Large Time Behavior, Bi-Hamiltonian Structure, and Kinetic Formulation for a Complex Burgers Equation

Journal Article (Journal Article)

We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex plane. These solutions converge to a steady state given by Wigner's semicircle law. A unique global weak solution with nonnegative initial data to the Dyson equation is obtained, and some explicit solutions are given by Wigner's semicircle laws. We also construct a bi-Hamiltonian structure for the system of real and imaginary components of the complex Burgers equation (coupled Burgers system). We establish a kinetic formulation for the coupled Burgers system and prove the existence and uniqueness of entropy solutions. The coupled Burgers system in Lagrangian variable naturally leads to two interacting particle systems, the Fermi–Pasta–Ulam–Tsingou model with nearest-neighbor interactions, and the Calogero–Moser model. These two particle systems yield the same Lagrangian dynamics in the continuum limit.

Full Text

Duke Authors

Cited Authors

  • Gao, Y; Liu, JG

Published Date

  • May 21, 2020

Published In

Volume / Issue

  • 79 / 1

Start / End Page

  • 120 - 123

Electronic International Standard Serial Number (EISSN)

  • 1552-4485

International Standard Serial Number (ISSN)

  • 0033-569X

Digital Object Identifier (DOI)

  • 10.1090/QAM/1573

Citation Source

  • Scopus