Gardner, Clifford Korteweg-de Vries equation and generalizations. IV: The Korteweg-de Vries equation as a Hamiltonian system. (English) Zbl 0283.35021 J. Math. Phys. 12, No. 8, 1548-1551 (1971). Summary: It is shown that if a function of \(x\) and \(t\) satisfies the Korteweg-de Vries equation and is periodic in \(x\), then its Fourier components satisfy a Hamiltonian system of ordinary differential equations. The associated Poisson bracket is a bilinear antisymmetric operator on functionals. On a suitably restricted space of functionals, this operator satisfies the Jacobi identity. It is shown that any two of the integral invariants discussed in Paper II of this series [R. Miura et al., J. Math. Phys. 9, 1204–1209 (1968; Zbl 0283.35019)] have a zero Poisson bracket. Cited in 4 ReviewsCited in 146 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 35A15 Variational methods applied to PDEs 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 70H05 Hamilton’s equations 70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics Citations:Zbl 0283.35019 PDFBibTeX XMLCite \textit{C. Gardner}, J. Math. Phys. 12, 1548--1551 (1971; Zbl 0283.35021) Full Text: DOI References: [1] DOI: 10.1098/rspa.1965.0019 · Zbl 0125.44202 [2] DOI: 10.1063/1.1664701 · Zbl 0283.35019 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.