Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Modeling Parabolic Pulse Formation in Mode-Locked Laser Cavities

Abstract: Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporal-structures observed in a broad range of applications and natural phenomena. In- deed, they have been predicted or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion regime with strong nonlinearity, optical amplifiers, and mode- locked lasers. These self-similar structures are typ- ically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physi- cal effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic profiles: the zero-dispersion similari- ton. The zero-dispersion similariton is the final so- lution state of the system, not a long-time, inter- mediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed by a nonlin- ear diffusion equation with a rapidly-varying, mean- zero diffusion coefficient. Indeed, the leading-order behavior is shown to be governed by the porous me- dia (nonlinear diffusion) equation whose solution is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the zero-dispersion similariton which is, to leading- order, of the Barenblatt form. This is the first an- alytic model proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed in mode-locked lasers.

Electrically induced vortex flow at a point electrode and azimuthal rotation

Abstract: The previous chapter dealt with the flows arising when an electric current radially diverges through the fluid from a small electrode. In spite of the simplicity of the physical model with a point current source, which was successfully combined with the similarity solution of equations of motion and magnetic field, it was found to be impossible to describe the flows for magnitudes of electric current higher than a certain critical magnitude. The source of failure may be sought in the similarity of the equations, which greatly restrict the form of the motions. However it should be recalled that the similarity is not an artificial assumption to describe the flows at a point electrode, it is derived by the dimensional analysis of the given physical quantities entering the problem [26]. Moreover, as was demonstrated in Section 2.10, the description of analogous flows without the assumption of similarity, e.g. in a closed hemispherical container, also led to serious difficulties. Consequently, the cause of difficulties should be sought in the physical statement of the problem, which probably does not take into account an essential mechanism limiting the growth of velocities in the flow when the critical magnitude of electric current is reached. The limiting mechanism cannot be related to the flow induced electric current since, even for excessive magnitudes of electrical condictivity (by 6–7 orders of magnitude compared to real materials) the critical magnitude of the electric current is not appreciably increased.