  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.
