Bogolyubov Institute for Theoretical Physics,

14-B Metrologichna Str.,

Kyiv 03143, UKRAINE;

Institute of Geophysics,

63-B Bohdan Khmel'nyts'kyy Str.,

Kyiv 01054, UKRAINE

E-mail: vakhnenko@bitp.kiev.ua

**Integrable nonlinear lattice associated with the
third-order spectral problem**

**Abstract:**

We propose a nonlinear model on a regular infinite one-dimensional
lattice. It describes the three component dynamical system with modulated
on-site masses and is shown to admit a zero-curvature representation. The
associated auxiliary spectral problem is basically of third order
and gives rise to fairly complicated subdivision

into domains of regularity of Jost functions in the plane of complex
spectral parameter. As a result, both the direct and the inverse scattering
problems turn out to be

substantially nontrivial. The Caudrey version of the direct and inverse
scattering technique for the needs of model integration is adapted. The
simplest soliton solution is found.