波解。对于三个非线性离子声波模型,其控制方程分别为三个偏微分方程系统,它们的行波系统也都属于第一类奇异行波系统,通过研究行波系统的分支,我们证明了存在合适的参数组使得这些奇异行波系统有孤立波解、周期波解、伪尖波解、周期尖波解以及不同形式的破缺波解,从而完善了文[1-3]的研究结果。最后,对于五类高阶非线性方程,利用动力系统理论,我们讨论了该类方程的行波解,在Cosgrove所得公式的基础上,获得了无限多的孤立波解和拟周期波解,且给出了精确的参数表达式,同时证明了这些方程也存在无限多的双峰孤立波解,并给出了这些孤立波解存在的参数范围和几何解释。更多还原

【Abstract】 In this dissertation,the approach of dynamical system is used to discuss the ex?act travelling wave solutions and their bifurcations for some nonlinear wave equations,which have important applications in mathematics and physics.These equations include Raman soliton equations,as well as certain coupled nonlinear equations,ion-acoustic wave models and higher-order nonlinear equations.The rich dynamical behaviors and the bifurcations with the parameters of the traveling wave systems corresponding to these nonlinear equations are analyzed in detail.The exact parametric representations of different traveling wave solutions are obtained by some complex calculation.Raman soliton model in nanoscale optical waveguides,with metamaterials,having Kerr-law nonlinearity and parabolic-law nonlinearity are investigated by the method of dynamical systems and bifurcations,respectively.The exact travelling wave solutions and their bifurcations for these equations are discussed.Because the functions φ(ξ)in the solutions q(x,t)=φ(x-vt)exp(i(-kx+ωt))satisfy a singular planar dynamical system having two singular straight lines.By using the bifurcation theory of dynami-cal systems to the equations ofφ(ξ)under 23 different parameter conditions,bifurca-tions of phase portraits and 92 different exact traveling wave solutions including solitary wave solutions,periodic wave solutions,kink and anti-kink solutions,periodic peakons,peakons as well as compactons for the system are given.In the case of parabolic-law nonlinearity,it is much more difficult to study the exact travelling wave solutions and their bifurcations for the equation including extra quartic nonlinear item.According to the bifurcation theory of dynamical systems,the system is investigated more carefully,the 28 representative phase portraits are drew,and 62 different traveling wave solutions of the system such as solitary wave solutions,periodic wave solutions,kink and anti-kink solutions,periodic peakons,peakons,pseudo peakons and compactons as well as their exact parameter expressions are obtained.In addition,some coupled nonlinear equations,ion-acoustic wave models and high-order nonlinear equations are studies respectively.For the coupled nonlinear equations system,its travelling wave system is the first class singular traveling wave system de-pending on 9 parameters.By using the bifurcation theory and the method of singular traveling wave systems,it is showed that there exist parameter groups such that this sin-gular system has kink and anti-kink wave solutions,periodic wave solutions,periodic peakons and compactons as well as different solitary wave solutions.For the three ion-acoustic wave models which are governed by three partial differential equation systems respectively,their travelling wave equations also are the first class singular traveling wave systems depending on different parameter groups.By studying the bifurcations of these dynamical systems,it is showed that there exist parameter groups such that these singular systems have solitary wave solutions,periodic wave solutions,pseudo peakons,periodic peakons,as well as different compactons,which complete the stud?ies of the three papers[1-3].Lastly,for the five high-order nonlinear equations,the exact traveling wave solutions are studied by using the theory of dynamical systems.Based on Cosgrove’s work,infinitely many soliton solutions and quasi-periodic solu-tions are presented in an explicit form.The existence of uncountably infinite many double-humped solitary wave solutions is proved,and the parameters range as well as geometrical explanation of solitary wave solutions are discussed.更多还原