Thermalized solution of the Galerkin-truncated Burgers equation: From the birth of local structures to thermalization

Discrepancy between truncated and exact solutions of the inviscid Burgers equation is studied by the pseudo-spectral method with setting all the Fourier modes with the wavenmubers beyond a truncated wavenumber KG to be zero. A localized short-wavelength oscillation, called as a “tyger”, appears at occurrence of the shock in the truncated solution. The “tyger” shows very different shapes depending on the way of truncation of the nonlinear term. Moreover, the birth of the “tyger” is related to a period-doubling bifurcation which is illustrated by a map constructed by an iterative method at the center of the “tyger”. In order to study the process of stability loss of the truncated wave solution, a perturbed wave is derived. The truncated wave solution loses its stability in every oscillator mode of the perturbed wave. Finally, the long-term process of thermalization is displayed by the perturbed wave coupled with a frozen wave profile containing a symmetric pair of shocks. Thermalization appears from the both sides of small structures around the center without symmetry breaking. The phenomenon of the birth of “a tyger” and its following thermalization can be understood from the view of stability loss of the truncated wave solution.