Multi-hump solutions of some singularly-perturbed equations of KdV type
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Bibliographic record
Abstract
This paper studies the existence of multi-hump solutionswith oscillations at infinity for a class ofsingularly perturbed 4th-order nonlinear ordinary differentialequations with $\epsilon > 0$ as a small parameter.When $\epsilon =0$, the equation becomes an equation of KdV type and hassolitary-wave solutions. For $\epsilon > 0 $ small, it is proved that such equations have single-hump (also called solitary wave or homoclinic) solutions with small oscillationsat infinity, which approach to the solitary-wave solutions for $\epsilon = 0$ as $\epsilon$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 isalgebraically small with respect to $\epsilon$ as $\epsilon \rightarrow 0$. The idea of the proof may be generalized to prove the existence of symmetric solutions of $2^n$-humps with $n=2,3,\dots,$for the equations. However, this method cannot be applied to show the existence of general nonsymmetric multi-hump solutions.
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