Law of Large Numbers and Central Limit Theorem for random sets of solitons for the Korteweg-de Vries equation

Series
Stochastics Seminar
Time
Thursday, February 13, 2025 - 3:30pm for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Manuela Girotti – Emory University – manuela.girotti@emory.eduhttps://mathemanu.github.io
Organizer
Benjamin McKenna

N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the Korteweg-de Vries equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas".

 

We study random configurations of N soliton solutions q_N(x,t) of the KdV equation. The randomness appears in the scattering (linear) problem, which is used to solve the PDE: the complex eigenvalues are chosen to be (1) i.i.d. random variables sampled from a probability distribution with compact support on the complex plane, or (2) sampled from a random matrix law. 

Next, we consider the scattering problem for the expectation of the random measure associated to the spectral data, in the limit as N -> + infinity. The corresponding solution q(x,t) of the KdV equation is a soliton gas. 

We are then able to prove a Law of Large Numbers and a Central Limit Theorem for the differences q_N(x,t)-q(x,t).

 

This is a collection of works (and ongoing collaborations) done with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF), A. Minakov (U. Karlova), J. Najnudel (Bristol).