Capillary wave equation in a quarter plane

We study the initial–boundary value problem (IBV problem)(IBV problem) for the capillary wave equation {iut+|∂x|32u=|u|2u,t>0,x>0;u(x,0)=u0(x),x>0,u(0,t)=h(t),t>0, where |∂x|32u=12π∫0∞sign  (x−y)|x−y|uyy(y)dy. We prove the global in time existence of solutions of IBV problem for the capillary wave equation with inhomogeneous Dirichlet boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions.