Details

Autor(en) / Beteiligte

• Deng, Shengfu
• Sun, Shu-Ming

Titel

Multi-hump solutions with small oscillations at infinity for stationary Swift-Hohenberg equation

Ist Teil von

• Nonlinearity, 2017-02, Vol.30 (2), p.765-809

Ort / Verlag

IOP Publishing

Erscheinungsjahr

2017

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• The paper considers the stationary Swift-Hohenberg equation cw−(∂x2+k02)2w−w3=0, where c  >  0 is a constant, k02=c−μ, and μ>0 is a small parameter. In this case, the linear operator has a pair of real eigenvalues and a pair of purely imaginary eigenvalues. It can be proved that the equation has homoclinic (or single hump) solutions approaching to periodic solutions as |x|→+∞ (called single-hump generalized homoclinic solutions). This paper provides the first rigorous proof of existence of homoclinic solutions with two humps which tend to periodic solutions at infinity (or two-hump generalized homoclinic solutions) by pasting two appropriate single-hump generalized homoclinic solutions together. The dynamical system approach is used to reformulate the problem into a classical dynamical system problem and then the solution is decomposed into a decaying part and an oscillatory part at positive infinity. By adjusting some free constants and modifying the single-hump generalized homoclinic solution near negative infinity, it is shown that the solution is reversible with respect to a point near negative infinity. Therefore, the translational invariant and reversibility properties of the system yield a two-hump generalized homoclinic solution. The method may be applied to prove the existence of 2k-hump solutions for any positive integer k.