摘要
非线性发展方程精确解和可积性的研究有助于理解孤子理论的本质属性和代数结构,而且对相应自然现象的合理解释及实际应用将起到重要的作用。
本学位论文基于Hirota所提出的双线性方法,深入、系统地研究了孤子可积系统的求解问题。揭示了双线性结构的代数特征;总结了Hirota双线性方法的各种推广及应用;探讨了Hirota双线性方法与其它孤子方程求解方法之间的广泛联系以及与可积性的内在关系;充分展示了Hirota双线性方法是孤子方程求解的强有力的工具。孤子方程存在形形色色的精确解以及多种精确解都可通过Hirota双线性方法推得的事实反映了孤子理论的丰富性、多样性、统一性特征。主要工作分为解的表示形式和多种精确解的推导两大部分。
第一部分系统地总结了双线性微分算子的性质以及孤子方程的双线性化工作。介绍解的Wronskian和Pfaffian表示形式,包括Grammian型解的表示形式;总结了Wronskian和Pfaffian的若干重要的性质和关系式。分别从以下三个方面展开研究:
对一些非等谱方程,主要是非等谱MKdV方程、具非均匀项的MKdV方程,分别利用Hirota双线性方法和Wronskian技巧得到了它们多孤子解的表示形式。
指出Wronskian和Pfaffian解的验证最终都化归为Plucker关系式或Jacobi恒等式,双线性结构实际上就是Pfaffian恒等式。
给出了KP、BKP方程及其相应Backlund变换Pfaffian解的表示形式。说明了通过Pfaffian化可导出其解仍具有Pfaffian形式新的可积方程的事实。
第二部分研究非线性发展方程多种精确解的推导。
对Hirota双线性方法作了直接的推广性发展,并具体地研究了两类浅水波方程和Ito方程,获得具有不同于经典孤子解带有奇异性的一类精确解,借助图形还具体地对类2-孤子解之间的碰撞问题进行了分析研究。
对带参数的双线性Backlund变换,经适当的修正可得到另一形式的Backlund变换。利用经修正后的双线性Baicklund变换可求得孤子方程新的精确解。具体地求得了浅水波方程和5阶KdV方程N-孤子解新的表示形式以及新的精确解,对于5阶KdV方程还得到了目前研究甚多的positon解和negaton解。
借助于双线性Backlund变换,并对参数作适当的选取,我们得到了可导出KdV方程、Vakhnenko方程许多精确解的非线性叠加公式。
应用双线性Backlund变换,同样地对其作合理的改造并选取适当的参数,从
The search on the explicit solution and integrability of nonlinear evolution equations are helpful in clarifying the underlying algebraic structure and play an important role in reasonable explaining of the corresponding natural phenomenon and application.In this dissertation, based on the Hirota bilinear method, the exact solution and physical problem are systematically investigated and the more abundant structure in bilinear formalism are revealed as well as the relations to other direct methods and integrability are discussed. It is evident that Hirota bilinear method is a powerful tool for solving a wide class of nonlinear evolution equation. More remarkable is that various physically important solutions to the soliton equations can be presented explicitly by means of Hirota bilinear method. This dissertation may be divided into two parts.Part I is devoted to summarize some properties of Hirota's bilinear operators, the properties of Wronskian and Pfaffian that appears in the expression of the N-soliton solution of the soliton equation. It is shown that the solutions of most of the soliton equations are given in terms of the determinants with a Wronskian and Grammian structure, or Pfaffian. The bilinear forms of the equations are reduced to the algebraic identities for determinants or the identities of Pfaffian.By using Hirota bilinear method and Wronskian technique, we consider the iV-soliton solutions of the nonisospectral MKdV equation and the MKdV equation with nonconformity, respectively.In addition, soliton equations whose solutions are expressed by Pfaffian are briefly discussed. Included are KP equation, BKP equation and theirs Backlund transformation in bilinear form. By applying the pfaffianization technique to the soliton equation, a new integrable model with Pfaffian solution could be generated.Part II is mainly focused on studying and the construction of various type of explicit exact solutions for nonlinear evolutions on the basis of bilinear formalism.The Hirota bilinear method is generalized and investigated, where the N-soliton-like solutions with singular slowly decaying at infinity for the shallow water waves equations and Ito equation are obtained. Further, we also explored the interaction behaviors and find singularity.
引文
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