**Title**: *Focusing media with non-zero background: universality, solitons and dispersive shocks
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**Abstract**:
The behavior of focusing media with non-zero background has received increased interest in recent years, revealing a number of interesting phenomena. This talk aims at presenting an overview of some recent results on this subject. Specifically:
1. I will characterize the nonlinear stage of modulational instability induced by localized perturbations of a uniform background. Using the inverse scattering transform (IST) for the focusing nonlinear Schrodinger (NLS) equation, one can prove that the long-time asymptotics is universal, in the sense that a large class of initial conditions give rise to the same behavior: two outer quiescent states separated by a central, wedge-shaped region described by modulated periodic oscillations, whose features can be described analytically.
2. I will show that another kind of universality also exists, in that the above behavior is not limited to the NLS equation, but is instead shared by many different continuous and discrete nonlinear system affected by modulational instability.
3. I will show that the interactions between solitons and the oscillatory wedge gives rise to three different outcomes, all of which can be completely characterized analytically: soliton transmission, soliton trapping, and a mixed regime in which a soliton-generated wake is also produced.
4. I will discuss a broad class of Riemann-problems in these kinds of systems, and I will show how some of these problems, which give rise to the formation of dispersive shock waves (DSW), are effectively described by Whitham modulation theory.
5. Finally, I will discuss the interactions between solitons and these DSW structures, and I will show how all the soliton properties (amplitude, velocity, width, internal structure) change upon passing through the DSW, but can still be correctly captured by the IST.