Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing.

We investigate the dynamics of traveling oscillating solitons of the cubic nonlinear Schrödinger (NLS) equation under an external spatiotemporal forcing of the form f(x,t)=aexp[iK(t)x]. For the case of time-independent forcing, a stability criterion for these solitons, which is based on a collective coordinate theory, was recently conjectured. We show that the proposed criterion has a limited applicability and present a refined criterion which is generally applicable, as confirmed by direct simulations. This includes more general situations where K(t) is harmonic or biharmonic, with or without a damping term in the NLS equation. The refined criterion states that the soliton will be unstable if the "stability curve" p(v), where p(t) and v(t) are the normalized momentum and the velocity of the soliton, has a section with a negative slope. In the case of a constant K and zero damping, we use the collective coordinate solutions to compute a "phase portrait" of the soliton where its dynamics is represented by two-dimensional projections of its trajectories in the four-dimensional space of collective coordinates. We conjecture, and confirm by simulations, that the soliton is unstable if a section of the resulting closed curve on the portrait has a negative sense of rotation.

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.

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

Stationary and oscillatory bound states, or complexes, of the damped-driven solitons are numerically path-followed in the parameter space. We compile a chart of the two-soliton attractors, complementing the one-soliton attractor chart.

Wobbling kinks in varphi(4) theory.

We present a uniform asymptotic expansion of the wobbling kink to any order in the amplitude of the wobbling mode. The long-range behavior of the radiation is described by matching the asymptotic expansions in the far field and near the core of the kink. The complex amplitude of the wobbling mode is shown to obey a simple ordinary differential equation with nonlinear damping. We confirm the t(-1/2)-decay law for the amplitude, which was previously obtained on the basis of energy considerations.

Resonantly driven wobbling kinks.

The amplitude of oscillations of the freely wobbling kink in the varphi(4) theory decays due to the emission of second-harmonic radiation. We study the compensation of these radiation losses (as well as additional dissipative losses) by the resonant driving of the kink. We consider both direct and parametric driving at a range of resonance frequencies. In each case, we derive the amplitude equations which describe the evolution of the amplitude of the wobbling and the kink's velocity. These equations predict multistability and hysteretic transitions in the wobbling amplitude for each driving frequency--the conclusion verified by numerical simulations of the full partial differential equation. We show that the strongest parametric resonance occurs when the driving frequency equals the natural wobbling frequency and not double that value. For direct driving, the strongest resonance is at half the natural frequency, but there is also a weaker resonance when the driving frequency equals the natural wobbling frequency itself. We show that this resonance is accompanied by the translational motion of the kink.

Exceptional discretizations of the sine-Gordon equation.

The method of one-dimensional maps was recently introduced as a means of generating exceptional discretizations of the phi(4) theory, i.e., discrete phi(4) models which support kinks centered at a continuous range of positions relative to the lattice. In this paper, we employ this method to obtain exceptional discretizations of the sine-Gordon equation (i.e., exceptional Frenkel-Kontorova chains). We also use one-dimensional maps to construct a discrete sine-Gordon equation supporting kinks which move with arbitrary velocities without emitting radiation.

Moving solitons in the discrete nonlinear Schrödinger equation.

Using the method of asymptotics beyond all orders, we evaluate the amplitude of radiation from a moving small-amplitude soliton in the discrete nonlinear Schrödinger equation. When the nonlinearity is of the cubic type, this amplitude is shown to be nonzero for all velocities and therefore small-amplitude solitons moving without emitting radiation do not exist. In the case of a saturable nonlinearity, on the other hand, the radiation is found to be completely suppressed when the soliton moves at one of certain isolated "sliding velocities." We show that a discrete soliton moving at a general speed will experience radiative deceleration until it either stops and remains pinned to the lattice or--in the saturable case--locks, metastably, onto one of the sliding velocities. When the soliton's amplitude is small, however, this deceleration is extremely slow; hence, despite losing energy to radiation, the discrete soliton may spend an exponentially long time traveling with virtually unchanged amplitude and speed.

Interactions of parametrically driven dark solitons. I. Néel-Néel and Bloch-Bloch interactions.

