Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation

Rémi Carles; David Lannes

Bulletin de la Société Mathématique de France (2003)

  • Volume: 131, Issue: 2, page 289-306
  • ISSN: 0037-9484

Abstract

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We consider a system of two linear conservative wave equations, with a nonlinear coupling, in space dimension three. Spherical pulse like initial data cause focusing at the origin in the limit of short wavelength. Because the equations are conservative, the caustic crossing is not trivial, and we analyze it for particular initial data. It turns out that the phase shift between the incoming wave (before the focus) and the outgoing wave (past the focus) behaves like ln ε , where ε stands for the wavelength.

How to cite

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Carles, Rémi, and Lannes, David. "Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation." Bulletin de la Société Mathématique de France 131.2 (2003): 289-306. <http://eudml.org/doc/272341>.

@article{Carles2003,
abstract = {We consider a system of two linear conservative wave equations, with a nonlinear coupling, in space dimension three. Spherical pulse like initial data cause focusing at the origin in the limit of short wavelength. Because the equations are conservative, the caustic crossing is not trivial, and we analyze it for particular initial data. It turns out that the phase shift between the incoming wave (before the focus) and the outgoing wave (past the focus) behaves like $\ln \varepsilon $, where $\varepsilon $ stands for the wavelength.},
author = {Carles, Rémi, Lannes, David},
journal = {Bulletin de la Société Mathématique de France},
keywords = {nonlinear geometric optics; short pulses; caustic; long range scattering},
language = {eng},
number = {2},
pages = {289-306},
publisher = {Société mathématique de France},
title = {Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation},
url = {http://eudml.org/doc/272341},
volume = {131},
year = {2003},
}

TY - JOUR
AU - Carles, Rémi
AU - Lannes, David
TI - Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation
JO - Bulletin de la Société Mathématique de France
PY - 2003
PB - Société mathématique de France
VL - 131
IS - 2
SP - 289
EP - 306
AB - We consider a system of two linear conservative wave equations, with a nonlinear coupling, in space dimension three. Spherical pulse like initial data cause focusing at the origin in the limit of short wavelength. Because the equations are conservative, the caustic crossing is not trivial, and we analyze it for particular initial data. It turns out that the phase shift between the incoming wave (before the focus) and the outgoing wave (past the focus) behaves like $\ln \varepsilon $, where $\varepsilon $ stands for the wavelength.
LA - eng
KW - nonlinear geometric optics; short pulses; caustic; long range scattering
UR - http://eudml.org/doc/272341
ER -

References

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  1. [1] D. Alterman & J. Rauch – « Nonlinear geometric optics for short pulses », 178 (2002), no. 2, p. 437–465. Zbl1006.35015MR1879833
  2. [2] R. Carles – « Geometric optics with caustic crossing for some nonlinear Schrödinger equations », 49 (2000), no. 2, p. 475–551. Zbl0970.35143MR1793681
  3. [3] —, « Geometric optics and long range scattering for one-dimensional nonlinear Schrödinger equations », 220 (2001), no. 1, p. 41–67. Zbl1029.35211MR1882399
  4. [4] R. Carles & J. Rauch – « Focusing of Spherical Nonlinear Pulses in ℝ 1 + 3 , II. Nonlinear Caustic », to appear in Rev. Mat. Iberoamericana. Zbl1094.35081MR2124490
  5. [5] —, « Absorption d’impulsions non linéaires radiales focalisantes dans ℝ 1 + 3 », 332 (2001), no. 11, p. 985–990. Zbl0988.35120MR1838124
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  8. [8] —, « Focusing of Spherical Nonlinear Pulses in ℝ 1 + 3 , III. Sub and Supercritical cases » », Preprint, 2002. Zbl1095.35010MR1866035
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  10. [10] J. Hunter & J. Keller – « Caustics of nonlinear waves », Wave motion9 (1987), p. 429–443. Zbl0645.35062MR909884
  11. [11] J.-L. Joly, G. Métivier & J. Rauch – « Focusing at a point and absorption of nonlinear oscillations », 347 (1995), no. 10, p. 3921–3969. Zbl0857.35087MR1297533
  12. [12] —, « Several recent results in nonlinear geometric optics », Partial differential equations and mathematical physics (Copenhagen, 1995; Lund, 1995), Birkhäuser Boston, Boston, MA, 1996, p. 181–206. Zbl0854.35115MR1380991
  13. [13] —, Caustics for dissipative semilinear oscillations, vol. 144, no.685, American Mathematical Society, Providence, 2000. Zbl0963.35114MR1682244
  14. [14] P. Lax – « Asymptotic solutions of oscillatory initial value problems », 24 (1957), p. 627–646. Zbl0083.31801MR97628
  15. [15] D. Ludwig – « Uniform asymptotic expansions at a caustic », 19 (1966), p. 215–250. Zbl0163.13703MR196254

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