# Approaching Electroencephalographic Pathological Spikes in Terms of Solitons
## 3 Equations for Solitons
Bottom Figure: Time evolution of the soliton position x(t) interpreted in terms of wave streamlines obeying NLSE; modified from [34]. (**D**) Peregrine soliton solution based on a periodic-wave background generated by an NLSE variant; modified from [4]. (**E**) Soliton solution of the integrable focusing NLSE with spectral data consisting of a pair of conjugate poles of order 2n; modified from [35]. (**F**) Soliton trains (packets) in shallow water wind waves. The wave train (black) is achieved by low-pass filtering of the data (red); modified from [29]. (**G**) Single soliton and soliton trains detected for a month in the solar wind’s pressure; modified from [36]. (**H**) Weakly nonlinear two-soliton propagation in shallow water described by KdVE; modified from [[https://demonstrations.wolfram.com/SolitonsFromTheKortewegDeVriesEquation/](https://demonstrations.wolfram.com/SolitonsFromTheKortewegDeVriesEquation/)]([https://demonstrations.wolfram.com/SolitonsFromTheKortewegDeVriesEquation/](https://demonstrations.wolfram.com/SolitonsFromTheKortewegDeVriesEquation/)) (accessed on 4 March 2024) (**I**) NLSE-derived soliton profile perturbed by a periodic potential; modified from [[https://demonstrations.wolfram.com/SolutionOfANonlinearSchroedingerEquation/](https://demonstrations.wolfram.com/SolutionOfANonlinearSchroedingerEquation/)]([https://demonstrations.wolfram.com/SolutionOfANonlinearSchroedingerEquation/](https://demonstrations.wolfram.com/SolutionOfANonlinearSchroedingerEquation/)) (accessed on 4 March 2024). Solitons in the Electrical Domain and in EEG Spikes  
Solitons may be produced in the context of the electrical domain, both at the macro-, meso- and micro-levels of observation. The occurrence of electrostatic solitons is ubiquitous in macroscopic contexts such as two-fluid astrophysical plasmas, planetary magnetospheres and lunar wakes [3]. In solar wind, the interplanetary magnetic fields provide favorable conditions for solitons formation [36]. In space plasmas, nonlinear electrostatic solitary waves can be interpreted in terms of ion- and electron-acoustic solitons propagating parallel to the ambient magnetic field. Concerning the micro-levels of observation, electromagnetic fields and saturable nonlinear dielectrics may influence solitary waves' shape and behavior [37]. Self-reinforcing magnetic solitons have been generated in Bose-Einstein condensates from atoms with different spins [38]. In nematic and cholesteric liquid crystals, external electric fields drive the collective reorientation of nematic molecules. This process generates solitons representing self-trapped \"bullets\" that move with high speed, perpendicularly to the electric field and to the initial alignment direction [14; 39]. The above-mentioned nonlinear wave equations have been widely used to cope with electric as well as external uniform magnetic fields inside the discrete nonlinear electrical lattices used for electrical transmission [40]. Once the solitons can be generated in an electromagnetic milieu, the question is whether they could occur in the hemispheric electric field detected by scalp EEG. The fact that NLSE may govern soliton propagation on spherical surfaces of electromagnetic fields traveling in coiled optical fibers [41] provides an indirect hint that solitons can be uncovered inside spherical electric fields [42]. Further, it has been shown that electric or magnetic monopole pairs can interact with Coulombic fields equipped with the SO(3) rotations typical of spherical surfaces [43]. Therefore, the nervous electric activity detected on the scalp could be assessed by using nonlinear wave equations with solitons solutions.