The periodic defocusing Ablowitz-Ladik equation and the geometry of Floquet CMV matrices
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In this work, we show that the periodic defocusing Ablowitz- Ladik equation can be expressed as an isospectral deformation of Floquet CMV matrices. We then introduce a Poisson Lie group whose underlying group is a loop group and show that the set of Floquet CMV matrices is a Coxeter dressing orbit of this Poisson Lie group. By using the group-theoretic framework, we establish the Liouville integrability of the equation by constructing action-angle variables, we also solve the Hamiltonian equations generated by the commuting flows via Riemann- Hilbert factorization problems.