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\begin{abstract}
Localized wave phenomena in discrete nonlinear Schrodinger (DNLS) lattices play a central role in applications ranging from nonlinear optics to energy transport in complex media. In such systems, the interplay between nonlinearity, discreteness, and extended interactions gives rise to rich families of spatially localized states, commonly referred to as discrete solitons.
In this talk, we revisit the existence of stationary localized waves in DNLS lattices with non-nearest neighbour interactions from a dynamical systems perspective. Building on recent results, where such structures are constructed via invariant manifold techniques and homoclinic dynamics, we highlight the geometric mechanisms underlying their formation and organization.
We then introduce a complementary computational approach aimed at promoting high-accuracy numerical approximations to mathematically reliable solutions. Without entering into technical details, we outline how modern validation techniques can be used to rigorously certify the existence of localized states while providing quantitative control of approximation errors.
The combined geometric and computational viewpoint offers a flexible framework for the analysis of nonlinear lattice models and opens the way to the systematic validation of localized wave phenomena in discrete media.
\end{abstract}
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