Chaos and integrability in a nonlinear wave equation
This paper analyses the integrability of the equation \[ u_{tt}- u_{xx}- au+ u\left({2\over \pi L}\int^{\pi L}_ 0 u^ 2(x)dx\right)^ \alpha= 0,\tag{1} \] where \(a\), \(\alpha\) are positive constants, on the interval \([0,\pi L]\) with Dirichlet boundary conditions. The term integrability is taken to mean Hamiltonian (or Liouville) integrability. The main tool used is the Fourier-sine expansion of the solution of (1) in \(x\). The coefficients of such an expansion satisfy a Hamiltonian system of ODEs in time. The author studies this system for the space of solutions that have a finite Fourier decomposition. For \(\alpha= 2,3,\dots\), the author proves that the 2-mode system possesses chaotic solutions for \(L> 1/\sqrt a\). (A heuristic argument that this holds for any \(\alpha\neq 1,1/2,1/3,\dots\) is also given.) However, for the case \(\alpha= 1\), it is shown that the system of ODEs is integrable for any (finite) number of modes. (An explicit set of commuting first integrals is presented and it is shown that the solution \(u\) are quasi-periodic in time, i.e. functions of periodic functions of \(t\).) (1) is therefore termed integrable in this case. The proof of nonintegrability for \(\alpha= 2,3,\dots\) relies on a nonintegrability result of \textit{L. M. Lerman} [Sel. Math. Sov. 10, 297- 306 (1991; Zbl 0743.58017)], and the proof of integrability in the case \(\alpha= 1\) uses the construction of integrals by \textit{T. Cazenave}, \textit{A. Haraux} and \textit{F. B. Weissler} [J. Dyn. Differ. Equations 5, 129-154 (1993; Zbl 0783.35003)]. The results are interesting because for the case \(\alpha= 1\), (1) approximates the Sine-Gordon equation which is well-known to be integrable.
scientific article; zbMATH DE number 553858
scientific article