Skip to main content
Log in

AnS matrix theory for classical nonlinear physics

  • Part I. Invited Papers Dedicated To John Archibald Wheeler
  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

The basic concepts appropriate for anS matrix theory for classical nonlinear physics are formulated here. These concepts are illustrated by a discussion of shock wave diffraction patterns. Other information concerning solutions of non-linear conservation laws is surveyed, so that a coherent picture of this theory can be seen. Within thisS matrix framework, a number of open problems as well as a few solved ones will be discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. A. Wheeler,Phys. Rev. 52, 1107 (1937).

    Google Scholar 

  2. T.-P. Liu,Commun. Math. Phys. 55, 163 (1977).

    Google Scholar 

  3. T.-P. Liu,Commun. Pure Appl. Math. 30, 585 (1977).

    Google Scholar 

  4. D. Marchesin and P. Paes-Leme, to appear,Adv. Hyperbolic Partial Differential Equations, Computers and Math. with Applications.

  5. B. Lindquist, to appear,Adv. Hyperbolic Partial Differential Equations, Computers and Math. with Applications.

  6. B. Lindquist, To appear,SIAM J. Anal.

  7. P. Lax,Commun. Pure Appl. Math. 7, 159 (1954).

    Google Scholar 

  8. O. Oleinik,Usp. Mat. Nauk (N.S.) 12, 3 (1957); English Translation:Am. Math. Soc. Transl. Ser. 2 27, 95 (1963).

    Google Scholar 

  9. J. Smoller,Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York, 1983).

    Google Scholar 

  10. F. Helfferidge and G. Klein,Multicomponent Chromatography: Theory of Interference (Marcel Dekker, New York, 1970).

    Google Scholar 

  11. B. Keyfitz and H. Kranser,J. Differ. Equ. 27, 444 (1978).

    Google Scholar 

  12. Eli Isaacson, To appear,J. Comput. Phys.

  13. D. Marchesin, P. Paes-Leme, D. Schaeffer, and M. Shearer, work in progress.

  14. W. Fickett and W. C. Davis,Detonation (University of California Press, Berkeley, 1979).

    Google Scholar 

  15. J. Glimm, C. Klingenberg, O. McBryan, B. Plohr, D. Sharp, and S. Yaniv,Adv. Appl. Math. 6, 259 (1985).

    Google Scholar 

  16. J. Guckenheimer,Arch. Ration. Mech. Anal. 59, 281 (1975).

    Google Scholar 

  17. D. Wagner,J. Math. Anal. 14, 534 (1983).

    Google Scholar 

  18. A. M. Abd-el-Fattah and L. F. Henderson,J. Fluid Mech. 86, 15 (1978).

    Google Scholar 

  19. A. M. Abd-el-Fattah and L. F. Henderson,J. Fluid Mech. 89, 79 (1978).

    Google Scholar 

  20. G. Ben-Dor and I. I. Glass,AIAA J. 16, 1146 (1978).

    Google Scholar 

  21. R. Deschambault and I. I. Glass,J. Fluid Mech. 131, 22 (1983).

    Google Scholar 

  22. R. G. Jahn,J. Fluid Mech. 1, 457 (1956).

    Google Scholar 

  23. J. Rauch and M. Reed,Commun. Math. Phys. 81, 203 (1981).

    Google Scholar 

  24. J. Glimm,Commun. Pure Appl. Math. 18, 697 (1965).

    Google Scholar 

  25. B. Engquist and S. Osher,Math. Comput. 36, 321 (1981).

    Google Scholar 

  26. P. Woodward and P. Colella,J. Comput. Phys. 54, 115 (1984).

    Google Scholar 

  27. P. Collela and P. Woodward,J. Comput. Phys. 54, 174 (1984).

    Google Scholar 

  28. R. D. Richtmyer and K. W. Morton,Difference Methods for Initial-Value Problems, 2nd edn. (Interscience Publishers, New York, 1967).

    Google Scholar 

  29. G. Moretti, Report No. PIBAL-72-37, Polytechnic Institute of Brooklyn, 1972.

  30. Roger Lazarus, private communication.

  31. J. Glimm, E. Isaacson, D. Marchesin, and O. McBryan,Adv. Appl. Math. 2, 91 (1981).

    Google Scholar 

  32. I. L. Chern, J. Glimm, O. McBryan, B. Plohr, and S. Yaniv, to appear,J. Comput. Phys.

  33. J. Glimm, B. Lindquist, O. McBryan, and L. Padmanabhan, “A Front Tracking Reservoir Simulator, Five-Spot Validation Studies and the Water Coning Problem,” inThe Mathematics of Reservoir Simulation, R. E. Ewing, ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1983), pp. 107–135.

    Google Scholar 

  34. J. Glimm, E. Isaacson, B. Lindquist, O. McBryan, and S. Yaniv, “Statistical Fluid Dynamics: the Influence of Geometry on Surface Instabilities,” inThe Mathematics of Reservoir Simulation, R. E. Ewing, ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1983), pp. 137–160.

    Google Scholar 

  35. J. Glimm, O. McBryan, R. Menikoff, and D. H. Sharp, to appear,SIAM J. Sci. Stat. Comput.

  36. C. Gardner, work in progress.

  37. J. Glimm and O. McBryan, in preparation.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to John A. Wheeler on the occasion of his 75th birthday.

Supported in part by the National Science Foundation, grant DMS-831229.

Supported in part by the Aplied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract DE-AC02-76ER03077.

Supported in part by the Army Research Office, grant DAAG29-83-K-0007.

Work supported by the U.S. Department of Energy.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Glimm, J., Sharp, D.H. AnS matrix theory for classical nonlinear physics. Found Phys 16, 125–141 (1986). https://doi.org/10.1007/BF01889377

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01889377

Keywords

Navigation