On the periodic solution of the Burgers equation: a unified approach

Abstract

For the most part, the various known expressions for the periodic solution to the Burgers equation have been derived by a mixture of ad hoc argument and approximation methods. These include the well-known solutions due to R. D. Fay and D. F. Parker. Remarkably, these solutions were obtained without recourse to the celebrated Hopf-Cole transformation which linearizes the Burgers equation (to the heat conduction equation). We present here an alternative approach to the periodic problem which uses the fact that the Riemann theta functions ‘solve’ the Burgers equation via the Hopf-Cole transformation, and thereby provide a unifying framework in which all previous representations of the periodic solution find their natural place. Yet another expression for the periodic solution, in the form of an amplitude modulated sine wave, is given and would appear to be new. The limiting forms of the periodic solution as t → 0 (embryonic ‘sawtooth’ profile) and t →∞ (‘old-age’ sinusoidal profile) are examined in detail. The representation due to D. F. Parker, which expresses the periodic wave as a superposition of spreading Taylor shock profiles, is discussed within the context of a nonlinear superposition principle . The composite asymptotic expansion due to M. B. Lesser and D. G. Crighton is examined in the light of our results. The curious behaviour of this expansion in the embryonic region ( t ≪ 1), whereby a first-order approximation yields an exact periodic solution (the Fay solution), is explained.