高维Sine-Gordon方程的精确解和动力学性质的研究

Higher dimensional Sine-Gordon(SG) equations are important nonlinear evolution equations, they appear in many physical fields, such as nonlinear optics, biological physics, quantum physics, nonlinear lattice, superconductors and so on. Explicit solutions and their dynamical properties are useful methods to study nonlinear evolution equations. So the research on explicit solutions and dynamical properties of the higher dimensional SG equations have important theoritical meaning and widely application prospects.. Based on the Bell polynomial method, we consider the bilinear derivative equations of the higher dimensional SG equations. Based on different test functions of the bilinear derivative equations, together with a "long wave" limit and Taylor series methods, we get multi-solitary wave solutions, multi-Lump wave solutions, periodic wave solutions, and inteactional solutions of three kinds of waves. Moreover, based on the asymptotic analysis theory, we consider the dynamics properties of explicit solutions and verify the correctness of the solutions and their properties by Mathematica software.. We have made substantial progress in the preliminary study of this project. The final results will extend the exsisting methods and reseach tools of the higher dimensional SG equations，and the solutions can explain more experiments and physical phenomena.

高维Sine-Gordon(SG)方程是一类具有重要意义的非线性发展方程，它在非线性光学、生物物理、量子物理、非线性晶格、超导体等物理领域有着广泛应用，精确解和动力学性质是研究刻画方程的有效手段，因此高维Sine-Gordon方程精确解和动力学性质的研究具有重要的理论意义和广泛的应用前景。. 本项目首先基于Bell多项式法研究高维SG方程的双线性导数方程，然后通过双线性导数方程不同的测试函数，结合长波极限法，Taylor展开法求得方程的多孤波解，多Lump波解，周期波解，以及三种波的作用解，并结合相关渐近分析理论讨论精确解的动力学性质，同时利用Mathematica软件验证解及其性质的正确性。 . 本项目的前期研究已取得了较大进展，其最终研究结果将扩展高维SG方程现有的研究方法和研究手段，且求得的精确解可以进一步解释一些实验现象和物理现象。

高维非线性发展方程在流体力学、非线性光学、生物物理、量子物理、非线性晶格、超导体等物理领域有着广泛应用，精确解和动力学性质是研究刻画方程的有效手段，因此高维非线性发展方程的精确解和动力学性质的研究具有重要的理论意义和广泛的应用前景，例如高维Sine-Gordon方程，2+1-维Sawada-Kotera 方程，2+1-维广义破裂孤子方程，Kudryashov-Sinelshchikov 方程，4+1-维Fokas方程等。. 本项目通过双线性导数方程不同的测试函数，结合长波极限法，Taylor展开法求得几类高维非线性方程的多孤波解，多Lump波解，周期波解，以及三种波的作用解，并结合相关渐近分析理论讨论精确解的动力学性质，同时利用Mathematica软件验证解及其性质的正确性。. 本项目已经顺利研究了几类高维非线性发展方程的精确解及其性质，其最终研究结果也已经扩展到高维SG方程。