Commuting flows and infinite-dimensional tori: sine-Gordon. (English) Zbl 1386.35372
The author considers in this paper the periodic sine-Gordon equation
\[
u_{tt}-u_{xx}=\sin u,
\]
where \(u(x,t)\) is a real valued function in the class of smooth one functions. He explains why the complete set of conserved functionals for sine-Gordon is an infinite-dimensional torus and identifies the generic flow (generated by this equation) with straight-line almost periodic motion on an infinite-dimensional torus. The paper is organized as follows: the first section is an introduction to the subject. Section 2 contains preliminaries and covers complete integrability of nonlinear evolution equations. Moreover, the author gives an infinite-dimensional version of the well known Arnold-Liouville theorem [V. I. Arnol’d, Sib. Mat. Zh. 4, 471–474 (1963; Zbl 0131.39102)] or [V. I. Arnold, Mathematical methods of classical mechanics. New York - Heidelberg - Berlin: Springer-Verlag. (1978; Zbl 0386.70001)]. Finally, section 3 is devoted to applying the results obtained in section 2 to the study of the sine-Gordon equation.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |
| 35B10 | Periodic solutions to PDEs |
References:
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| [3] | Lamb Jr., G.L.: Higher conservation laws in ultrashort optical pulse propagation. Phys. Lett. 32A, 251 (1970) · doi:10.1016/0375-9601(70)90306-3 |
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| [6] | Liouville, J., No article title, J. Math., 20, 137 (1855) |
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| [9] | Schwarz, M.: Commuting flows and invariant Tori: Korteweg-de Vries. Adv. Math. 89 (1991) · Zbl 0753.35087 |
| [10] | Schwarz, M.: Nonlinear Schrodinger infinite-dimensional Tori and neighboring Tori. J. Nonlinear Math. Phys. 10, 65-77 (2003) · Zbl 1021.35111 · doi:10.2991/jnmp.2003.10.1.5 |
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