MULTI-HUMP SOLUTIONS OF SOME SINGULARLY-PERTURBED EQUATIONS OF KDV TYPE

作者：Choi J W^*; Lee D S; Oh S H; Sun S M; Whang S I

来源：Discrete and Continuous Dynamical Systems, 2014, 34(12): 5181-5209.

DOI：10.3934/dcds.2014.34.5181

• Multi-hump waves
• singularly perturbed equations

This paper studies the existence of multi-hump solutions with oscillations at infinity for a class of singularly perturbed 4th-order nonlinear ordinary differential equations with epsilon %26gt; 0 as a small parameter. When epsilon = 0, the equation becomes an equation of KdV type and has solitary-wave solutions. For epsilon %26gt; 0 small, it is proved that such equations have single-hump (also called solitary wave or homoclinic) solutions with small oscillations at infinity, which approach to the solitary-wave solutions for epsilon = 0 as c goes to zero. Furthermore, it is shown that for small epsilon %26gt; 0 the equations have two-hump solutions with oscillations at infinity. These two-hump solutions can be obtained by patching two appropriate single-hump solutions together. The amplitude of the oscillations at infinity is algebraically small with respect to epsilon as epsilon -%26gt; 0. The idea of the proof may be generalized to prove the existence of symmetric solutions of 2(n)-humps with n = 2, 3, ... , for the equations. However, this method cannot be applied to show the existence of general nonsymmetric multi-hump solutions.

• 出版日期2014-12
• 单位美国弗吉尼亚理工大学(Virginia Tech)