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G.L. Alfimov, A.S. Korobeinkiov, C.J. Lustri, and D.E. Pelinovsky

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Standing lattice solitons in the discrete NLS equation with saturation

**Abstract:**

We consider standing lattice solitons for discrete nonlinear Schrodinger equation
with saturation (NLSS), where so-called transparent points were recently discovered. These
transparent points are the values of the governing parameter (e.g., the lattice spacing) for
which the Peierls-Nabarro barrier vanishes. In order to explain the existence of transparent
points, we study a solitary wave solution in the continuous NLSS and analyse the singularities
of its analytic continuation in the complex plane. The existence of a quadruplet of logarithmic
singularities nearest to the real axis is proven and applied to two settings: (i) the fourth-order
differential equation arising as the next-order continuum approximation of the discrete NLSS
and (ii) the advance-delay version of the discrete NLSS.

In the context of (i), the fourth-order differential equation generally does not have solitary
wave solutions. This is because the two-dimensional center manifold at the zero equilibrium
coexist with the one-dimensional stable and unstable manifolds. Nevertheless, we show that
solitary waves solutions exist for specific values of governing parameter that form an infinite
sequence. We present an asymptotic formula for the distance between two subsequent elements
of the sequence in terms of the small parameter of lattice spacing. To derive this formula, we
used two dierent analytical techniques: the semi-classical limit of oscillatory integrals and the
beyond-all-order asymptotic expansions. Both produced the same result that is in excellent
agreement with our numerical data.

In the context of (ii), we also derive an asymptotic formula for values of lattice spacing for
which approximate standing lattice solitons can be constructed. The asymptotic formula is
in excellent agreement with the numerical approximations of transparent points. However, we
show that the asymptotic formulas for the cases (i) and (ii) are essentially dierent and that
the transparent points do not imply existence of continuous standing lattice solitons in the
advance-delay version of the discrete NLSS.

**Keywords:**

discrete nonlinear Schrodinger equation, lattice solitons, transparent points,
singular perturbation theory, oscillatory integrals, beyond-all-order methods.