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## A water tank and some math solve s 'soliton' puzzle - Futurity

However, the mathematical theory of solitons is now a well developed "science". Solitons have practical applications and they are also interesting subjects of theoretical study.

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Four interesting and surprising things about solitons are: Integrability : Before the discovery of solitons, mathematicians and people who used math were under the impression that we could not solve nonlinear partial differential equations However, solitons led us to recognize that through a combination of such diverse subjects as quantum physics and algebraic geometry, we can actually solve some nonlinear equations exactly, which gives us a tremendous "window" into what is possible in nonlinearity. Nonlinear Superposition : In linear equations there is a simple way to make new solutions from known ones, just multiply them by scalars and add!

This is known as superposition. Before the discovery of solitons, there was no analogue of this construction for nonlinear equations, but the way that a 2-soliton solution can be viewed as a combination though not a simple linear combination of two 1-soliton solutions leads to a recognition that at least for soliton equations there is a nonlinear superposition principle as well. This is now understood algebraically in terms of the symmetries known as Darboux-Backlund transformations and geometrically in terms of the structure of the Sato Grassmannian. If Zeeman splitting is also included, the system becomes nonintegrable.

We illustrate this by considering the random walk of a soliton in a disordered SOC landscape.

## First unchanging 'soliton' wave found in space

While at zero Zeeman splitting the soliton moves without scattering along linear trajectories in the random SOC landscape; at nonzero splitting it exhibits strong scattering by the SOC inhomogeneities. For a large Zeeman splitting, the integrability is restored.

In this sense, the Zeeman splitting serves as a parameter controlling the crossover between two different integrable limits. Kartashov 1 , V. Konotop 2 , M.

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For full-text searches, use our search page. Note: the article keyword search field and some other of the site's functionality would require Javascript, which however is turned off in your browser. In general, the temporal and spectral shape of a short optical pulse changes during propagation in a transparent medium due to self-phase modulation resulting from the Kerr effect and chromatic dispersion.

This kind of phenomenon was first observed in the context of water waves [1], but later also for light propagating in optical fibers [4]. Enter input values with units, where appropriate. After you have modified some values, click a "calc" button to recalculate the field left of it. For fibers, specify the dispersion and nonlinearity for one meter of fiber, for example.

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Attention: The buttons do not work, as Javascript is turned off in your browser! Under the mentioned conditions, the pulse can propagate as a fundamental soliton or solitary pulse with constant temporal and spectral shape. It only acquires a phase shift which is just half the nonlinear phase shift which the peak of the pulse would experience if only the nonlinearity alone were to act on it. This soliton phase shift is constant over time or frequency, i.

In most situations, it is therefore not relevant. As a quantitative example, solitons in standard telecom fibers single-mode fibers for the 1. This means that solitons can adiabatically adapt their shape to slowly varying parameters of the medium.

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Also, solitons can accommodate to some amount of higher-order dispersion; they then automatically adjust their shape to achieve the mentioned balance under the given conditions. If the pulse energy is the square of an integer number times the fundamental soliton energy, the pulse is a so-called higher-order soliton. Such pulses do not have a preserved shape, but their shape varies periodically, with the period being the above-mentioned soliton period.

Higher-order solitons can break up into fundamental solitons under the influence of higher-order dispersion and other disturbing effects. They are by far not as stable as fundamental solitons. In the latter situation, soliton-like pulses can be formed when the typically lumped pieces of dispersion and nonlinearity in the laser cavity are sufficiently weak.

Solitons are also applied in various techniques for pulse compression using optical fibers; examples are adiabatic soliton compression and higher-order soliton compression. When propagating in an optical fiber, soliton pulses are subject not only to the Kerr nonlinearity, but also to stimulated Raman scattering. For very short solitons with durations of e. This causes an overall spectral shift of the soliton towards longer wavelengths, i.

The strength of this effect depends strongly on the pulse duration, since shorter solitons exhibit a higher peak power and a broader optical spectrum. The latter is important because the Raman gain is weak for small frequency offsets. During propagation, the rate of the frequency shift often slows down, because the pulse energy is reduced and the pulse duration increased [17].