  Nonlinear surface-plasmon polaritons~(NSPPs) in nanophotonic waveguides
excite with dissimilar temporal properties due to input field modifications and
material characteristics, but they possess similar nonlinear spectral
evolution. In this work, we uncover the origin of this similarity and establish
that the spectral dynamics is an inherent property of the system that depends
on the synthetic dimension and is beyond waveguide geometrical dimensionality.
To this aim, we design an ultra-low loss nonlinear plasmonic waveguide, to
establish the invariance of the surface plasmonic frequency combs~(FCs) and
phase singularities for plasmonic peregrine waves and Akhmediev breather. By
finely tuning the nonlinear coefficient of the interaction interface, we
uncover the conservation conditions through this plasmonic system and employ
the mean-value evolution of the quantum NSPP field commensurate with the
Schr\"odinger equation to evaluate spectral dynamics of the plasmonic
FCs~(PFCs). Through providing suppressed interface losses and modified
nonlinearity as dual requirements for conservative conditions, we propose
exciting PFCs as equally spaced invariant quantities of this plasmonic scheme
and prove that the spectral dynamics of the NSPPs within the interaction
interface yields the formation of plasmonic analog of the synthetic photonic
lattice, which we termed \textit{synthetic plasmonic lattice}~(SPL).
