UC Berkeley

The Ginzburg-Landau equation with spatially periodic parametric forcing is used to

study localized states near a weakly subcritical steady-state bifurcation. First the speed and

stability of fronts near such a bifurcation are studied without spatial forcing, focusing on the

transition between pushed and pulled fronts. Exact nonlinear front solutions are constructed

and their stability properties investigated. In some cases, the exact solutions are stable but

are not selected from arbitrary small amplitude initial conditions. In other cases, the exact

solutions are unstable to modulational instabilities which select a distinct front. Chaotic

front dynamics may result and is studied using numerical techniques.

When periodic spatial forcing is added the Ginzburg-Landau equation exhibits bistabil-

ity between the trivial state and a nontrivial periodic state. It is shown that a family of

stationary localized states accumulate near the Maxwell point of the homogeneous problem.

Numerical continuation is used to show that under appropriate conditions these localized

states are organized within a snakes-and-ladders structure. This phenomenon is named

forced snaking. The stability properties of these states are determined and it is shown that

longer lengthscale forcing leads to stationary trains consisting of a finite number of strongly

localized, weakly interacting pulses exhibiting foliated snaking.

The phenomenon of forced snaking is not only relevant in dissipative systems and is

introduced in the study of gap solitons. These solitons are described by the cubic-quintic

Gross-Pitaevskii equation with a spatially periodic potential. The stability of the forced

snaking solutions in the gap soliton context is determined. It is shown that multi-pulse solu-

tions of all parities are stabilized when the spatial scale of the periodic forcing is sufficiently

large, effectively quenching the self interactions between the pulses. Finally it is shown that

the solitons unbind from the potential when subjected to sufficiently large perturbations and

a strongly nonlinear theory is derived to capture the dynamics during this transition.