Abstract:A concept of a new type of singular solutions to hyperbolic systems of conservation laws is introduced. It is so-called δ^{n}-shock wave, where δ^{n}is n-th derivative of the delta function.We introduce a definition of δ'-shock wave type solution for the system

uWithin the framework of this definition, the Rankine--Hugoniot conditions for δ'-shock are derived and analyzed from geometrical point of view. We prove δ'-shock_{t}+ f(u)_{x}=0, v_{t}+(f'(u)v)_{x}=0, w_{t}+(f''(u)v2+f'(u)w)_{x}=0.balance relationsconnected witharea transportation. A solitary δ'-shock wave type solution to the Cauchy problem of the system of conservation lawsuwith piecewise continuous initial data is constructed._{t}+(u^{2})_{x}=0, v_{t}+2(uv)_{x}=0, w_{t}+2(v^{2}+uw)_{x}=0These results show that solutions of hyperbolic systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well.

**Paper:**- Available as PDF (320 Kbytes), Postscript (608 Kbytes) or gzipped PostScript (256 Kbytes; uncompress using gunzip).
**Author(s):**- E. Yu. Panov, <pey@novsu.ac.ru>
- V. M. Shelkovich, <February 7>
**Publishing information:****Comments:****Submitted by:**- <shelkv@vs1567.spb.edu> March 15 2005.

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Conservation Laws Preprint Server <conservation@math.ntnu.no> 2005-02-15 13:36:49 UTC