# Approaching Electroencephalographic Pathological Spikes in Terms of Solitons
## 3 Equations for Solitons  
In mathematical terms, solitons stand for one of the countless possible solutions for different types of nonlinear wave equations [6]. In particular, the Korteweg-de Vries equation (KdVE), originally developed to describe weakly nonlinear wave propagation in shallow water, admits analytic solutions representing solitons. KdVE equation can be written as  
\[\partial_{t}\Phi+\partial^{3}x\Phi-6\Phi\partial_{x}\Phi=0\]  
where \(\partial_{x}\) and \(\partial_{t}\) are partial derivatives of x and t, \(\Phi\) stands for the height displacement of the water surface from its equilibrium height. KdVE implies an equilibrium between dispersion \(\partial^{3}x\Phi\) and advection \(\Phi\partial_{x}\Phi\), the latter standing for the particle transport provided by bulk motion and velocity of the fluid. KdVE can describe soliton systems in which the particles are driven by fluid pressure [28] as well as soliton turbulence in ocean waves [29]. The Benjamin-Bona-Mahony equation (BBME) is a KdVE variant for modeling long surface gravity waves of small amplitude, propagating uni-directionally in 1+1 dimensions. Yet, a particle-based fluid trajectory description of multi-soliton interactions can be achieved by the Kadomtsev-Petviashvili equation (KPE), i.e., a two-dimensional version of KdVE. Apart from KdVE, BBME, and KPE, other wave equations display soliton solutions. The most prominent is the one-dimensional nonlinear Schrodinger equation (NLSE), an example of integrable systems of nonlinear partial differential equations owning an infinite set of conservation laws. NLSE stands for a universal tool to model the dynamical evolution in dispersive media of weakly nonlinear, varying packets of quasi-monochromatic waves [30]. Among the manifold versions, the linearly damped and driven NLSE can be expressed in the dimensionless form:  
\[\mathrm{i}u_{t}+\frac{1}{2}u_{xx}+\,|u\,|^{2}u=f-iyu\]  
where f = f(x,t) is the driving (or forcing) of the system, iyu is the linear damping of strength \(\gamma>0\)[31]. Depending on the chosen physical setting, different NLSE formulations provide different results according to the values of the dispersive nonlinear coefficients [30]. NLSE has a wide range of applications, including, e.g., the slow evolution of wave amplitude in shallow waters of uniform depth, the irregular wave trains propagating in deep water, superconductivity devices, nonlinear optics, etc. [32]. Concerning Bose-Einstein condensates, the soliton's behavior can be described by a variant of the NLSE referred to as the Gross-Pitaevskii equation (GPE) [30,33]. In this case, the velocities of single particles are determined by the phase of the wave function. Figure 1 illustrates a few examples of soliton solutions to various nonlinear wave equations. Therefore, every attempt to look for hints of nervous solitons requires electric solitary waves to obey explicit nonlinear wave equations. Figure 1: Features of different types of solitons. (**A**) A cnoidal wave is a periodic traveling-wave solution of KdVE. It refers to surface waves whose wavelength is large compared to the water depth; modified from [[https://demonstrations.wolfram.com/CnoidalWavesFromKortewegDeVriesEquation/](https://demonstrations.wolfram.com/CnoidalWavesFromKortewegDeVriesEquation/)]([https://demonstrations.wolfram.com/CnoidalWavesFromKortewegDeVriesEquation/](https://demonstrations.wolfram.com/CnoidalWavesFromKortewegDeVriesEquation/)) (accessed on 4 March 2024) (**B**) Formation of a rogue wave in the deep ocean, modeled using LNSE; modified from [[https://demonstrations.wolfram.com/RogueOceanWaves/](https://demonstrations.wolfram.com/RogueOceanWaves/)]([https://demonstrations.wolfram.com/RogueOceanWaves/](https://demonstrations.wolfram.com/RogueOceanWaves/)) (accessed on 4 March 2024) (**C**) Top figure: example of dynamic behavior of an optical soliton with spatio-temporal dispersion, based on variable-coefficient NLSE.