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Zbl 1200.39001
Zhou, Zhan; Yu, Jianshe
On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. (English)
[J] J. Differ. Equations 249, No. 5, 1199-1212 (2010). ISSN 0022-0396

The authors consider the nonlinear discrete periodic system $$a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n-\omega u_n=\sigma f_n(u_n),\quad n\in\mathbb{Z},$$ where $f_n(u)$ is continuous in $u$ and with saturable nonlinearity for each $n\in\mathbb{Z}$, $f_{n+T}(u)=f_n(u)$, $\{a_n\},\{b_n\}$ are real valued $T$-periodic sequences. They are interested in the existence of nontrivial homoclinic solutions for this equation; this problem appears when one looks for the discrete solitons of the periodic discrete nonlinear Schrödinger equations. A new sufficient condition guaranteeing the existence of homoclinic solutions is obtained by using critical point theory. It is proved that it is also necessary in some special cases. Moreover, the rate of decay is established.
[Pavel Rehak (Brno)]

MSC 2000:: *39A12 Discrete version of topics in analysis
39A70 Difference operators
39A23
37C29 Homoclinic and heteroclinic orbits

Keywords: homoclinic solutions; discrete nonlinear periodic systems; critical point theory; periodic approximation; discrete solitons; discrete nonlinear Schrödinger equations; homoclinic solutions

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