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A more detailed account is given by Whitham [29]. 115) t = 0, for the classical heat equation. 116) 40 1. 117) dt The uniqueness of this solution for suitably restricted * (x 1 ) was demonstrated by Cole [40]. We now proceed to specific cases. (i) Initial Step Data In this case, the evolution of a shock wave into a steady state is considered. t = - 2sh(V 1X 1/2v) +exp[V 1x 1/2vJerf[(x l + V 1t)/2(vt)1/2J +exp[-U 1x 1/2vJerf[(x1-V 1t)/2(vt)1/2J VI 2 ch(V l X1/2v) + exp[V 1X1/2v J erf'[tx! + V 1t)/2(vt)1/2J - exp[ -U 1x 1/2vJerf[(x l - V 1t)/2(vt)1/2]. *

Thus, it may be regarded as an important canonical form. The rich tapestry of physical applications of the Korteweg-deVries equation is apparent. That this important nonlinear equation admits a Backlund transformation is as remarkable as the consequences thereof. Thus; not only can the Backlund transformation be used to generate multi-soliton solutions of the Korteweg-deVries equation, but it may also be adduced to construct both the associated inverse scattering problem and a multiplicity of conservation laws.

42 1. r BACKLUND TRANSFORMATIONS AND NONLINEAR EQUATIONS r The area under the positive phase of the pulse profile is u dx' = -2v U'IU'-l dx! where R(t) is a Reynolds number. As t dominant. 113) is _ 8nv ~ u - A n~l nexp 2n2vt} 1)/[ {-4n . (2nnx ~ A2 sin A. 129) corresponding to a periodic wave (Whitham [29J). 112) due to a distribution of heat sources set at a distance A apart. Some progress has been made in direct extensions of the Cole-Hopf reduction, notably by Chu [41J, Sachdev [179J, and Tasso and Teichmann [180].