Large-Time Asymptotic Behavior of the Generalized Burgers Equation on the Infinite Line

Abstract

This study examines a generalized Burgers equation (GBE) with step-like initial data on the infinite line. For the classical Burgers equation, the Cole–Hopf transformation facilitates the derivation of exact solutions, whose asymptotic behavior is systematically analyzed. As this transformation is not applicable to general flux functions, numerical simulations based on the method of lines are employed to investigate the large-time asymptotic behavior of the GBE. The findings reveal that the asymptotic profile evolves into either a traveling or a stationary wave, depending on the initial condition. Numerical experiments validate the large-time asymptotic behavior obtained from the analytical solution of the classical Burgers equation and demonstrate the reliability of the numerical method used. In cases where exact solutions are unavailable, the large-time asymptotic behavior is investigated through numerical analysis.

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