Existence and stability of solitary wave solutions of the Benjamin equation (Q1283888)

Rigorous results are obtained that establish the existence and stability of traveling-wave solitary-wave solutions to the nonlocal Benjamin-Ono equation, \[ \eta_t +2\eta\eta_x - l\mathcal {H}\eta_{xx} - \eta_{xxx} = 0, \] where \(\eta(x,t)\) is a real-valued function, \(l\) is a constant, and \(\mathcal{H}\) is a symbol standing for the Hilbert transform. The solitary-wave solutions are sought for as \(\eta(x + Ct)\) with a constant velocity \(C\). For the proof of the existence and stability of the solution, a variational representation of the underlying equation, generated by a constrained functional, is used. In terms of this representation, the velocity naturally appears as a Lagrangian multiplier. The existence of the minimizer to the constrained functional is proved by means of Lions' concentration compactness method. In particular, it is shown that the velocities of the traveling solitary waves satisfy an inequality \(C\geq l^2/4\).