A model for micro-front dynamics using a ϕ4 equation.

In this article, we propose a numerical model based on the ϕ4 equation to simulate the dynamics of a front inside a microchannel that features an imperfection at a sidewall to different flow rates. The micro-front displays pinning-depinning phenomena without damped oscillations in the aftermath. To model this behavior, we propose a ϕ4 model with a localized external force and a damping coefficient. Numerical simulations with a constant damping coefficient show that the front displays pinning-depinning phenomena showing damped oscillations once the imperfection is overcome. Replacing the constant damping coefficient with a parabolic spatial function, we reproduce accurately the experimental front-defect interaction.

Stability of bubble-like fluxons in disk-shaped Josephson junctions in the presence of a coaxial dipole current.

We investigate analytically and numerically the stability of bubble-like fluxons in disk-shaped heterogeneous Josephson junctions. Using ring solitons as a model of bubble fluxons in the two-dimensional sine-Gordon equation, we show that the insertion of coaxial dipole currents prevents their collapse. We characterize the onset of instability by introducing a single parameter that couples the radius of the bubble fluxon with the properties of the injected current. For different combinations of parameters, we report the formation of stable oscillating bubbles, the emergence of internal modes, and bubble breakup due to internal mode instability. We show that the critical germ depends on the ratio between its radius and the steepness of the wall separating the different phases in the system. If the steepness of the wall is increased (decreased), the critical radius decreases (increases). Our theoretical findings are in good agreement with numerical simulations.

Localized Faraday patterns under heterogeneous parametric excitation.

Faraday waves are a classic example of a system in which an extended pattern emerges under spatially uniform forcing. Motivated by systems in which uniform excitation is not plausible, we study both experimentally and theoretically the effect of heterogeneous forcing on Faraday waves. Our experiments show that vibrations restricted to finite regions lead to the formation of localized subharmonic wave patterns and change the onset of the instability. The prototype model used for the theoretical calculations is the parametrically driven and damped nonlinear Schrödinger equation, which is known to describe well Faraday-instability regimes. For an energy injection with a Gaussian spatial profile, we show that the evolution of the envelope of the wave pattern can be reduced to a Weber-equation eigenvalue problem. Our theoretical results provide very good predictions of our experimental observations provided that the decay length scale of the Gaussian profile is much larger than the pattern wavelength.

Bubblelike structures generated by activation of internal shape modes in two-dimensional sine-Gordon line solitons.

Nonlinear waves that collide with localized defects exhibit complex behavior. Apart from reflection, transmission, and annihilation of an incident wave, a local inhomogeneity can activate internal modes of solitons, producing many impressive phenomena. In this work we investigate a two-dimensional sine-Gordon model perturbed by a family of localized forces. We observed the formation of bubblelike and droplike structures due to local internal shape mode instabilities. We describe the formation of such structures on the basis of a one-dimensional theory of activation of internal modes of sine-Gordon solitons. An interpretation of the observed phenomena, in the context of phase transition theory, is given. Implications in Josephson junctions with a current dipole device are discussed.

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.

Formation of a two-kink soliton pair in perturbed sine-Gordon models due to kink-internal-mode instabilities.

The existence of two-kink soliton solutions in polynomial potentials was first reported by Bazeia et al. in a special type of scalar field systems [Phys. Rev. Lett. 91, 241601 (2003)]. A general feature of these potentials is that they possess two minima and a local metastable minimum between them. In the present work we investigate the appearance of this special kind of soliton in the sine-Gordon model under the perturbation of a space-dependent force. We show that a pair of solitons is emitted during the process of kink breakup by internal mode instabilities. A possible explanation of these phenomena is an interplay between the solitons repelling interaction and the external force, resulting in a separation or a packing of several kinks.

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.