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Korteweg-de Vries equation and generalizations. IV: The Korteweg-de Vries equation as a Hamiltonian system. (English) Zbl 0283.35021
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.

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
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[1] DOI: 10.1098/rspa.1965.0019 · Zbl 0125.44202 · doi:10.1098/rspa.1965.0019
[2] DOI: 10.1063/1.1664701 · Zbl 0283.35019 · doi:10.1063/1.1664701
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