Exact Harmonic Dimensional Reduction and Conformal Lifting for Multicomponent $(3+1)$ Nonlinear Schr\"odinger Systems
A harmonic dimensional reduction framework is developed for $(3+1)\mathrm{D}$ systems of coupled nonlinear Schr\"odinger-type equations with stationary transverse trapping potentials. The central result is a lifting lemma: if the transverse phase functions are harmonic and the trapping potential exactly cancels the squared phase gradient, the full $(3+1)\mathrm{D}$ system reduces identically to a closed $(1+1)\mathrm{D}$ integrable hierarchy, and every solution of the reduced system lifts to an exact solution of the original multidimensional model. The framework is applied to four systems. For the scalar Gross--Pitaevskii equation, Kuznetsov--Ma breathers are embedded in $(3+1)\mathrm{D}$ geometries carrying vortex lattices with finite, non-singular density at the cores. For the two-component Manakov system, the phase-inversion ansatz yields exact vector solutions with vanishing mass current and non-trivial transverse spin current modulated by the longitudinal breather. For the three-component spinor $F=1$ Bose--Einstein condensate, a symmetric Kuznetsov--Ma breather and a spin-exchange rogue wave are constructed, the latter exhibiting transient density amplification by a factor of nine in the $m_F=0$ channel. For the Maxwell--Bloch system, self-induced transparency solitons, two-soliton elastic collisions, and Kuznetsov--Ma breathers are lifted to full $(3+1)\mathrm{D}$ geometry, with population inversion remaining transversely uniform despite arbitrary phase winding in the cross-section.