Natasha Devine

Integrable Turbulence and Rogue Waves: Breathers or Solitons?

Turbulence in dynamical systems is one of the most intriguing phenomena of modern science. Integrable systems offer the possibility to understand, to some extent, turbulence. Recent numerical and experimental data suggest that the probability of the appearance of rogue waves in a chaotic wave state in such systems increases when the initial state is a random function of sufficiently high amplitude. We provide explanations for this effect.

Dissipative ring solitons with vorticity

We study dissipative ring solitons with vorticity in the frame of the (2+1)-dimensional cubic-quintic complex Ginzburg-Landau equation. In dissipative media, radially symmetric ring structures with any vorticity m can be stable in a finite range of parameters. Beyond the region of stability, the solitons lose the radial symmetry but may remain stable, keeping the same value of the topological charge. We have found bifurcations into solitons with n-fold bending symmetry, with n independent on m. Solitons without circular symmetry can also display (m + 1)-fold modulation behaviour. A sequence of bifurcations can transform the ring soliton into a pulsating or chaotic state which keeps the same value of the topological charge as the original ring.

Rogue waves and other solutions of single and coupled Ablowitz–Ladik and nonlinear Schrödinger equations

We provide a simple technique for finding the correspondence between the solutions of Ablowitz?Ladik and nonlinear Schr?dinger equations. Even though they belong to different classes, in that one is continuous and one is discrete, there are matching solutions. This fact allows us to discern common features and obtain solutions of the continuous equation from solutions of the discrete equation. We consider several examples. We provide tables, with selected solutions, which allow us to easily match the pairs of solutions. We show that our technique can be extended to the case of coupled Ablowitz?Ladik and nonlinear Schr?dinger (i.e. Manakov) equations. We provide some new solutions.

Transformations of continuously self-focusing and continuously self-defocusing dissipative solitons

Dissipative media admit the existence of two types of stationary self-organized beams: continuously self-focused and continuously self- defocused. Each beam is stable inside of a certain region of its existence. Beyond these two regions, beams loose their stability, and new dynamical behaviors appear. We present several types of instabilities related to each beam configuration and give examples of beam dynamics in the areas adjacent to the two regions. We observed that, in one case beams loose the radial symmetry while in the other one the radial symmetry is conserved during complicated beam transformations.

Modulation instability, Cherenkov radiation, and Fermi–Pasta–Ulam recurrence

The work of J.M.S.C. is supported by the Ministerio de Economia y Competitividad under contract FIS2009-09895. The authors acknowledge the support of the Australian Research Council (Discovery Project No. DP110102068). N.A. is a recipient of the Alexander von Humboldt Award.

How Cherenkov radiative losses can improve optical frequency combs

Broader optical frequency combs on a photonic chip can help refine time standards [Also see Report by Brasch et al.] The idea of a frequency comb seems relatively simple, yet substantial technical efforts are required for one to be generated with the high accuracy and stability needed for metrology applications. The ideal frequency comb would be a set of discrete equidistant frequency (f) components separated by intervals Δf. Typically Δf is in the microwave range so that the separation of the comb “teeth” can be measured and controlled electronically. However, stabilization of Δf is not sufficient for applications; having a nearly octave-wide comb is necessary for “self-referencing” the comb (1, 2). On page 357 of this issue, Brasch et al. (3) show how to create a wideband comb spectrum to help realize this goal.

Dissipative solitons with extreme spikes: bifurcation diagrams in the anomalous dispersion regime

Dissipative solitons with extreme spikes (DSESs), previously thought to be rare solutions of the complex cubic–quintic Ginzburg–Landau equation, occupy in fact a significant region in its parameter space. The variation of any of its five parameters results in a rich structure of bifurcations. We have constructed several bifurcation diagrams that reveal periodic and chaotic dynamics of DSESs. There are various routes to the chaotic behavior of DSESs, including a sequence of period-doubling bifurcations. It is well known that the complex cubic–quintic Ginzburg–Landau equation can serve as a master equation for the description of passively mode-locked lasers. Our results may lead to the observation of DSESs in laser systems.

How Cherenkov radiative losses can improve optical frequency combs: Broader optical frequency combs on a photonic chip can help refine time standards

Broader optical frequency combs on a photonic chip can help refine time standards [Also see Report by Brasch et al.] The idea of a frequency comb seems relatively simple, yet substantial technical efforts are required for one to be generated with the high accuracy and stability needed for metrology applications. The ideal frequency comb would be a set of discrete equidistant frequency (f) components separated by intervals Δf. Typically Δf is in the microwave range so that the separation of the comb “teeth” can be measured and controlled electronically. However, stabilization of Δf is not sufficient for applications; having a nearly octave-wide comb is necessary for “self-referencing” the comb (1, 2). On page 357 of this issue, Brasch et al. (3) show how to create a wideband comb spectrum to help realize this goal.

Rogue waves of the nonlinear Schrodinger equation with even symmetric perturbations

We show that a rogue wave solution of the nonlinear Schr?dinger equation (NLSE) can survive even-parity perturbations of the equation, such as the addition of a quintic term and fourth-order dispersion. We present a solution which is accurate to the first order for such a perturbation. Our numerical simulations confirm the rogue wave existence when the parameter of perturbation |?|?

Dissipative solitons with extreme spikes in the normal and anomalous dispersion regimes

Prigogine’s ideas of systems far from equilibrium and self-organization (Prigogine & Lefever. 1968 J. Chem. Phys. 48, 1695–1700 (doi:10.1063/1.1668896); Glansdorff & Prigogine. 1971 Thermodynamic theory of structures, stability and fluctuations. New York, NY/London, UK: Wiley) deeply influenced physics, and soliton science in particular. These ideas allowed the notion of solitons to be extended from purely integrable cases to the concept of dissipative solitons. The latter are qualitatively different from the solitons in integrable and Hamiltonian systems. The variety in their forms is huge. In this paper, one recent example is considered—dissipative solitons with extreme spikes (DSESs). It was found that DSESs exist in large regions of the parameter space of the complex cubic–quintic Ginzburg–Landau equation. A continuous variation in any of its parameters results in a rich structure of bifurcations. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)’.