Nonlinear Pulse Propagation
Journées équations aux dérivées partielles (2001), article no. 11, 11 p.

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.

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     author = {Rauch, Jeffrey},
     title = {Nonlinear {Pulse} {Propagation}},
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     pages = {1--11},
     publisher = {Universit\'e de Nantes},
     year = {2001},
     doi = {10.5802/jedp.595},
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     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.595/}
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Rauch, Jeffrey. Nonlinear Pulse Propagation. Journées équations aux dérivées partielles (2001), article  no. 11, 11 p. doi : 10.5802/jedp.595. http://www.numdam.org/articles/10.5802/jedp.595/

[A] D. Alterman, Diffractive nonlinear geometric optics for short pulses, Ph.D. Thesis, University of Michigan, May 1999.

[AR1] D. Alterman and J. Rauch, Diffractive short pulse asymptotics for nonlinear wave equations, Phys. Lett. A. 264(5) 2000, pp. 390-395. | MR | Zbl

[AR2] D. Alterman and J. Rauch, Nonlinear geometric optics for short pulses, Journal of Differential Equations, to appear. | MR | Zbl

[AR3] D. Alterman and J. Rauch, The linear diffractive pulse equation, Methods and Applications of Analysis 7(2001), to appear. | MR | Zbl

[BL] Baraill and D. Lannes, In preparation.

[C1] R. Carles, Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J. 49(2000) 475-551. | MR | Zbl

[C2] R. Carles, Focusing on a line for nonlinear Schrödinger equations in ℝ 2 , Asymptotic Analysis 24(2000) 255-276. | MR | Zbl

[CR1] R. Carles and J. Rauch, Focusing of spherical nonlinear pulses in ℝ 1+3 , Proc. AMS (2001), to appear | Zbl

[CR2] R. Carles and J. Rauch, Absorption d’impulsions non-linéaires radiales focalisantes dans ℝ 1+3 , Note CRAS, to appear. | MR | Zbl

[CR3] R. Carles and J. Rauch, Diffusion d’impulsions non-linéaires radiales focalisantes dans ℝ 1+3 , Note CRAS to appear. | MR | Zbl

[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. | Numdam | MR | Zbl

[Du] E. Dumas, Univ. Rennes I Thesis, Fall 2000.

[JMR1] J.-L. Joly, G. Métivier and J. Rauch, Diffractive nonlinear geometric optics with rectification, Indiana Math. J. 47 1998 1167-1241. | MR | Zbl

[JMR2] J.-L. Joly, G. Métivier and J. Rauch, Transparent nonlinear geometric optics and Maxwell-Bloch equations, J. Diff. Eq. 166(2000), 175-250. | MR | Zbl

[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. | MR | Zbl

[MR] A. Majda and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves I, Stud. Appl. Math., 71 1984, 149-179. | MR | Zbl

[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.

[S] S. Schochet, Fast singular limits of hyperbolic partial differential equations, J. Diff. Eq. 114(1994, 474-512 | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

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