Soliton complexity in the damped-driven nonlinear Schrödinger equation: stationary to periodic to quasiperiodic complexes

Transition from nonradiative to radiative oscillons in parametrically driven systems

Nonequilibrium systems exhibit particle-type solutions. Oscillons are one of the best-known localized states of systems with time-dependent forcing or parametrically driven systems. We investigate the transition from nonradiative to radiative oscillons in the parametrically driven sine-Gordon model in two spatial dimensions. The bifurcation takes place when the strength of the forcing (frequency) increases (decreases) above a certain threshold. As a result of this transition, the oscillon emits radially symmetric evanescent waves. Numerically, we provide the phase diagram and show the supercritical nature of this transition. For small oscillations, based on the amplitude equation approach, the sine-Gordon equation with time-dependent forcing is transformed into the parametrically driven damped nonlinear Schrödinger model in two spatial dimensions. This amplitude equation exhibits a transition between nonradiative to radiative localized structures, consistently. Both models show quite good agreement.

Two-dimensional localized chaotic patterns in parametrically driven systems

We study two-dimensional localized patterns in weakly dissipative systems that are driven parametrically. As a generic model for many different physical situations we use a generalized nonlinear Schrödinger equation that contains parametric forcing, damping, and spatial coupling. The latter allows for the existence of localized pattern states, where a finite-amplitude uniform state coexists with an inhomogeneous one. In particular, we study numerically two-dimensional patterns. Increasing the driving forces, first the localized pattern dynamics is regular, becomes chaotic for stronger driving, and finally extends in area to cover almost the whole system. In parallel, the spatial structure of the localized states becomes more and more irregular, ending up as a full spatiotemporal chaotic structure.

Traveling pulse on a periodic background in parametrically driven systems

Macroscopic systems with dissipation and time-modulated injection of energy, parametrically driven systems, can self-organize into localized states and/or patterns. We investigate a pulse that travels over a one-dimensional pattern in parametrically driven systems. Based on a minimal prototype model, we show that the pulses emerge through a subcritical Andronov-Hopf bifurcation of the underlying pattern. We describe a simple physical system, a magnetic wire forced with a transverse oscillatory magnetic field, which displays these traveling pulses.

Periodic and chaotic solitons in a semiconductor laser with saturable absorber

In a semiconductor laser with saturable absorber, solitons may spontaneously drift and/or oscillate. We study three different regimes characterized by strong intensity oscillations, both periodic and chaotic. We show that (i) soliton dynamics may be similar to that of passively Q-switched lasers, (ii) solitons may drift and oscillate simultaneously, and (iii) chaotic solitons may coexist with stationary ones and with the laser off solution.

Dissipation-driven behavior of nonpropagating hydrodynamic solitons under confinement

We have identified a physical mechanism that rules the confinement of nonpropagating hydrodynamic solitons. We show that thin boundary layers arising on walls are responsible for a jump in the local damping. The outcome is a weak dissipation-driven repulsion that determines decisively the solitons' long-time behavior. Numerical simulations of our model are consistent with experiments. Our results uncover how confinement can generate a localized distribution of dissipation in out-of-equilibrium systems. Moreover, they show the preponderance of such a subtle effect in the behavior of localized structures. The reported results should explain the dynamic behavior of other confined dissipative systems.

Phase shielding soliton in parametrically driven systems

Parametrically driven extended systems exhibit dissipative localized states. Analytical solutions of these states are characterized by a uniform phase and a bell-shaped modulus. Recently, a type of dissipative localized state with a nonuniform phase structure has been reported: the phase shielding solitons. Using the parametrically driven and damped nonlinear Schrödinger equation, we investigate the main properties of this kind of solution in one and two dimensions and develop an analytical description for its structure and dynamics. Numerical simulations are consistent with our analytical results, showing good agreement. A numerical exploration conducted in an anisotropic ferromagnetic system in one and two dimensions indicates the presence of phase shielding solitons. The structure of these dissipative solitons is well described also by our analytical results. The presence of corrective higher-order terms is relevant in the description of the observed phase dynamical behavior.

Dissipative localized States with shieldlike phase structure

A novel type of parametrically excited dissipative solitons is unveiled. It differs from the well-known solitons with constant phase by an intrinsically dynamical evolving shell-type phase front. Analytical and numerical characterizations are proposed, displaying quite a good agreement. In one spatial dimension, the system shows three types of stationary solitons with shell-like structure whereas in two spatial dimensions it displays only one, characterized by a π-phase jump far from the soliton position.

Time-periodic solitons in a damped-driven nonlinear Schrödinger equation

Time-periodic solitons of the parametrically driven damped nonlinear Schrödinger equation are obtained as solutions of the boundary-value problem on a two-dimensional spatiotemporal domain. We follow the transformation of the periodic solitons as the strength of the driver is varied. The resulting bifurcation diagrams provide a natural explanation for the overall form and details of the attractor chart compiled previously via direct numerical simulations. In particular, the diagrams confirm the occurrence of the period-doubling transition to temporal chaos for small values of dissipation and the absence of such transitions for larger dampings. This difference in the soliton's response to the increasing driving strength can be traced to the difference in the radiation frequencies in the two cases. Finally, we relate the soliton's temporal chaos to the homoclinic bifurcation.