Breathers for nonlinear wave equations

Michael W. Smiley

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1988)

  • Volume: 82, Issue: 3, page 431-435
  • ISSN: 1120-6330

Abstract

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The semilinear differential equation (1), (2), (3), in ℝ × Ω with Ω ∈ ℝ N , (a nonlinear wave equation) is studied. In particular for Ω = ℝ 3 , the existence is shown of a weak solution u ⁢ ( t , x ) , periodic with period T , non-constant with respect to t , and radially symmetric in the spatial variables, that is of the form u ⁢ ( t , x ) = ν ⁢ ( t , | x | ) . The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of Cesari.

How to cite

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Smiley, Michael W.. "Breathers for nonlinear wave equations." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.3 (1988): 431-435. <http://eudml.org/doc/287218>.

@article{Smiley1988,
abstract = {The semilinear differential equation (1), (2), (3), in $\mathbb\{R\} \times \Omega$ with $\Omega \in \mathbb\{R\}^\{N\}$, (a nonlinear wave equation) is studied. In particular for $\Omega = \mathbb\{R\}^\{3\}$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x) = \nu(t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of Cesari.},
author = {Smiley, Michael W.},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Periodicity; Breathers; Distributional solution; Weighted Hilbert space; Method of alternative problems; solution manifold; semilinear wave equations; breathers},
language = {eng},
month = {9},
number = {3},
pages = {431-435},
publisher = {Accademia Nazionale dei Lincei},
title = {Breathers for nonlinear wave equations},
url = {http://eudml.org/doc/287218},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Smiley, Michael W.
TI - Breathers for nonlinear wave equations
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/9//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 3
SP - 431
EP - 435
AB - The semilinear differential equation (1), (2), (3), in $\mathbb{R} \times \Omega$ with $\Omega \in \mathbb{R}^{N}$, (a nonlinear wave equation) is studied. In particular for $\Omega = \mathbb{R}^{3}$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x) = \nu(t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of Cesari.
LA - eng
KW - Periodicity; Breathers; Distributional solution; Weighted Hilbert space; Method of alternative problems; solution manifold; semilinear wave equations; breathers
UR - http://eudml.org/doc/287218
ER -

References

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  1. CESARI, L., Functional analysis, nonlinear differential equations, and the alternative method, in «Non-linear Functional Analysis and Differential Equations», ( Cesari, , Kannan, Schuur, eds.), Marcel Dekker, New York, 1976, pp. 1-197. Zbl0343.47038MR487630
  2. CORON, J.M., Période minimale pour une corde vibrante de longueur infinite, C.R. Acad. Sci. Paris Ser. A.294 (1982), 127-129. Zbl0942.35512MR651803
  3. LEVINE, H.A., Minimal periods for solutions of semilinear wave equations in exterior domains and for solutions of the equations of nonlinear elasticity, J. of Math. Anal. and Appl. (to appear). Zbl0676.35063MR960819DOI10.1016/0022-247X(88)90155-2
  4. PALEY, R., WIENER, N., Fourier Transforms in the Complex Domain. «A.M.S. Colloquium Publications», Vol. 19, Providence, R.I., 1934. Zbl0011.01601
  5. SMILEY, M.W., Eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance, J. of Math. Anal. and Appl.122 no. 1 (1987), 129-151. Zbl0624.35015MR874965DOI10.1016/0022-247X(87)90350-7
  6. SMILEY, M.W., Time-periodic solutions of wave equations on ℝ 1 and ℝ 3 , Math. Meth. in Appl. Sci. (to appear) 3. Zbl0669.35059MR909918
  7. SMILEY, M.W., Breathers and forced oscillations of nonlinear wave equations on ℝ 3 , (submitted to), J. für die reine und angewandte Mathematik. Zbl0666.35066

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