Generalized Fourier transform and action-angle variables for the Lakshmanan–Porsezian–Daniel equation

In this work, we develop the generalized Fourier transform and construct action-angle variables for the Lakshmanan-Porsezian-Daniel (LPD) equation. This framework mathematically characterizes ultrashort femtosecond soliton propagation in dispersive nonlinear optical fibers. By establishing completeness relations for squared eigenfunctions in the spectral problem, we prove that solutions of the Lax system span the Schwartz function space, thereby rigorously formulating the generalized Fourier transform. Furthermore, explicit Poisson brackets construction for scattering data generates canonical action-angle variables governed by spectral parameters. Crucially, the Hamiltonian formulation reveals a duality: conservation laws admit equivalent representations through both spectral parameters and potential functions, establishing a direct correspondence between scattering data and nonlinear potentials. This framework not only deepens the analytical understanding of the LPD equation’s integrability but also provides theoretical foundations for controlling optical solitons in defocusing media with higher-order nonlinear effects.