Some Exact Solutions of Reaction-Diffusion Systems and Derivative Reaction-Diffusion Systems and the Application

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.Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . .1 2 Examples of nonlinear evolution equation from U-V condition . . . . . . . . . . . . . . . . . . . . . .2 2.1 Review on 2 2 AKNS-ZS system with a linear spectral parameter in Beals-Coifman set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.1 Scattering problem . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.2 Associated evolution equations . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Review on 2 2 AKNS-ZS system with a quadratic spectral parameter . . . . . . . . . . . . . . . . . 3 2.2.1 Scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.2 Associated evolution equations . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Examples of 2 2 U-V equations with linear and quadratic spectral parameters . . . . . . . . . . . . . 4 2.4 Comparison of One-Soliton Solutions via Different Methods . . . . . . . . . . . . . . . . . . .6 3 NLS & DNLS with "quantum potential" 8 3.1 DNLS with "quantum potential" . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Modi ed NLS with "Quantum Potential" . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Some plots of exact solutions . . . . . . . . . . . . . . . . . . . . . . . 13 4 Conclusion . . . . . . . . . . . . . . . . . . 14 Appendix . . . . . . . . . . . . . . . . . . 15 A Equivalent Relations . . . . . . . . . . . . . . . . . 15 B Some computation results for the asymptotic expansion of MJM.......................................17 C Hirota Bilinear Method.......20 D Possible application (Colin Rogers' remarks).........21 D.1 Some equation in plasma.......................21 D.2 NLS with "quantum potential".............21 D.3 Reaction-Diffusion System related to NLS with "quantum potential"..................22 Bibliography ..............................2