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Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear electrodynamics


Authors: Miguel A. Alejo and Claudio Muñoz
Journal: Proc. Amer. Math. Soc. 146 (2018), 2225-2237
MSC (2010): Primary 37K15, 35Q53; Secondary 35Q51, 37K10
DOI: https://doi.org/10.1090/proc/13947
Published electronically: January 8, 2018
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Abstract: We study decay of small solutions of the Born-Infeld equation in 1+1 dimensions, a quasilinear scalar field equation modeling nonlinear electromagnetism, as well as branes in String theory and minimal surfaces in Minkowski space-times. From the work of Whitham, it is well known that there is no decay because of arbitrary solutions traveling to the speed of light just as linear wave equation. However, even if there is no global decay in 1+1 dimensions, we are able to show that all globally small $ H^{s+1}\times H^s$, $ s>\frac 12$ solutions do decay to the zero background state in space, inside a strictly proper subset of the light cone. We prove this result by constructing a Virial identity related to a momentum law, in the spirit of works by Kowalczyk, Martel, and the second author, as well as a Lyapunov functional that controls the $ \dot H^1 \times L^2$ energy.


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  • [1] Miguel A. Alejo, Claudio Muñoz, and José M. Palacios, On the variational structure of breather solutions I: Sine-Gordon equation, J. Math. Anal. Appl. 453 (2017), no. 2, 1111-1138. MR 3648277, https://doi.org/10.1016/j.jmaa.2017.04.056
  • [2] Metin Arik, Fahrünisa Neyzi, Yavuz Nutku, Peter J. Olver, and John M. Verosky, Multi-Hamiltonian structure of the Born-Infeld equation, J. Math. Phys. 30 (1989), no. 6, 1338-1344. MR 995779, https://doi.org/10.1063/1.528314
  • [3] M. Born and L. Infeld, Foundation of the new field theory, Proc. Roy. Soc. A 144 (1934): 425-451.
  • [4] Simon Brendle, Hypersurfaces in Minkowski space with vanishing mean curvature, Comm. Pure Appl. Math. 55 (2002), no. 10, 1249-1279. MR 1912097, https://doi.org/10.1002/cpa.10044
  • [5] Yann Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 65-91. MR 2048567, https://doi.org/10.1007/s00205-003-0291-4
  • [6] J. C. Brunelli and Ashok Das, A Lax representation for the Born-Infeld equation, Phys. Lett. B 426 (1998), no. 1-2, 57-63. MR 1628305, https://doi.org/10.1016/S0370-2693(98)00265-2
  • [7] Dongho Chae and Hyungjin Huh, Global existence for small initial data in the Born-Infeld equations, J. Math. Phys. 44 (2003), no. 12, 6132-6139. MR 2023572, https://doi.org/10.1063/1.1621057
  • [8] A. A. Chernitskii, Born-Infeld equations, Encyclopedia of Nonlinear Science, ed. Alwyn Scott. New York and London: Routledge, 2004, pp. 67-69, arXiv:0509087v1 (hep-th).
  • [9] Roland Donninger, Joachim Krieger, Jérémie Szeftel, and Willie Wong, Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space, Duke Math. J. 165 (2016), no. 4, 723-791. MR 3474816, https://doi.org/10.1215/00127094-3167383
  • [10] S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293-326. MR 837683
  • [11] De-Xing Kong, Qiang Zhang, and Qing Zhou, The dynamics of relativistic strings moving in the Minkowski space $ \mathbb{R}^{1+n}$, Comm. Math. Phys. 269 (2007), no. 1, 153-174. MR 2274466, https://doi.org/10.1007/s00220-006-0124-z
  • [12] Michał Kowalczyk, Yvan Martel, and Claudio Muñoz, Kink dynamics in the $ \phi^4$ model: asymptotic stability for odd perturbations in the energy space, J. Amer. Math. Soc. 30 (2017), no. 3, 769-798. MR 3630087, https://doi.org/10.1090/jams/870
  • [13] Michał Kowalczyk, Yvan Martel, and Claudio Muñoz, Nonexistence of small, odd breathers for a class of nonlinear wave equations, Lett. Math. Phys. 107 (2017), no. 5, 921-931. MR 3633030, https://doi.org/10.1007/s11005-016-0930-y
  • [14] Joachim Krieger and Hans Lindblad, On stability of the catenoid under vanishing mean curvature flow on Minkowski space, Dyn. Partial Differ. Equ. 9 (2012), no. 2, 89-119. MR 2965741, https://doi.org/10.4310/DPDE.2012.v9.n2.a1
  • [15] Hans Lindblad, A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time, Proc. Amer. Math. Soc. 132 (2004), no. 4, 1095-1102. MR 2045426, https://doi.org/10.1090/S0002-9939-03-07246-0
  • [16] Yvan Martel and Frank Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl. (9) 79 (2000), no. 4, 339-425. MR 1753061, https://doi.org/10.1016/S0021-7824(00)00159-8
  • [17] Yvan Martel and Frank Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal. 157 (2001), no. 3, 219-254. MR 1826966, https://doi.org/10.1007/s002050100138
  • [18] Yvan Martel and Frank Merle, Asymptotic stability of solitons of the subcritical gKdV equations revisited, Nonlinearity 18 (2005), no. 1, 55-80. MR 2109467, https://doi.org/10.1088/0951-7715/18/1/004
  • [19] Yvan Martel, Frank Merle, and Tai-Peng Tsai, Stability in $ H^1$ of the sum of $ K$ solitary waves for some nonlinear Schrödinger equations, Duke Math. J. 133 (2006), no. 3, 405-466. MR 2228459, https://doi.org/10.1215/S0012-7094-06-13331-8
  • [20] Frank Merle and Pierre Raphael, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2) 161 (2005), no. 1, 157-222. MR 2150386, https://doi.org/10.4007/annals.2005.161.157
  • [21] C. Muñoz, F. Poblete, and J. C. Pozo, Scattering in the energy space for Boussinesq equations, preprint 2017 arXiv:1707.02616.
  • [22] Wladimir Neves and Denis Serre, Ill-posedness of the Cauchy problem for totally degenerate system of conservation laws, Electron. J. Differential Equations (2005), No. 124, 25. MR 2181268
  • [23] Erwin Schrödinger, A new exact solution in non-linear optics. (Two-wave-system), Proc. Roy. Irish Acad. Sect. A. 49 (1943), 59-66. MR 0008743
  • [24] Atanas Stefanov, Global regularity for the minimal surface equation in Minkowskian geometry, Forum Math. 23 (2011), no. 4, 757-789. MR 2820389, https://doi.org/10.1515/FORM.2011.027
  • [25] G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954

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Additional Information

Miguel A. Alejo
Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, Brasil
Email: miguel.alejo@ufsc.br

Claudio Muñoz
Affiliation: Departamento de Ingeniería Matemática and CMM UMI 2807-CNRS Universidad de Chile, Santiago, Chile
Email: cmunoz@dim.uchile.cl

DOI: https://doi.org/10.1090/proc/13947
Keywords: Born-Infeld equation, scattering, decay estimates, Virial
Received by editor(s): July 9, 2017
Received by editor(s) in revised form: August 22, 2017
Published electronically: January 8, 2018
Additional Notes: The first author was partially funded by Product. CNPq grant no. 305205/2016-1, Universal 16 CNPq grant no. 431231/2016-8 and MathAmSud/Capes EEQUADD collaboration Math16-01.
The second author was partially supported by Fondecyt no. 1150202, Millennium Nucleus Center for Analysis of PDE NC130017, Fondo Basal CMM, and MathAmSud EEQUADD collaboration Math16-01.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society

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