Multi-hump solutions of some singularly-perturbed equations of KdV type

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 ∈ > 0 as a small parameter. When ∈ = 0, the equation becomes an equation of KdV type and has solitary-wave solutions. For ∈ > 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 ∈ = 0 as e goes to zero. Furthermore, it is shown that for small ∈ > 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 e as ∈ → 0. The idea of the proof may be generalized to prove the existence of symmetric solutions of 2ⁿ-humps with n = 2, 3,..., for the equations. However, this method cannot be applied to show the existence of general nonsymmetric multi-hump solutions.