by a continuation-Newton technique. The temporal discretization of the periodic initial-value problem for the Benjamin equation was effected by an explicit predictor-corrector scheme. They identified another branch of solitary wave solutions of the Benjamin equation, the depression solitary waves (resembling analogous solutions of the Euler equations), and tested their stability by using them as initial values in their fully discrete scheme for the time-dependent equation. They observed that the initial profile propagated without change for some time, gradually developed an instability due to the perturbative effect of the numerical scheme, and resolved itself into two pulses resembling usual (elevation) solitary waves of the Benjamin equation plus small-amplitude dispersive oscillations. (A linearized stability analysis, also performed in [@CA], yields that the depression solitary waves are linearly unstable.)

In a recent paper [@DDM1], we made a study of several incremental continuation techniques for approximating solitary waves of the Benjamin equation that satisfy (\[E15\]). (The values of $\alpha, \beta, \delta$ and $c_{s}$ were fixed, and $\gamma$ was used as continuation parameter.) A standard pseudospectral (collocation) method yielded the underlying discrete nonlinear system. We found that Newton’s method, combined with a suitably preconditioned conjugate gradient technique for solving the attendant linear system at each 