Commuting flows and infinite-dimensional tori: sine-Gordon
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 [\textit{V. I. Arnol'd}, Sib. Mat. Zh. 4, 471--474 (1963; Zbl 0131.39102)] or [\textit{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.
scientific article; zbMATH DE number 6815940