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

Abstract

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 > 0 as a small parameter. When epsilon = 0, the equation becomes an equation of KdV type and has solitary-wave solutions. For epsilon > 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 > 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 -> 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.