The deterministic and statistical Burgers equation

Fourier-Lagrangian representations of the UV-region inviscid-limit solutions of the equations of Burgers (1939) are developed for deterministic and random initial conditions. The Fourier-mode amplitude behavior of the deterministic case is characterized by complex singularities with fast decrease, power-law preshocks with k indices of about -4/3, and shocks with k to the -1. In the random case, shocks are associated with a k to the -2 spectrum which overruns the smaller wavenumbers and appears immediately under Gaussian initial conditions. The use of the Hopf-Cole solution in the random case is illustrated in calculations of the law of energy decay by a modified Kida (1979) method. Graphs and diagrams of the results are provided.