Exact $q$-exponential Multi-Mode Solutions with Independent Centres and Power-Law Relaxation in the Plastino-Plastino Equation

Abstract: We present the first exact, multi-mode solutions to the Plastino-Plastino nonlinear diffusion equation with arbitrary power-law drift. By allowing each $q$-exponential mode to have its own independent, time-dependent centre, all inter-mode couplings in the drift term vanish, yielding fully separable evolution equations for centre motion, probability content, and (for the attractor mode) width. Transient modes exhibit constant width and decay via exact q-exponential (power-law) relaxation, while a single attractor mode irreversibly absorbs the entire probability flux, with fixed amplitude and time-growing width, driving the system to the known stationary q-exponential state from arbitrary initial conditions. The hierarchy closes exactly without approximation. These analytic solutions unify Tsallis nonextensive thermodynamics, fractal-space diffusion, and multi-scale relaxation dynamics, with direct applications to heavy-quark jets in quark-gluon plasma, Lévy flights in fractal media, and urban population redistribution. All previous exact results are recovered as special cases.