We study interactions between the dark solitons of the parametrically driven nonlinear Schrödinger equation, Eq. 1 . When the driving strength, h , is below sqrt[gamma(2)+1/9], two well-separated Néel walls may repel or attract. They repel if their initial separation 2z(0) is larger than the distance 2zu between the constituents in the unstable stationary complex of two walls. They attract and annihilate if 2z(0) is smaller than 2zu. Two Néel walls with h lying between sqrt[gamma(2)+1/9] and a threshold driving strength hsn attract for 2z(0)<2zu and evolve into a stable stationary bound state for 2z(0)>2zu. Finally, the Néel walls with h greater than hsn attract and annihilate-irrespective of their initial separation. Two Bloch walls of opposite chiralities attract, while Bloch walls of like chiralities repel-except near the critical driving strength, where the difference between the like-handed and oppositely handed walls becomes negligible. In this limit, similarly handed walls at large separations repel while those placed at shorter distances may start moving in the same direction or transmute into an oppositely handed pair and attract. The collision of two Bloch walls or two nondissipative Néel walls typically produces a quiescent or moving breather.

Interactions of parametrically driven dark solitons. II. Néel-Bloch interactions.

The interaction between a Bloch and a Néel wall in the parametrically driven nonlinear Schrödinger equation is studied by following the dissociation of their unstable bound state. Mathematically, the analysis focuses on the splitting of a fourfold zero eigenvalue associated with a pair of infinitely separated Bloch and Néel walls. It is shown that a Bloch and a Néel wall interact as two classical particles, one with positive and the other one with negative mass.

Translationally invariant discrete kinks from one-dimensional maps.

For most discretizations of the phi4 theory, the stationary kink can only be centered either on a lattice site or midway between two adjacent sites. We search for exceptional discretizations that allow stationary kinks to be centered anywhere between the sites. We show that this translational invariance of the kink implies the existence of an underlying one-dimensional map phi(n+1) =F (phi(n)) . A simple algorithm based on this observation generates three families of exceptional discretizations.

Two-dimensional solitons on the surface of magnetic fluids.

We report an observation of a stable solitonlike structure on the surface of a ferrofluid, generated by a local perturbation in the hysteretic regime of the Rosensweig instability. Unlike other pattern-forming systems with localized 2D structures, magnetic fluids are characterized by energy conservation; hence their mechanism of soliton stabilization is different from the previously discussed gain-loss balance mechanism. The radioscopic measurements of the soliton's surface profile suggest that locking on the wavelength defined by the nonmonotonic dispersion curve is instrumental in its stabilization.

Complexes of stationary domain walls in the resonantly forced Ginsburg-Landau equation.

The parametrically driven Ginsburg-Landau equation has well-known stationary solutions-the so-called Bloch and Ne el, or Ising, walls. In this paper, we construct an explicit stationary solution describing a bound state of two walls. We also demonstrate that stationary complexes of more than two walls do not exist.

Multistable pulselike solutions in a parametrically driven Ginzburg-Landau equation.

It is well known that pulselike solutions of the cubic complex Ginzburg-Landau equation are unstable but can be stabilized by the addition of quintic terms. In this paper we explore an alternative mechanism where the role of the stabilizing agent is played by the parametric driver. Our analysis is based on the numerical continuation of solutions in one of the parameters of the Ginzburg-Landau equation (the diffusion coefficient c), starting from the nonlinear Schrödinger limit (for which c=0). The continuation generates, recursively, a sequence of coexisting stable solutions with increasing number of humps. The sequence "converges" to a long pulse which can be interpreted as a bound state of two fronts with opposite polarities.

Parametrically driven dark solitons.

We show that unlike the bright solitons, the parametrically driven kinks are immune from instabilities for all dampings and forcing amplitudes; they can also form stable bound states. In the undamped case, the two types of stable kinks and their complexes can travel with nonzero velocities.

Two- and three-dimensional oscillons in nonlinear faraday resonance.

We study 2D and 3D localized oscillating patterns in a simple model system exhibiting nonlinear Faraday resonance. The corresponding amplitude equation is shown to have exact soliton solutions which are found to be always unstable in 3D. On the contrary, the 2D solitons are shown to be stable in a certain parameter range; hence the damping and parametric driving are capable of suppressing the nonlinear blowup and dispersive decay of solitons in two dimensions. The negative feedback loop occurs via the enslaving of the soliton's phase, coupled to the driver, to its amplitude and width.

Traveling solitons in the parametrically driven nonlinear Schrödinger equation.

We show that the (undamped) parametrically driven nonlinear Schrödinger equation has wide classes of traveling soliton solutions, some of which are stable. For small driving strengths stable nonpropagating and moving solitons co-exist while strongly forced solitons can only be stable when moving sufficiently fast.