×

Quasiperiodic soliton solutions to nonlinear Klein-Gordon equations on \(R^ 2\). (English) Zbl 0666.35076

We prove the existence and the regularity of time-quasiperiodic solutions to certain nonlinear Klein-Gordon equations on \(R^ 2\). Those solutions decay exponentially rapidly toward a constant equilibrium solution as the spatial variable goes to plus or minus infinity, and their spectrum consists of integral linear combinations of finitely many Q-independent prescribed frequencies. They are obtained by taking the diagonal section of certain exponentially decaying multiperiodic functions which we construct as solutions to auxiliary Cauchy problems associated with certain infinite-dimensional systems on tori. The main difficulty to overcome in our construction is a small divisor problem. It stems from the fact that the spectrum of the infinitesimal generator associated with the linearized equations consists of the union of a countable point spectrum and of a continuous spectrum in which the point spectrum is everywhere dense. The solution to the above problem requires no use of KAM - or related methods.
Reviewer: P.A.Vuillermot

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35B10 Periodic solutions to PDEs
35P05 General topics in linear spectral theory for PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Corduneanu, C.: Almost-periodic functions, New York: Wiley 1968 · Zbl 0175.09101
[2] Favard, J.: Leçons sur les fonctions presque-périodiques. Paris: Gauthiers-Villars 1933
[3] Levitan, B.M., Zhikov, V.V.: Almost-periodic functions and differential equations. Cambridge University Press 1982 · Zbl 0499.43005
[4] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Math. Sciences, vol. 44. Berlin Heidelberg New York: Springer 1983 · Zbl 0516.47023
[5] Pöschel, J.: On elliptic lower dimensional tori in hamiltonian systems. Math. Z.202, 559–608 (1989) · Zbl 0662.58037 · doi:10.1007/BF01221590
[6] Rudin, W.: Fourier analysis on groups. New York: Wiley 1960 · Zbl 0099.32201
[7] Scarpellini, B., Vuillermot, P.A.: Variétés stables et instables pour certaines équations des ondes semilinéaires dans \(\mathbb{R}\)2. C.R. Acad. Sci. Paris, Sér. I, Math.306, 33–36 (1988) · Zbl 0654.35069
[8] Scarpellini, B., Vuillermot, P.A.: Smooth manifolds for semilinear wave equations on \(\mathbb{R}\)2: On the existence of almost-periodic breathers. J. Diff. Equations77 1, 123–166 (1989) · Zbl 0702.35169 · doi:10.1016/0022-0396(89)90160-5
[9] Vuillermot, P.A.: Variétés lisses associées à certains systèmes dynamiques et solitons périodiques pour les équations de Klein-Gordon nonlinéaires sur \(\mathbb{R}\)2. C.R. Acad. Sci. Paris, Sér. I, Math.307, 639–642 (1988) · Zbl 0655.58032
[10] Vuillermot, P.A.: Periodic soliton solutions to nonlinear Klein-Gordon equations on \(\mathbb{R}\)2. Differential and Integral Equations, in press · Zbl 0731.35065
[11] Vuillermot, P.A.: Problèmes de Cauchy multipériodiques et solitons quasipériodiques pour les equations de Klein-Gordon nonlinéaires sur \(\mathbb{R}\)2. C.R. Acad. Sci. Paris, Sér. I, Math.308, 215–218 (1989) · Zbl 0687.35091
[12] Vuillermot, P.A.: Limit quasiperiodic Soliton solutions to nonlinear Klein-Gordon Equations on \(\mathbb{R}\)2. Adv. Math. (to be submitted) · Zbl 0666.35076
[13] Wayne, E.: Periodic and quasi-periodic solutions of nonlinear, wave equations via KAM theory. Pennsylvania State University Preprint no. 88027 (1988)
[14] Weinstein, A.: Periodic nonlinear waves on a half-line. Commun. Math. Phys.99, 385–388 (1985); erratum. ibid Weinstein, A.: Periodic nonlinear waves on a half-line. Commun. Math. Phys.107, 1 (1986) · Zbl 0585.35003 · doi:10.1007/BF01240354
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.