Ill-Posedness of the Cubic Nonlinear Half-Wave Equation and Other Fractional NLS on the Real Line

Abstract

In this paper, we study ill-posedness of cubic fractional nonlinear Schrodinger equations. First, we consider the cubic nonlinear half-wave equation (NHW) on R. In particular, we prove the following ill-posedness results: (i) failure of local uniform continuity of the solution map in Hs(R) for s∈(0,1/2), and also for s=0 in the focusing case; (ii) failure of C3- smoothness of the solution map in L2(R); (iii) norm inflation and, in particular, failure of continuity of the solution map in Hs(R), s 2.