  In the study, we address the mathematical problem of proton migration in the
Earth's mantle and suggest a prototype for exploring the Earth's interior to
map the effects of superionic proton conduction. The problem can be
mathematically solved by deriving the self-consistent electromagnetic field
potential U(x,t) and then reconstructing the distribution function f(x, v, t).
Reducing the Vlasov-Maxwell system of equations to non-linear sh-Gordon
hyperbolic and transport equations, the propagation of a non-linear wavefront
within the domain, and transport of the boundary conditions in the form of a
non-linear wave are examined. By computing a 3D model and through
Fourier-analysis, the spatial and electrical characteristics of potential U(x,
t) are investigated. The numerical results are compared to the Fourier
transformed quantities of the potential (V) obtained through field observations
of the electric potential (Kuznetsov method). The non-stationary solutions for
the forced oscillation of a two-component system, and therefore, the
oscillatory strengths of two types of charged particles can be usefully
addressed by the proposed mathematical model. Moreover, the model, along with
data analysis of the electric potential observations and probabilistic seismic
hazard maps, can be used to develop an advanced seismic risk metric.
