Stefan C. Mancas,
University of Central Florida
Spatiotemporal Structure of
Dissipative Solitons in the Cubic–Quintic Ginzburg–Landau Equations
Abstract: Comprehensive numerical simulations of pulse
solutions of the Cubic–Quintic Ginzburg– Landau Equation (CGLE) reveal
various intriguing and entirely novel classes of solutions. In
particular, there are five new classes of pulse or solitary waves
solutions, viz. pulsating, creeping, snake, erupting, and chaotic
solitons that are not stationary in time. Rather, they are spatially
confined pulse–type structures whose envelopes exhibit complicated
temporal dynamics. The numerical simulations also reveal very
interesting bifurcations sequences of these pulses as the parameters of
the CGLE are varied. In this talk, we focus on the conditions for the
occurrence of the five categories of dissipative solitons, as well the
dependence of both their shape and their stability on the various
parameters of the CGLE, viz. the system parameters. We develop and
discuss a variational formalism within which to explore the various
classes of dissipative solitons. Given the complex dynamics this
formulation is significantly generalized over all earlier approaches.
Also, the Euler–Lagrange equations are treated in a completely novel
way. Rather than consider the stable fixed points which correspond to
the well-known stationary solitons or plain pulses, we use dynamical
systems theory to focus on more complex attractors viz. periodic,
quasiperiodic, and chaotic ones. This approach, which has been
partially explored, fails only to address the fifth class of
dissipative solitons, viz. the exploding or erupting solitons. Results
will be presented for the pulsating and snake soliton cases.
Advisor: S. Roy Choudhury (UCF)