“Turbulent steady states in the nonlinear Schrödinger equation”

Giorgio Krstulovic (Observatoire de la Côte d’Azur) will present at the 2022 Simons Collaboration on Wave Turbulence Annual Meeting on December 1st and 2nd in New York.

“Turbulent steady states in the nonlinear Schrödinger equation”:

The nonlinear Schrodinger (NLS) equation, also known as the Gross-Pitaevskii equation, is one of the most common equations in physics. Its applications go from the propagation of light in nonlinear media to the description of gravity waves and Bose-Einstein condensates. In general, the NLS equation describes the evolution of nonlinear waves. Such waves interact and transfer energy and other invariants along scales in a cascade process. This phenomenon is known as wave turbulence and is described by the (weak) wave turbulence theory (WWT). One of the most significant achievements of WWT is the complete analytical characterization of turbulent steady states obtained in the long time limit when the system contains forcing and dissipative terms acting on well-separated scales. 

In the first part of my talk, I will present recent theoretical developments on the steady-state solutions of the 3D NLS equation in the four-wave regime. Those new predictions are validated using high-resolution numerical simulations of the NLS equation and its associated wave kinetic equation. In the second part, I will address the three-wave interaction regime of the NLS equation in which waves become non-dispersive. I will start by reviewing some of the mathematical issues of the WWT for acoustic waves in 2D and 3D. Then, I will present a new theory for 2D NLS in the acoustic regime, which is in excellent agreement with numerical simulations without adjustable parameters.