Existence of Traveling Wave Solutions for a Nonlinear Dissipative-Dispersive Equation
-
摘要: 非线性耗散-色散方程出现在很多物理现象中．基于动力系统理论，利用几何奇摄动法，当耗散项系数充分小时，研究了该方程行波解的存在性．结果表明，在常微分方程组的一个三维系统中，行波依靠二维的慢流变形而存在．然后利用Melnikov方法，在该流形中建立了同宿轨道的存在性，它与方程的孤立波解相对应．进一步，给出了某些数值计算，得到该波轨道的近似．Abstract: A dissipative-dispersive nonlinear equation which appears in many physical phenomena is considered.By using dynamical systems method,specifically the geometric singular perturbation method,the existence of traveling wave solutions of the equation when the dissipative terms have sufficiently small coefficients was investigated.It was shown that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ODEs.Then,by using the Melnikov method,the existence of a homoclinic orbit in this manifold,which corresponds to a solitary wave solution of the equation,was established.Furthermore,some numerical computations were presented to show approximations of such wave orbits.
-
Key words:
- dissipative-dispersive equation /
- singular perturbations /
- traveling waves
-
[1] Kliakhandler I L, Porubov A V, Velarde M G.Localized finite-amplitude disturbances and selection of solitary waves[J].Phys Rev E,2000,62：4959-4962. doi: 10.1103/PhysRevE.62.4959 [2] Lou S Y, Huang G X, Ruan H Y. Exact solitary waves in a convecting fluid[J].J Phys A,1991,24(11): L587-L590. [3] Porubov A V. Exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluid[J].J Phys A,1993,26(17): L797-L800. [4] Velarde M G, Nekorkin V I, Maksimov A G. Further results on the evolution of solitary waves and their bound states of a dissipative Korteweg-de Vries equation[J].Internat J Bifurcation Chaos,1995,5(3): 831-839. doi: 10.1142/S0218127495000612 [5] Fenichel N. Geometric singular perturbation theory for ordinary differential equations [J].J Differantial Equations,1979,31(1): 53-98. doi: 10.1016/0022-0396(79)90152-9 [6] Jones C K R T. Geometric singular perturbation theory[A].In: Johnson R , Ed.Dynamical Systems[C].Berlin, Heidelberg: Springer-Verlag, 1995. [7] Ruan S G, Xiao D M. Stability of steady states and existence of travelling waves in a vector-disease model[J].Proc Roy Soc Edinburgh, Sect A,2004,134(5): 991-1011. doi: 10.1017/S0308210500003590 [8] Ktrychko Y N, Bartuccelli M V, Blyuss K B. Persistence of traveling wave solutions of a fourth order diffusion system[J].J Comput Appl Math,2005,176(2): 433-443. doi: 10.1016/j.cam.2004.07.028 [9] Mansour M B A. Existence of traveling wave solutions in a hyperbolic-elliptic system of equations[J].Comm Math Sci,2006,4:731-739. [10] Guckenheimer J, Holmes P.Nonlinear Oscillations, Dynamical System, and Bifurcation of Vector Fields[M]. New York: Springer-Verlag, 1983. -
计量
- 文章访问数: 2960
- HTML全文浏览量: 162
- PDF下载量: 591
- 被引次数: 0
下载:
渝公网安备50010802005915号