Universal behavior of modulationally unstable media


Abstract in English

Evidence is presented of universal behavior in modulationally unstable media. An ensemble of nonlinear evolution equations, including three partial differential equations, an integro-differential equation, a nonlocal system and a differential-difference equation, is studied. Collectively, these systems arise in a variety of applications in the physical and mathematical sciences, including water waves, optics, acoustics, Bose-Einstein condensation, and more. All these models exhibit modulational instability, namely, the property that a constant background is unstable to long-wavelength perturbations. In this work, each of these systems is studied analytically and numerically for a number of different initial perturbations of the constant background, and it is shown that, for all systems and for all initial conditions considered, the dynamics gives rise to a remarkably similar structure comprised of two outer, quiescent sectors separated by a wedge-shaped central region characterized by modulated periodic oscillations. A heuristic criterion that allows one to compute some of the properties of the central oscillation region is also given.

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