Linear Instability of Breathers for the Focusing Nonlinear Schrödinger Equation

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed-point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below ...

Erratum to: Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

Self-Focusing in the Damped Nonlinear Schrödinger Equation

We analyze the effect of damping (absorption) on critical self-focusing. We identify a threshold value δth for the damping parameter δ such that when δ > δth damping arrests blowup. When δ < δth, the solution blows up at the same asymptotic rate as the undamped nonlinear Schrödinger equation.

Peak-height formula for higher-order breathers of the nonlinear Schrödinger equation on nonuniform backgrounds.

Given any background (or seed) solution of the nonlinear Schrödinger equation, the Darboux transformation can be used to generate higher-order breathers with much greater peak intensities. In this work, we use the Darboux transformation to prove, in a unified manner and without knowing the analytical form of the background solution, that the peak height of a high-order breather is just a sum of...

Stroboscopic Averaging for the Nonlinear Schrödinger Equation

In this paper, we are concerned with an averaging procedure, – namely Stroboscopic averaging [SVM07, CMSS10] –, for highly-oscillatory evolution equations posed in a (possibly infinite dimensional) Banach space, typically partial differential equations (PDEs) in a high-frequency regime where only one frequency is present. We construct a highorder averaged system whose solution remains exponenti...