TY - JOUR

T1 - Multi-hump solutions of some singularly-perturbed equations of KdV type

AU - Choi, Jeongwhan

AU - Lee, D. S.

AU - Oh, S. H.

AU - Sun, S. M.

AU - Whang, S. I.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Multi-hump waves

KW - Singularly perturbed equations

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U2 - 10.3934/dcds.2014.34.5181

DO - 10.3934/dcds.2014.34.5181

M3 - Article

AN - SCOPUS:84902651103

VL - 34

SP - 5181

EP - 5209

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 12

ER -