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.