  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 the 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: Fermi-Pasta-Ulam-Tsingou model with
nearest-neighbor interactions, and Calogero-Moser model. These two particle
systems yield the same Lagrangian dynamics in the continuum limit.
