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On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. (English) Zbl 1200.39001

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.
Reviewer: Pavel Rehak (Brno)

MSC:

39A12 Discrete version of topics in analysis
39A70 Difference operators
39A23 Periodic solutions of difference equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
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