Single-Phase Averaging for the Ablowitz–Ladik Chain

The Bogolyubov–Whitham averaging method is applied to the Ablowitz–Ladik chain
$$\begin{align*} -i\dot q_n-(1-q_nr_n)(q_{n-1}+q_{n+1})+2q_n&=0, \\ -i\dot r_n+(1-q_nr_n)(r_{n-1}+r_{n+1})-2r_n&=0 \end{align*}$$
in the single-phase case. We consider an averaged system and prove that the Hamiltonian property is preserved under averaging. The single-phase solutions are written in terms of elliptic functions and, in the “focusing” case, Riemannian invariants are obtained for modulation equations. The characteristic rates of the averaged system are stated in terms of complete elliptic integrals and the self-similar solutions of the system are obtained. Results of the corresponding simulations are given.