Whitham method for the Benjamin-Ono-Burgers equation and dispersive shocks

Whitham method for the Benjamin-Ono-Burgers equation and dispersive shocks.

The Whitham modulation equations for the parameters of a periodic solution are derived using the generalized Lagrangian approach for the case of the damped Benjamin-Ono equation. The structure of the dispersive shock is considered in this method.

Perturbation theory for the Benjamin–Ono equation

We develop a perturbation theory for the Benjamin–Ono (BO) equation. This perturbation theory is based on the inverse scattering transform for the BO equation, which was originally developed by Fokas and Ablowitz and recently refined by Kaup and Matsuno. We find the expressions for the variations of the scattering data with respect to the potential, as well as the dual expression for the variat...

Temporal Asymptotic Behavior of Solutions of the Benjamin-ono-burgers Equation

We study certain similarity solutions of the Benjamin-Ono-Burgers (BOB) equation and their role in the asymptotic behavior of the general solution. For small initial data in L 1 (R) we prove that a solution of the BOB equation exists in BC(R + ; L 1 (R)) and depends continuously on its initial data. Results about existence, uniqueness, regularity, and spatial asymptotics of solutions of a simil...

Benjamin-Ono Equation on a Half-Line

Sharp ill-posedness result for the periodic Benjamin-Ono equation

We prove the discontinuity for the weak L(T)-topology of the flowmap associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in Hs(T) as soon as s < 0 and thus completes exactly the well-posedness result obtained in [12]. AMS Subject Classification : 35B20, 35Q53.