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Focusing of spherical nonlinear pulses in \(\mathbb R^{1+3}\). II: Nonlinear caustic. (English) Zbl 1094.35081

Summary: For parts I and III see [Proc. Am. Math. Soc. 130, No. 3, 791–804 (2002; Zbl 0953.35089) and Tôhoku Math. J. (2) 56, No. 3, 393–410 (2004; Zbl 1095.35010)].
We study spherical pulse like families of solutions to semilinear wave equations in space time of dimension \(1+3\) as the pulses focus at a point and emerge outgoing. We emphasize the scales for which the incoming and outgoing waves behave linearly but the nonlinearity has a strong effect at the focus. The focus crossing is described by a scattering operator for the semilinear equation, which broadens the pulses. The relative errors in our approximate solutions are small in the \(L^\infty\) norm.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A45 Diffraction, scattering
78A05 Geometric optics
35B33 Critical exponents in context of PDEs
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References:

[1] Alterman, D. and Rauch, J., Nonlinear Geometric Optics for Short Pulses. J. Differential Equations 178 (2002), no. 2, 437-465. · Zbl 1006.35015 · doi:10.1006/jdeq.2001.4016
[2] Carles, R.: Comportements précisés pr‘ es d’une caustique en optique géométrique non linéaire. Comm. Partial Differential Equations 23 (1998), no. 11-12, 1929-1967. · Zbl 0938.35108 · doi:10.1080/03605309808821407
[3] Carles, R.: Focusing on a line for nonlinear Schrödinger equations in R2. Asymptot. Anal. 24 (2000), no. 3-4, 255-276. · Zbl 0977.35131
[4] Carles, R.: Geometric optics with caustic crossing for some nonlinear Schrödinger equations. Indiana Univ. Math. J. 49 (2000), no. 2, 475-551. · Zbl 0970.35143 · doi:10.1512/iumj.2000.49.1804
[5] Carles, R.: Geometric Optics and Long Range Scattering for One- Dimensional Nonlinear Schrödinger Equations. Comm. Math. Phys. 220 (2001), no. 1, 41-67. · Zbl 1029.35211 · doi:10.1007/s002200100438
[6] Carles, R. and Rauch, J.: Absorption d’impulsions non linéaires radi- ales focalisantes dans R1+3. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 11, 985-990. · Zbl 0988.35120 · doi:10.1016/S0764-4442(01)01970-X
[7] Carles, R. and Rauch, J.: Diffusion d’impulsions non linéaires radiales focalisantes dans R1+3. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 12, 1077-1082. · Zbl 0985.35045 · doi:10.1016/S0764-4442(01)01971-1
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