MULTI-HUMP SOLUTIONS OF SOME SINGULARLY-PERTURBED EQUATIONS OF KDV TYPE
- Authors
- Choi, J. W.; Lee, D. S.; Oh, S. H.; Sun, S. M.; Whang, S. I.
- Issue Date
- 12월-2014
- Publisher
- AMER INST MATHEMATICAL SCIENCES
- Keywords
- Multi-hump waves; singularly perturbed equations
- Citation
- DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v.34, no.12, pp.5181 - 5209
- Indexed
- SCIE
SCOPUS
- Journal Title
- DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
- Volume
- 34
- Number
- 12
- Start Page
- 5181
- End Page
- 5209
- URI
- https://scholar.korea.ac.kr/handle/2021.sw.korea/96637
- DOI
- 10.3934/dcds.2014.34.5181
- ISSN
- 1078-0947
- 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.
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