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

Jeongwhan Choi, D. S. Lee, S. H. Oh, S. M. Sun, S. I. Whang

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 ∈ > 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.

Original languageEnglish
Pages (from-to)5181-5209
Number of pages29
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume34
Issue number12
DOIs
Publication statusPublished - 2014 Jan 1

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Singularly Perturbed
Korteweg-de Vries Equation
Solitary Wave Solution
Infinity
Oscillation
Solitons
Homoclinic Solutions
Symmetric Solution
Nonlinear Ordinary Differential Equations
Small Parameter
Ordinary differential equations
Zero

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics
  • Analysis

Cite this

Multi-hump solutions of some singularly-perturbed equations of KdV type. / Choi, Jeongwhan; Lee, D. S.; Oh, S. H.; Sun, S. M.; Whang, S. I.

In: Discrete and Continuous Dynamical Systems- Series A, Vol. 34, No. 12, 01.01.2014, p. 5181-5209.

Research output: Contribution to journalArticle

Choi, Jeongwhan ; Lee, D. S. ; Oh, S. H. ; Sun, S. M. ; Whang, S. I. / Multi-hump solutions of some singularly-perturbed equations of KdV type. In: Discrete and Continuous Dynamical Systems- Series A. 2014 ; Vol. 34, No. 12. pp. 5181-5209.
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