Nonlinear stability of a one-dimensional Boussinesq equation

In this paper we shall study nonlinear stability and instability of solitary waves of the nonlinear one-dimensional Boussinesq type equation or Benney-Luke equation Φtt − Φxx + aΦxxxx − bΦxxtt +ΦtΦxx + 2ΦxΦxt = 0, (1.1) where a and b are positive numbers such that a − b = σ − 1/3 (σ is named the Bond number)