【作者】 王伟；

【导师】 姚若侠；

【作者基本信息】 陕西师范大学 ， 计算机软件与理论， 2021， 博士

【摘要】 作为当代发展迅速的前沿科学,非线性科学对各个学科都产生了深远的影响,非线性偏微分系统已在几乎每个科学领域得到应用.在非线性科学的研究过程中,伴随着计算机代数理论的发展和符号计算系统的出现,使非线性偏微分系统得到了飞速的发展.理论研究与实验研究、数值计算与符号计算的相互融合,成为促进理论创新和重大科学发现的主要手段.特别是符号计算在符号处理、公式推演及精准计算等方面的优势,使其被广泛应用于非线性偏微分系统的研究工作中.虽然非线性偏微分系统的研究已经取得了很大发展,但借助符号计算系统,仍然有很多未知的内容在不断探索和研究中获得新发现.其中,非线性偏微分系统的各种不变性、可积性,以及精确解析解的求解等方面的研究仍然是十分重要的工作.本文以几类非线性偏微分系统为研究对象,借助符号计算系统Maple,展开非线性偏微分系统的不变性、可积性及可解性的研究.论文主要工作如下:1.基于等价活动标架方法,从微分不变量和syzygies、向量场和Lie代数、全局递推公式等方面,主要对群作用下无穷小生成子的高阶延拓和微分不变量全局递推公式展开研究.构造用于群变化作用下的无穷小生成子的高阶延拓公式,给出高阶规范化微分不变量的相关定义定理,获得高阶规范化微分不变量,重构相应的Taylor级数展开式,获得子流形上所选匹配点的规范化形式.2.从李点对称群、广义对称、对称约化和守恒可积性方面研究Whitham-Broer-Kaup(WBK)方程,基于另一种加倍其他约化过程效果的直接代数替换法.获得新的李点对称、李对称群和相应对称约化及一些新的相似解.同时给出了广义对称和一个高阶对称,证明在对称意义下WBK方程的可积性.最后,证明WBK方程无穷多守恒律的存在性,并公式化给出一些守恒密度和守恒流.3.围绕高阶非线性物理模型Sawada-Kotera(SK)方程的行波解展开研究,首次构造了(1+1)维SK方程新颖的孤子分子、孤子晶格、扭结反扭结分子、峰平台孤子分子、少周期脉冲孤子、双峰及三峰孤子等丰富的行波结构.针对一个(2+1)维SK(2DSK)类型方程,通过对数型变换v=2(lnf)xx和v=4(lnf)xx,得到一个三线性方程和一个五线性方程.利用双线性导数算子,构造了该类型2DSK型方程对应的非等价双线性化形式,获得的两个双线性方程可由标准2DSK方程和变形的Kadomtsev-Petviashvili(vKP)方程的双线性化形式组合构成.最后,根据两对Hirota双线性方程,分别获得了 2DSK型方程的两族多孤子和孤子分子.4.根据分数阶Caputo、Riemann-Liouville导数及相关修正导数,针对分数阶微分方程(fractional differential equations,FDEs)处理现实世界现象时具有的某些缺点,结合Jumarie修正的Riemann-Liouville分数阶导数的定义及其相关性质,借助(G’/G)-展开法和齐次平衡法,获得了时空分数阶Korteweg-deVries-Burgers(KdVB)方程新的精确行波解.同时,根据KdVB方程、KdV方程及Burgers方程三者之间的关系,进一步获得时空分数阶KdV方程和Burgers方程的精确行波解.更多还原

【Abstract】 As a rapidly developed frontier science in contemporary times,nonlinear science has had a profound impact on every disciplines,and nonlinear partial differential systems have been applied in almost every scientific field.In the research process of nonlinear science,along with the development of computer algebra theory and the appearance of symbolic computation system,nonlinear partial differential systems have developed rapidly.The integration of theoretical research and experimental research,numerical calculations and symbolic computations have become the main means tools to promote scientific and technological progress and major scientific discoveries.In particular,the advantages of symbolic computation in symbol processing and formula deduction and precise calculation have made symbolic computation widely used in the research of nonlinear partial differential systems.Although the research of nonlinear partial differential systems has made great progress,with the help of symbolic computation systems,there are still many unknown areas that are constantly being explored and studied.Among those areas,the invariability and integrability of nonlinear partial differential systems,as well as the solution of exact analytical solutions,are still very important research topics.In this paper,we take several types of nonlinear partial differential systems as the research object,with the help of the symbolic computation system Maple,we study the invariability,integrability and solvability of nonlinear partial differential systems.The main contents of this paper consist of the following parts:1.Based on the equivalence moving frames method and from the perspectives of differential invariants,syzygies,vector fields,Lie algebra,global recursive formulas,etc.,we mainly expand our research on the high-order prolongation of infinitesimal generators and the global recursive formulas of differential invariants under group actions.By constructing high-order prolongation formulaes for infinitesimal generators under the action of group transformations and the relevant definition theorems of high-order normalisation differential invariants,we reconstruct the corresponding Taylor series expansion,and obtain the normalisation form of the selected matching points on the sub-manifold.2.We analyze the Whitham-Broer-Kaup(WBK)equation from the standpoints of Lie point symmetry group,generalized symmetry,and symmetry reduction and conservation laws.We obtain a better Lie point symmetries,Lie symmetry groups,the related symmetry reductions and some similarity solutions by proposing a different direct algebraic substitution method that doubles the effect of other reduction processes.At the same time,we provide results on generalized symmetries and a higher-order symmetry.We also prove the integrability of the WBK equation in the sense of symmetry.Finally,we prove the existence of the infinite conservation laws for the WBK equation,and formulate some conservation densities and related fluxes.3.For the traveling wave solutions of the high-order nonlinear physical model Sawada-Kotera(SK)equation,we at the first time;obtain the structure of soliton molecules,soliton lattice,kink anti-kink molecules,peak-plateau soliton molecules,few-cycle-pulse solitons,double-peak and triple-peak solitons,and other rich traveling wave structures of(1+1)-dimensional SK equation.For the(2+1)-dimensional SK(2DSK)type equations,we obtain a trilinear equation and a quintic linear equation through the logarithmic transformation v=2(lnf)xx and v=4(lnf)xx.By using bilinear differential operators,we construct the corresponding non-equivalent bilinearizations of the 2DSK type equations and obtain the two pairs of bilinear equations that can be constructed from both the bilinear form of a standard 2DSK equation and a variant Kadomtsev-Petviashvili(vKP)equation.Finally,according to the two pairs of Hirota bilinear form equations,we obtain two hierarchies of multiple solitons and soliton molecules of the 2DSK type equation.4.Focusing on the fractional Caputo,Riemann-Liouville derivative and the related modified derivatives,realizing the shortcomings of the fractional differential equations(FDEs)in deal with the real-world phenomena,considering the Jumarie modified Riemann-Liouville fractional derivative,and utilizing the definition and related properties of(G’/G)-expansion method,we obtain a new exact traveling wave solutions of the space-time fractional Korteweg-deVries-Burgers(KdVB)equation.At the same time,we further obtain the exeact traveling wave solutions of the spacetime fractional KdV equation and the Burgers equation based on the relationships among the KdVB equation,the Korteweg-deVries(KdV)equation and the Burgers equation.更多还原