Nonlinear Pulse Propagation

Jeffrey Rauch

Journées équations aux dérivées partielles (2001)

  • page 1-11
  • ISSN: 0752-0360

Abstract

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This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.

How to cite

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Rauch, Jeffrey. "Nonlinear Pulse Propagation." Journées équations aux dérivées partielles (2001): 1-11. <http://eudml.org/doc/93408>.

@article{Rauch2001,
abstract = {This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.},
author = {Rauch, Jeffrey},
journal = {Journées équations aux dérivées partielles},
keywords = {wave trains versus pulses; short pulse solutions; scales of geometric optics; diffractive geometric optics},
language = {eng},
pages = {1-11},
publisher = {Université de Nantes},
title = {Nonlinear Pulse Propagation},
url = {http://eudml.org/doc/93408},
year = {2001},
}

TY - JOUR
AU - Rauch, Jeffrey
TI - Nonlinear Pulse Propagation
JO - Journées équations aux dérivées partielles
PY - 2001
PB - Université de Nantes
SP - 1
EP - 11
AB - This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.
LA - eng
KW - wave trains versus pulses; short pulse solutions; scales of geometric optics; diffractive geometric optics
UR - http://eudml.org/doc/93408
ER -

References

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  1. [A] D. Alterman, Diffractive nonlinear geometric optics for short pulses, Ph.D. Thesis, University of Michigan, May 1999. Zbl1036.35049
  2. [AR1] D. Alterman and J. Rauch, Diffractive short pulse asymptotics for nonlinear wave equations, Phys. Lett. A. 264(5) 2000, pp. 390-395. Zbl0947.35033MR1739455
  3. [AR2] D. Alterman and J. Rauch, Nonlinear geometric optics for short pulses, Journal of Differential Equations, to appear. Zbl1006.35015MR1879833
  4. [AR3] D. Alterman and J. Rauch, The linear diffractive pulse equation, Methods and Applications of Analysis 7( 2001), to appear. Zbl1001.35015MR1869285
  5. [BL] Baraill and D. Lannes, In preparation. 
  6. [C1] R. Carles, Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J. 49( 2000) 475-551. Zbl0970.35143MR1793681
  7. [C2] R. Carles, Focusing on a line for nonlinear Schrödinger equations in ℝ 2 , Asymptotic Analysis 24( 2000) 255-276. Zbl0977.35131MR1797772
  8. [CR1] R. Carles and J. Rauch, Focusing of spherical nonlinear pulses in ℝ 1 + 3 , Proc. AMS ( 2001), to appear 
  9. [CR2] R. Carles and J. Rauch, Absorption d’impulsions non-linéaires radiales focalisantes dans ℝ 1 + 3 , Note CRAS, to appear. Zbl0988.35120MR1838124
  10. [CR3] R. Carles and J. Rauch, Diffusion d’impulsions non-linéaires radiales focalisantes dans ℝ 1 + 3 , Note CRAS to appear. Zbl0985.35045MR1847483
  11. [DJMR] P. Donnat, J.-L. Joly, G. Métivier and J. Rauch, Diffractive nonlinear geometric optics, Séminaire Equations aux Dérivées Partielles, Ecole Polytechnique, Paris, 1995-1996. Zbl0910.35076MR1604366
  12. [Du] E. Dumas, Univ. Rennes I Thesis, Fall 2000. 
  13. [JMR1] J.-L. Joly, G. Métivier and J. Rauch, Diffractive nonlinear geometric optics with rectification, Indiana Math. J. 47 1998 1167-1241. Zbl0928.35093MR1687154
  14. [JMR2] J.-L. Joly, G. Métivier and J. Rauch, Transparent nonlinear geometric optics and Maxwell-Bloch equations, J. Diff. Eq. 166( 2000), 175-250. Zbl1170.78311MR1779260
  15. [Ma] A. Majda, Nonlinear geometric optics for hyperbolic systems of conservation laws, Oscillation theory, computation, methods of compensated compactness, IMA Vol. Math. Appl. 2, Springer, New York, 1986, pp. 115-165. Zbl0622.65117MR869824
  16. [MR] A. Majda and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves I, Stud. Appl. Math., 71 1984, 149-179. Zbl0572.76066MR760229
  17. [R] J. E. Rothenberg, Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses, Optics Letters, 17 1992, 1340-1342. 
  18. [S] S. Schochet, Fast singular limits of hyperbolic partial differential equations, J. Diff. Eq. 114( 1994, 474-512 Zbl0838.35071MR1303036
  19. [Y1] A. Yoshikawa, Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation Trans. A.M.S., 340 1993, 103-126. Zbl0794.35099MR1208881
  20. [Y2] A. Yoshikawa, Asymptotic expansions of the solutions eto a class of quasilinear hyperbolic initial value problems, J. Math. Soc. Japan, (47)1995, 227-252. Zbl0855.35076MR1317281

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