Soliton dynamics under fractional action-induced non-Markovian dissipation and memory effects

We present a coherent variational framework based on the fractional actionlike variational approach (FALVA) in which fractionality enters solely through a time/propagation-weighted kernel rather than through fractional derivatives, and apply it to classical and quantum soliton dynamics in optical fibers. Beginning from a weighted action Sα=Γ−1(α)∫∫(z1−z)α−1Ldzdt, we derive modified Euler–Lagrange field equations that augment the standard nonlinear Schrödinger model with a scale-dependent imaginary potential i(1−α)/(z1−z)that encodes non-Markovian dissipation and memory. We show that a simple gauge transformation maps modifies the classical integrable nonlinear Schrödinger equation (NLSE), which preserves the inverse scattering data and ensures that soliton stability and elastic collision properties are inherited up to a global amplitude scaling. From the weighted action we formulate a generalized Noether theorem producing kernel-weighted conserved charges (weighted power, momentum, and Hamiltonian), and we give a covariant generalization of the construction suitable for field theories with a temporal kernel. Promoting the envelope to an operator field yields a Heisenberg evolution with the same scale-dependent dissipative term and, after tracing out bath degrees of freedom, an effective master equation with a Lindblad-like term whose rate scales as (1−α)/(z1−z), linking αto bath spectral properties. Finally, we describe practical experimental diagnostics to extract αfrom single-soliton amplitude scaling, Optical Signal-to-Noise Ratio (OSNR) trends in Wavelength-Division Multiplexing (WDM) measurements, and photon-number statistics in the few-photon regime, discuss implications for long-haul transmission (timing jitter, effective nonlinear length, four-wave mixing penalties), and outline numerical and microscopic routes to relate αto underlying scattering and phonon processes. The FALVA approach thus offers a parsimonious, variationally consistent, and experimentally testable single-parameter extension of standard models that captures memory and non-Markovian dissipation while preserving much of the analytical structure of integrable soliton theory.