具有五阶色散项的Van der Waals方程波前解的持续性  

Persitence of Wavefront Solutions of Van der Waals Equation with Fifth Order Dispersion Term

作　　者：杨思源 傅仰耿[1] YANG Siyuan;FU Yanggeng(School of Mathematical Science,Huaqiao Univercity,Quanzhou 362021,China)

机构地区：[1]华侨大学数学科学学院,福建泉州362021

出　　处：《四川轻化工大学学报（自然科学版）》2022年第6期90-96,共7页Journal of Sichuan University of Science & Engineering(Natural Science Edition)

基　　金：国家自然科学基金项目(11401229)。

摘　　要：Van der Waals方程是一类重要的非线性偏微分方程,能够描述非弹性碰撞粒子流的动力学行为。本文证明了在充分小色散情况下,具有五阶色散项的Van der Waals方程波前解的持续存在性。首先,由于其波前解实际对应三维空间的异宿轨,利用辐角原理计算了其平衡点的稳定和不稳定流形的维数。其次,由于三维空间异宿轨的存在性研究是一个困难的问题,利用几何奇异摄动理论证明慢系统的临界流形是法向双曲的,进而把三维问题转化为二维问题。最后,在未扰动系统存在异宿轨的情况下,利用隐函数定理证明扰动系统的稳定流形与不稳定流形横截相交,即异宿轨的持续存在性。Van der Waals equation is an important class of nonlinear partial differential equations, which can be used to describe the dynamical behaviors of fluids composed of inelastically colliding particles. In this paper,for sufficiently small dispersion, the persistence of the wavefront solutions for Van der Waals equation with fifth order dispersion term has been proven. Firstly, since the study on the wavefront solutions actually correspond to heteroclinic orbits of a three-dimensional space, the dimensions of the stable and unstable manifolds of the equilibria are calculated by using argument principle. Secondly, since the study on the existence of the heteroclinic orbits in three-dimensional space is a difficult problem, applying geometric singular perturbation theory, we show that the critical manifold of the slow system is normally hyperbolic, and then transform the three-dimensional problem into a two-dimensional problem. Finally, in the case of heteroclinic orbits in the undisturbed system, the implicit function theorem is used to prove the transversal intersection of the stable and unstable manifold for the different equilibrium,that is the existence of the heteroclinic orbits.

关 键 词：五阶色散项 Van der Waals方程 几何奇异摄动理论 波前解 持续性 

分 类 号：O175[理学—数学]