Neumann problem for the capillary wave equation

We study the initial-boundary value problem IBV problem for the cubic capillary wave equation iut+|u|2u=|∂x|32u,t>0,x>0;u(x,0)=u0(x),x>0,ux(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 nonlinear Fractional Schrödinger equation with inhomogeneous Neumann boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on the study of a linear theory and then using the fixed point argument. To get a linear theory we propose general method based on Riemann-Hilbert approach and theory Cauchy type integral equations. To get smooth solutions in L∞ we modify a method based on the factorization for the free Fractional  Schrödinger evolution group.