Generalized Fourier transform and action-angle variables for the local/nonlocal derivative nonlinear Schrödinger equations

In this work, we present the generalized Fourier transform and action-angle variables for the local/nonlocal derivative nonlinear Schrödinger equations. The generalized Fourier transform, derived from the completeness relation, serves as the expansion of any Schwartz function in terms of solutions to the Lax spectral problem. We establish the completeness relations for squared solutions to the Lax spectral problem and determine the Poisson brackets between the scattering data. Action-angle variables, expressed through the scattering data, are constructed based on these Poisson brackets. It is noteworthy that the analyticity of scattering data differs from that of the AKNS-type spectral problem, necessitating adjustments to the integral paths when deriving the completeness relations. Interestingly, the conservation laws for the equation can be expressed in terms of either the spectral parameter or the potential functions, revealing the connection between the potential functions and the scattering data.