Dionyssios Mantzavinos

Universal Nature of the Nonlinear Stage of Modulational Instability.

We characterize the nonlinear stage of modulational instability (MI) by studying the longtime asymptotics of the focusing nonlinear Schrödinger (NLS) equation on the infinite line with initial conditions tending to constant values at infinity. Asymptotically in time, the spatial domain divides into three regions: a far left and a far right field, in which the solution is approximately equal to its initial value, and a central region in which the solution has oscillatory behavior described by slow modulations of the periodic traveling wave solutions of the focusing NLS equation. These results demonstrate that the asymptotic stage of MI is universal since the behavior of a large class of perturbations characterized by a continuous spectrum is described by the same asymptotic state.

Hölder Continuity for the Fokas–Olver–Rosenau–Qiao Equation

It has been shown that the Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation is well-posed for initial data

The Korteweg–de Vries equation on the half-line

u_0\in H^s

The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation

u0∈Hs,

The Cauchy problem for a 4-parameter family of equations with peakon traveling waves

s>5/2

Long-Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

s>5/2, with its data-to-solution map