【作者】 于亚璇；

【导师】 张鸿庆；

【作者基本信息】 大连理工大学 ， 计算数学， 2006， 博士

【摘要】 本文主要作了以下三方面的研究:首先,借助于符号计算和吴方法,研究了非线性微分-差分方程的精确解,提出了双曲函数有理展开法和有理形式的展开法,并推广了非线性发展方程的椭圆函数有理展开法。其次,在Hirota双线性算子推广到超对称的情形下,给出了许多重要的超对称双线性恒等式,并应用它们求得了B(?)cklund变换和孤波解。最后,为了今后在更广泛的空间中研究非线性问题,我们讨论了局部凸空间中凸性与光滑性之间的关系。 第一章主要介绍了本文所涉及到的学科(包括孤立子理论、数学机械化、局部凸空间的凸性与光滑性等)的起源及发展过程,以及国内外学者在这些方面所做的工作和已经取得的一些成果。最后介绍了本文的主要工作。 第二章主要阐述了求解非线性发展方程的AC=BD模式及其应用。首先给出了c-D对和c-D可积系统的基本理论以及构造c-D对的方法.然后把AC=BD理论应用于微分-差分方程和微分方程的双线性形式,这样就给AC=BD理论增加了新的更丰富的内容。 第三章以符号计算软件Maple为工具研究了微分-差分方程的行波解,孤波解,周期解等。推广了双曲函数展开法,提出了微分-差分方程的双曲函数有理展开法,进一步提出有理形式的展开法,并应用这些方法研究了各类Toda晶格方程、Hybrid晶格方程、Ablowitz-Ladik晶格方程和Volterra晶格方程,得到了丰富的新的精确解。 第四章基于非线性发展方程求解代数化,算法化,机械化的指导思想,以吴方法和符号计算为工具,推广了求解非线性发展方程的椭圆函数有理展开法,求解了反对称NizhnikNovikov-Veselov方程、Davey-Stewartson方程和Hirota-Satsuma耦合KdV方程,得到了丰富的双周期解,周期解和三角函数解。 第五章首先简单回顾了Hirota双线性算子的定义和性质,并把Hirota双线性算子推广到超对称的情况,给出了许多新的重要的超对称双线性恒等式。然后,根据物理意义对方程进行了超对称延拓,并由此研究了N=1的超对称Sawada-Kotera-Ramani方程的B(?)cklund变换和孤波解。 第六章引入半范数族P的S-最简化和P-自反局部凸空间的新概念,证明了半范数族P和它的S-最简化不仅生成X上相同的局部凸分离拓扑,而且具有相同的凸性和光滑性,讨论了P-自反与自反的关系,并指出当X是赋范线性空间时,P-自反和自反是更多还原

【Abstract】 This dissertation has mainly done the following three aspects research: First, with the aid of symbolic computation and Wu method, the exact solutions of some nonlinear differential-difference equations have been studied. The hyperbolic function rational expansion method and the rational formal expansion method for them are put forward. And elliptic function rational expansion method for nonlinear evolutional equations is developed. Next, in Hirota bilinear operator extended to the supersymmetrical situation, many important supersymmetrical bilinear identical equations have been produced. And we applied them to obtain Backlund transformation and solitary wave solutions. Finally, for the following studies and in order to study the nonlinear problems in a widespread space, the relationship between the convexity and the smoothness is studied in the locally convex space.Chapter 1 mainly introduces the origin and development of several subjects related to this dissertation (including soliton theory, mechanization, the convexity and the smoothness in the locally space et. al.), as well as the work and achievements of the domestic and foreign scholars which have been obtained in these aspects. Our main works are presented at last.Chapter 2 is devoted to AC=BD model and its applications in nonlinear evolutional equations. First, basic notations, basic theory of C-D pair and C-D integrable systems to construct C-D pair are given out. Then, the theory of AC=BD is applied to differential-difference equations and bilinear formal differential equations. These greatly enlarge AC = BD theory and increase the new richer contents.In Chapter 3, the travelling wave solutions, soliton solutions, periodic solutions of differential-difference equations are studied based on symbolic computation Maple. The hyperbolic function expansion method is extended and the hyperbolic function rational expansion method is brought forward. Moreover rational formal expansion method is put forward. These methods are applied to every kinds of Toda lattice equations, Hybrid lattice equation, Ablowitz-Ladik lattice equation and Volterra lattice equation and many explicit exact solutions are obtained.Based on the ideas of solving nonlinear evolution equations, algebraic method, algorithm re-ality, mechanization, Chapter 4 extends the Jacobi elliptic function rational expansion method. With the help of symbolic computation and Wu method, more new explicit exact solutions, including soliton solutions, two periodic solutions, periodic solutions, of an asymmetric Nizhnik-Novikov-Veselov equation, the Davey-Stewartson equation and a generalized Hirota-Satsuma coupled KdV equations are constructed.In Chapter 5, the definition and properties of Hirota bilinear operator are briefly introduced. It is extended to the supersymmetric equations and many new supersymmetric identical equations are given out. Under the physical meaning, the N=l supersymmetric Sawada-Kotera-Ramani is put forward. Using Hirota’s bilinear operator, a Backhand transformation and super-soliton solutions are obtained.Chapter 6 introduces the concept of the S-simplest form of a seminorm family P and that of the P-reflexive locally convex space (X, Tp). The seminorm family P and every its S-simplest form not only generate the same locally convex separated topology on X but also have the same convexity and smoothness. Moreover, relationship between the P-reflexivity and the reflexivity are discussed, which shows that the P-reflexivity and the reflexivity are two equivalent concepts when X is a normed linear space. Under the condition of P-reflexivity, we establish that a dual pair (X, P) is uniformly smooth (uniformly convex) if and only if its strong dual pair (X1. P’) is uniformly convex (uniformly smooth).更多还原