  We analyze the stationary and traveling wave solutions to a family of
degenerate dispersive equations of KdV and NLS-type. In stark contrast to the
standard soliton solutions for non-degenerate KdV and NLS equations, the
degeneracy of the elliptic operators studied here allows for compactly
supported steady or traveling states. As we work in $1$ dimension, ODE methods
apply, however the models considered have formally conserved Hamiltonian, Mass
and Momentum functionals, which allow for variational analysis as well.
