Study On Symbolic Computation Of Soliton Solutions For Integrable Systems

Many phenomenon in natural science and engineering technology can be described with nonlinear mathematical physics models, while the analysis and understanding to these nonlinear mathematical physics models are boiled down to the solutions of non-linear evolution equations. However, duing to the complexity of the nonlinear evolution equations, there are often involved complicated differential and algebraic calculations and some is beyond human in seeking the solutions of nonlinear evolution equations. Thus a new topic is posed to the researchers in mathematical physics and computer field, namely put the computer to complete the complicated differential and algebraic calculations.In the dissertation, under the guidance of mathematical mechanization, both of the Darboux transformation method and Hirota bilinear transformation method are investi-gated by means of symbolic computation software Maple. The application and mecha-nization of them are discussed, respectively. There are two main parts in this dissertation:1. Both of the (2×2)-and (3×3)-matrix spectral problem are investigated, respec-tively. Some well-known soliton equations and new nonlinear evolution equations are devised from these matrix spectral problems. N fold Darboux transformations of some types of (2×2)-systems and (3×3)-system are constructed and a simplified algorithm which construct the multi-soliton solutions of nonlinear evolution equations is designed. Meanwhile, the software package dbtransformation about the simplified algorithm is de-veloped with the help of symbolic computation software Maple.2. Based on the Hirota bilinear transformation method, two software packages Bilin-earization and Multisoliton are developed which aim to construct the bilinear forms and N soliton solutions of nonlinear evolution equations, respectively. We believe that the two software packages could promote the study of integrability and exact solutions for soliton equations.In chapter1, an introduction is devoted to review the theory background and the current situation related the dissertation, which include soliton theory, exact solutions of nonlinear evolution equations and symbolic computation.In chapter2, the N fold Darboux transformations of (2×2)-AKNS system and (2×2)-BK system are constructed, respectively. Also, the N fold Darboux transfor-mations of some special equations, such as KdV hierarchy, mKdV hierarchy and NLS hierarchy are constructed. A simplified algorithm which construct the multi-soliton solu-tions of nonlinear evolution equations is designed and the corresponding software pack-age dbtransformation is developed which succeed in working out the complicated formula manipulation and avoiding redundant operation as much as possible. The (2×2)-AKNS system and BK system are investigated by appling the algorithm and different forms of soliton solutions, such as double soliton solution, pursueing soliton solution and single-peak collision soliton solution.In chapter3. a new Lax pair related to (3×3)-matrix spectral problem is construct-ed, corresponding hierarchy is devised from the Lax pair which not only including the well-known equations, such as KdV equation, mKdV equation and NLS equation, but also including the complicated higher order equation, such as HNLS equation. The Dar-boux transformation of the (3×3)-matrix spectral problem is constructed and the exact solutions of the nonlinear evolution equation which devised from the Lax pair is obtained.In chapter4, based on the idea of Hirota bilinear transformation method, the mechan-ical algorithmic about the bilinearization of nonlinear evolution equation was realized on symbolic computation software Maple and corresponding software package is developed named Bilinearization. Many examples are presented to illustrate the implementation of the package.In chapter5, Through analysis and investigation, we found out that the soliton so-lutions of the bilinear equations are usually expressed as exponential function. Based on this idea, we developed the software package Multisoliton which applied to construct N soliton solutions of nonlinear evolution equations by means of symbolic computation software Maple. The single soliton solution, double soliton solution and three soliton so-lution can be obtained by appling the algorithm. It provides an effective tool for the study of completely integrable equations and their exact solutions.In chapter6. the summary and discussion of this dissertation are given, as well as the outlook of future work is discussed.