Nonlinear wave motion governed by the modified Burgers equation

Abstract

The modified Burgers equation (MBE) ∂ V ∂ X + V 2 ∂ V ∂ τ = ε ∂ 2 V ∂ τ 2 has recently been shown by a number of authors to govern the evolution, with range X , of weakly nonlinear, weakly dissipative transverse waves in several distinct physical contexts. The only known solutions to the M B E correspond to the steady shock wave (analogous to the well-known Taylor shock wave in a thermoviscous fluid) or to a similarity form. It can, moreover, be proved that there can exist no Bäcklund transformation of the W B E onto itself or onto any other parabolic equation, and in particular, therefore, that no linearizing transformation of Cole-Hopf type can exist. Attempts to understand the physics underlying the M B E must then, for the moment, rest on asymptotic studies and direct numerical computation.