Convergence of finite difference schemes for the Benjamin-Ono equation
  In this paper, we analyze finite difference schemes for Benjamin-Ono equation, u_t = uu_x + Hu_{xx}, where H denotes the Hilbert transform. Both the decaying case on the full line and the periodic case are considered. If the initial data are sufficiently regular, fully discrete finite difference schemes shown to converge to a classical solution. Finally, the convergence is illustrated by several examples. 
Rajib Dutta, Helge Holden, Ujjwal Koley, and Nils Henrik Risebro
math.AP math.NA