International Journal of Modern Nonlinear Theory and Application, 2013, 2, 7-13 http://dx.doi.org/10.4236/ijmnta.2013.21002 Published Online March 2013 (http://www.scirp.org/journal/ijmnta) Classic and Non-Classic Soliton Like Structures for Traveling Nerve Pulses Fidel Contreras 1 , Hilda Cervantes 1 , Máximo Aguero 1 , Ma. de Lourdes Najera 2 1 Facultad de Ciencias, Universidad Autónoma del Estado de México, Toluca de Lerdo, México 2 Plantel Nezahualcoyotl, Universidad Autónoma del Estado de México, Toluca de Lerdo, México Email: maaguerog@uaemex.mx Received December 10, 2012; revised January 18, 2013; accepted January 28, 2013 ABSTRACT After some reduction procedure made on the nonlinear evolution equation for nerve pulses, based on thermodynamic principles, new classic and non-classic traveling solutions have been obtained. We have studied this model for par- ticular values in the parameter space, and obtained both the bell and compacton like solutions. These nonlinear traveling waves could be responsible for transmitting efficiently the necessary information along the axons. The non-classic structures named as compactons, due to their robust configuration, could be considered in some sense a more realistic type of nonlinear chargers of information. The last solutions do not have tails and as adiabatic waves could propagate along the nerve with constant velocity that could be equal, smaller or higher than the sound velocity. Keywords: Compactons; Nonlinear Waves; Solitons; Nerve Pulse; Molecular Waves 1. Introduction One of key fundamental problems in biophysics is to understand how nature makes to carry information with- out significant distortion along distances between two considerable long separated centers. Now it is well estab- lished that the dynamic of ionic currents through voltage channels is responsible for the change of the membrane potential in nerve tissues. As is well known the first measurements on ionic currents were performed by Hodgkin and Huxley (HH) in the 50s [1]. In 1966 Katz [2] proposed for traveling pulse the soliton traveling pul- se as a simplest model for this activity. Later, the Hodgkin and Huxley system was developed independently by Richard FitzHugh, and Jin-ichi Na- gumo [3,4]. Based on the work of Balthazar Van Der Pol, FitzHugh proposed a simplified neuronal model of Hodg- kin and Huxley. For its part, Nagumo suggested as ana- logous neuronal, a nonlinear electrical circuit, controlled by an equation system also similar to those of Van Der Pol currents. The proposed simplified analog of these authors, is called FitzHugh-Nagumo model. Being sus- ceptible fairly complete analysis, the FHN system allows a qualitative understanding of the phenomenon of excit- ability, from the point of view of dynamic systems [5] and constitutes a classical model of neurophysiology. The current importance of this model is beyond the scope of biophysics and neurophysiology, being of interest to researchers who need to understand the wide range of nonlinear phenomena accompanying the phenomenon of excitability. By introducing an approximate scheme to the famous model of Hodgkin-Huxley, Muratov in his paper [6] obtained solitary wave pulse for nerve conduc- tance and obtained the value of velocities that are in agreement with experimental results. Recently, Heimburg and coworkers have developed a model for nerve pulses that support soliton like solutions [7-10]. The model is constructed considering the nerve axon as a one dimensional cylinder with lateral density excitations moving along the axes that is represented by the coordinate z. This model shows the appearance of lipid phase transition slightly below physiological tem- peratures. Given measured values of the compression modulus as a function of lateral density and frequency, soliton properties can be determined by the velocity of the traveling waves. That is, resuming we can say that this theory is based on the lipid transition from a fluid to a gel phase at slightly below of body temperatures. The effects of nonlinearity and dispersion as is common would be responsible for appearance of soliton like struc- tures in nerve membrane in the gel state [10] and more recent results on soliton and periodical solutions can be found in [11]. On the other hand, as is well known, classical solitons Copyright © 2013 SciRes. IJMNTA