Zero mode-soliton duality and pKdV kinks in Boussinesq system for non-linear shallow water waves

A Boussinesq system for a non-linear shallow water is considered. The nonlinear and topological effects are examined through an associated matrix spectral problem. It is shown an equivalence relationship between the bound states and topological soliton charge densities which resembles a formula of the Atiyah-Patodi-Singer-type index theorem. The zero mode components describe a topologically protected Kelvin wave of KdV-type and a novel Boussinesq-type field. We show that either the $1+1$ dimensional pKdV kink or the Kelvin mode can be mapped to the bulk velocity potential in $2+1$ dimensions.