On the symplectic structure of the phase space for periodic KdV, Toda, and defocusing NLS
Let \(L^2_0= \{q\in C^2[ 0,1 ]\); \(\int^1_0 q(x) dx= 0\}\) be a space of periodic potentials of average 0. According to Gardner, the phase space \((L^2_0, \omega)\) for the KdV equation can be identified with the space of potentials \((p, q)\) from \(L^2_0\) with symplectic structure \[ \omega (p, q)= \sum_{n\geq 1} (\pi n)^{-1} (p_{2n}\wedge q_{2n-1}- p_{2n-1} \wedge q_{2n}), \] where \(p_n\) and \(q_n\) are Fourier modes for \(p\) and \(q\): \[ q(x)= 2\sum_{n\geq 1} q_{2n} \cos 2\pi nx+ q_{2n-1} \sin 2\pi nx. \] The main result of the paper is an implicit construction of symplectomorphisms \(\Omega: (L^2_0, \omega)\to (\ell^2_{1/2} (\mathbb{R}^2), \omega_0)\) where \(\ell^2_{1/2} (\mathbb{R}^2)\) consists of the sequences \((x_n, y_n)\), \(n\geq 1\), \(\sum n (x^2_n+ y^2_n)< \infty\), \(\omega_0= \sum_n (x_n \wedge y'_n- y_n \wedge x'_n)\). Moreover, in polar coordinates \(r_j, \alpha_j\) given by \(x_j= r_j \cos \alpha_j\), \(y_j= r_j \sin \alpha_j\), the quantities \({1\over2} r^2_j, \alpha_j\) are action-angle variables for the KdV equation. This theorem, besides the proof of the existence of global action-angle variables, allows to describe the structure of singularities for the fibration of the phase space by invariant tori. Analogous considerations are given for defocusing nonlinear Schrödinger equations and periodic Toda lattices.
scientific article; zbMATH DE number 878408
scientific article