Dmitry E. Pelinovsky and Vadim Zharnitsky

Averaging of dispersion-managed pulses: existence and stability

SIAM J. Appl. Math. 63, 745-776 (2003)

Abstract:
We consider existence and stability of dispersion-managed pulses in the two approximations of the periodic NLS equation: (i) a dynamical system for a Gaussian pulse and (ii) an average integral NLS equation. We apply normal form transformations for finite-dimensional and infinite-dimensional Hamiltonian systems with periodic coefficients. First-order corrections to the leading-order averaged Hamiltonian are derived explicitly for both approximations. Bifurcations of pulse solutions and their stability are studied by analysis of critical points of the first-order averaged Hamiltonians. The validity of the averaging procedure is verified and the presence of ground states corresponding to dispersion-managed pulses in the averaged Hamiltonian is established.

Keywords:
EXISTENCE AND STABILITY OF PULSES, OPTICAL SOLITONS, DISPERSION MANAGEMENT, AVERAGING THEORY, NORMAL FORM TRANSFORMATIONS, ERRORS and CONVERGENCE OF ASYMPTOTIC SERIES, PERIODIC NLS EQUATION, INTEGRAL NLS EQUATIONS, GAUSSIAN APPROXIMATION.