Conservation Laws for Pure–Cubic Optical Solitons with Self–Frequency Shift

We investigate conservation laws for a third-order dispersion-dominated (pure-cubic) nonlinear Schrödinger equation with derivative perturbations and several forms of self-phase modulation. The multiplier approach yields local density–flux relations, and integrals of localized profiles are used to study optical power, linear momentum, and an energy-like invariant where their hypotheses are satisfied. These quantities provide diagnostics of power leakage, carrier-wave transport, and dispersive–nonlinear energy exchange. Importantly, inverse-power (anti-cubic) nonlinearities with a nonzero singular coefficient cannot support the previously proposed zero-background hyperbolic-secant profiles: the singular response diverges in their tails. Therefore, such profiles must not be interpreted as physical bright soliton solutions or used to assign conserved quantities. The distinction between formal conservation identities and admissible finite-energy waveforms is made explicit.