A topical review of numerical and experimental studies of supercontinuum generation in photonic crystal fiber is presented over the full range of experimentally reported parameters, from the femtosecond to the continuous-wave regime. Results from numerical simulations are used to discuss the temporal and spectral characteristics of the supercontinuum, and to interpret the physics of the underlying spectral broadening processes. Particular attention is given to the case of supercontinuum generation seeded by femtosecond pulses in the anomalous group velocity dispersion regime of photonic crystal fiber, where the processes of soliton fission, stimulated Raman scattering, and dispersive wave generation are reviewed in detail. The corresponding intensity and phase stability properties of the supercontinuum spectra generated under different conditions are also discussed.

Supercontinuum generation in photonic crystal fiber

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

Statistical physics of vehicular traffic and some related systems

Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0334270000003891

Water waves, nonlinear Schrödinger equations and their solutions

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.

/pdf/physical-mechanisms-of-the-rogue-wave-phenomenon-iao5kemdkh.pdf

Physical Mechanisms of the Rogue Wave Phenomenon

Soliton solutions of a coupled Korteweg-de Vries equation

/pdf/b-initial-value-problems-of-one-dimensional-self-modulation-2172e1km0q.pdf

B. Initial Value Problems of One-Dimensional Self-Modulation of Nonlinear Waves in Dispersive Media

B Initial Value Problems of One-Dimensional self-Modulation of Nonlinear Waves in Dispersive Media (Part V. Initial Value Problems)

Two‐dimensional lump solutions which decay to a uniform state in all directions are obtained for the Kadomtsev–Petviashvili and a two‐dimensional nonlinear Schrodinger type equation. The amplitude of these solutions is rational in its independent variables. These solutions are constructed by taking a ’’long wave’’ limit of the corresponding N‐soliton solutions obtained by direct methods. The solutions describing multiple collisions of lumps are also presented.

Two‐dimensional lumps in nonlinear dispersive systems

We show a direct connection between a cellular automaton and integrable nonlinear wave equations. We also present the $N$-soliton formula for the cellular automaton. Finally, we propose a general method for constructing such integrable cellular automata and their $N$-soliton solutions.

/pdf/from-soliton-equations-to-integrable-cellular-automata-29nh7r36ku.pdf

Junkichi Satsuma

Papers

Soliton solutions of a coupled Korteweg-de Vries equation

B. Initial Value Problems of One-Dimensional Self-Modulation of Nonlinear Waves in Dispersive Media

B Initial Value Problems of One-Dimensional self-Modulation of Nonlinear Waves in Dispersive Media (Part V. Initial Value Problems)

Two‐dimensional lumps in nonlinear dispersive systems

From soliton equations to integrable cellular automata through a limiting procedure.