Homoclinic Orbits for the Coupled Schrödinger–Boussinesq Equation and Coupled Higgs Equation
Homoclinic Orbits for the Coupled Schrödinger–Boussinesq Equation and Coupled Higgs Equation
复制标题
DOI：
10.1143/jpsj.72.189
复制
发表时间：
2003-01
影响因子：
1.7
通讯作者：
Xing-Biao Hu;B. Guo;H. Tam
中科院分区：
文献类型：
--
作者：
Xing-Biao Hu;B. Guo;H. Tam
自1965年Zabusky和Kruskal提出孤子概念以来，在理论和应用科学中发现了越来越多的可积方程。2和3以及其中的参考文献）。这些可积方程通常表现出丰富的显式解，如孤子解、有理解等，最近的研究表明，非线性薛定谔方程（NLS）、sine-Gordon方程（SG）及其可积离散方程都具有一类重要的显式解--同宿解（见参考文献1）。（第4至7段）。此外，对于扰动NLS和扰动SG方程及其离散情况等的同宿轨道的存在性和相关问题的研究也取得了许多显着的成果。在文献中，即使对于可积方程，也很少有同宿解的显式表示。据我们所知，除了NLS，SG和它们的可积离散形式之外，其他已知的具有同宿解的可积方程是Davey-Stewartson方程，8）复mKdV 9）和离散mKdV。10)鉴于这一不令人满意的情况和同宿解的重要性，迫切需要寻求更多的具有同宿结构的可积方程。这篇短文的目的是给出两个新的具有同宿解的方程的例子：耦合薛定谔-布西涅斯克方程和耦合希格斯方程。基于Hirota的双线性方法构造了这两个方程的同宿轨道的解析表达式。第十一章
Since the soliton concept was introduced by Zabusky and Kruskal1) in 1965, more and more integrable equations have been found both in theory and in applied sciences (see, eg refs. 2 and 3 and reference therein). These integrable equations usually exhibit richness of explicit solutions, such as soliton solutions, rational solutions, and so on. Recently, it has been shown that nonlinear Schrödinger equation (NLS), sine-Gordon equation (SG) and their integrable discrete versions possess a class of important explicit solutions–the homoclinic solutions (see refs. 4–7). Besides, many remarkable results have been achieved towards studying the existence of homoclinic orbits and related problems for the perturbed NLS and perturbed SG equations and their discrete cases and etc. In the literature, it is rare to have explicit representation of homoclinic solutions even for integrable equations. To our knowledge, besides the NLS, SG and their integable discrete versions, other known integrable equations exhibiting homoclinic solutions are the Davey–Stewartson equation, 8) the complex mKdV9) and the discrete mKdV. 10) In view of this unsatisfactory situation and importance of homoclinic solutions, there is an urgercy to seek more integrable equations exhibiting homoclinic stuctures. The purpose of this short note is to give two new examples of equations exhibiting homoclinic solutions: the coupled Schrödinger–Boussinesq equation and the coupled Higgs equation. Analytic expressions of homoclinic orbits for these two equations are constructed based on Hirota’s bilinear method. 11)