EXISTENCE AND STABILITY OF SOLUTIONS TO A MODEL RQUATION FOR DISPERSION-MANAGED SOLITARY WAVES

Abstract

We study the existence and stability of solutions of the initial-value problem for the dispersion-managed nonlinear Schro ̈dinger (DMNLS) equation, a model equation for optical pulses in a dispersion-managed fiber. One interesting feature of the DMNLS equation is that the nonlinear term involves the non-local operator T (s) = e iD(s)x2 , where the periodic function D(s) governs the dispersive properties of the fiber. Another interesting feature is that even when the average dispersion \alpha is equal to zero, the equation is known to have solitary-wave solutions. For the Cauchy problem for the DMNLS equation with initial data in H^s with s>= 1, under weak assumptions on the variable dispersion and nonlinear coef ficients, we prove local well-posedness for all \alpha in R, and global well- posedness for \alpha= 0. We also use a Strichartz estimate on T(s) to establish global well-posedness for initial data in L2 for all \alpha in R, and local well-posedness for data in L^2 \intersect L^\infty in the case \alpha=0. We also revisit the proofs of existence and stability of solitary waves due to Zharnitsky in the case \alpha>0 and to Kunze in the case \alpha=0. We show that their arguments, based on a concentration compactness approach to a variational characterization of solitary waves, continue to be valid under weak assumptions on the dispersion and nonlinear coefficients.