A PRIORI ESTIMATE FOR A HIGHER ORDER NONLINEAR SCHRÖDINGER EQUATION IN NEGATIVE SOBOLEV SPACES

Abstract

The higer order Nonlinear Schrödinger equation is one of the general examples of dispersive nonlinear partial differential equations. We study a priori bounds for higher-order nonlinear Schrödinger in the Sobolev space. The equation models propagation of light pulses in optical fibers. We show that the solution of the Cauchy problem associated with this equation solution satisfies a priori upper bound which depends only on the time. The result is weak than the global well-posedness. We obtain a priori bound on range −1/8 < s < 0. The main methodology comes from recent work of Christ, Holmer and Tataru