## Abstract

The modified Burgers equation (MBE) $\frac{\partial V}{\partial X}+V^{2}\frac{\partial V}{\partial \tau}=\epsilon \frac{\partial ^{2}V}{\partial \tau ^{2}}$ 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 MBE 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 Backlund transformation of the MBE 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 MBE must then, for the moment, rest on asymptotic studies and direct numerical computation. Our aim in the present paper is to find asymptotic solutions to the MBE for small values of the dissipation coefficient $\epsilon $, but covering all values of the range and phase variables (X,$\tau $) of interest, this being achieved by systematic use of matched asymptotic expansion techniques. Two specific initial distributions are studied, in which V(0,$\tau $) is either an N-wave or a sinusoid. The corresponding problems for the ordinary Burgers equation, generalized to include cylindrical or spherical spreading effects, were treated in detail by this method elsewhere. A feature of particular interest in the present paper is that the steady Taylor-type shock waves, which separate the lossless portions of the wave form in the early stages after shock formation, develop an internal singularity at and beyond some finite range X$_{1}$ (X$_{1}$ = 10 for N-waves, X$_{1}$ = 9.601$\ldots $ for the sinusoid) unless the condition H+2G = 0 for X $\geq $ X$_{1}$ is imposed on the values H(X), G(X) of V on either side of the shock. This leads to a `refraction' of the characteristics of the lossless solution as they pass through a shock at ranges greater than X$_{1}$, and to multiple refractions in the periodic problem. The structure of the shock waves is analysed at all ranges, and it is shown how, at large ranges (X = O ($\epsilon ^{-2}$) for N-waves, X = O ($\epsilon ^{-1}$) for the sinusoid) the shocks thicken and merge with the lossless portions, leading to a phase of the motion governed by the full MBE. This phase is followed at still larger ranges by transition into old-age decay under linear mechanisms, and the form of the old-age functions is given. Computations of the wave form for the sinusoidal initial distribution are given that support the imposition of the criterion H + 2G = 0 for X > X$_{1}$. Appendix A gives a brief derivation of the MBE for the case of transverse electromagnetic waves in a nonlinear dielectric, and Appendix B provides a sketch of some interesting features and unresolved difficulties associated with higher-order calculations.