反應擴散方程及具微分項反應擴散方程之公式解及其應用

Abstract

本論文首先重述二階AKNS-ZS 系統之質譜問題及相關之進展方程。此方程之孤立子解可由退化之逆質譜變換求得，也可由Hirota 雙 線性法求得。 我們也重述具量子位能之NLS 及DNLS 與相關之反應擴散方程。我們也考慮混合型方程之情形。用Hirota 法求得1-dissipaton 解，這 已由Pashaev et al.求出[31]，我們於第三章3.3 節附上1-dissipaton 的函數圖。我們也求得2-dissipaton 解，就我們所知這結果是新的。

In this thesis, we will review some results on NLS and DNLS and the versions with 'quantum potential'. At first, we will review some results about two by two AKNS-ZS system with a linear and quadratic spectral parameter. Here exact solutions of NLS and DNLS could be obtained by degenerate inverse scattering transform, also by Hirota bilinearization method. We also review the reaction-diffusion systems related to NLS with 'quantum potential' and derivative reaction-diffusion systems related to DNLS with 'quantum potential'. Here we consider the mixed type case called modified NLS with 'quantum potential'. Some exact solutions (e.g. one-dissipatons, two-dissipatons) of the related mixed type Reaction-Diffusion system(MDRD) are constructed by Hirot a bilinearization method. Some plots of the one-dissipatons are given in section 3.