Kink solitary solutions to a hepatitis C evolution model

Tadas Telksnys, ,Research Group for Mathematical, and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Studentu 50-147, Kaunas LT-51368, Lithuania, Zenonas Navickas, Miguel A. F. Sanjuán, Romas Marcinkevicius, Minvydas Ragulskis, ,Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain, ,Department of Applied Informatics, Kaunas University of Technology, Studentu 50-407, Kaunas LT-51368, Lithuania, ,Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA, ,Department of Software Engineering, Kaunas University of Technology, Studentu 50-415, Kaunas LT-51368, Lithuania
2017 Discrete and continuous dynamical systems. Series B  
The standard nonlinear hepatitis C evolution model described in (Reluga et al. 2009 ) is considered in this paper. The generalized differential operator technique is used to construct analytical kink solitary solutions to the governing equations coupled with multiplicative and diffusive terms. Conditions for the existence of kink solitary solutions are derived. It appears that kink solitary solutions are either in a linear or in a hyperbolic relationship. Thus, a large perturbation in the
more » ... bation in the population of hepatitis infected cells does not necessarily lead to a large change in uninfected cells. Computational experiments are used to illustrate the evolution of transient solitary solutions in the hepatitis C model. 1 2 TELKSNYS, NAVICKAS, SANJUÁN, MARCINKEVICIUS AND RAGULSKIS 1. Introduction. Recent developments in computer hardware and software enable to use powerful symbolic computation techniques for the construction of nonlinear wave solutions to high-dimensional nonlinear evolution equations in mathematical physics. Solitary (or soliton) solutions represent solitary wave packets that do not change their shape when propagating at constant velocities [24]. Due to their unique properties, construction of solitary solutions is an important problem in nonlinear science [26, 4, 1]. A short overview of typical examples illustrating the discovery of solitary solutions in nonlinear evolution problems is presented below. The dynamic pressure of an irrotational solitary wave propagating at the surface of water over a flat bed is studied in [6]. Properties of bright and dark solitary solutions in strongly magnetized warm plasmas are considered in [5]. Solitary wave solutions to a system of coupled complex Newell-Segel-Whitehead equations are constructed in [9]. Gray/dark solitons in nonlocal nonlinear media are analytically studied using the symmetry reduction method in [8]. A closed-form analytical solution, including bright and dark solitons, to the driven nonlinear Schrödinger equation is constructed in [22]. Solitary solutions are often encountered in coupled differential equations. Next we mention several typical examples. Dark-bright soliton solutions to a coupled Schrödinger system with equal, repulsive cubic interactions are considered in [2]. In [23], exact bright one-and two-soliton solutions to a particular type of coherently coupled Schrödinger equations are constructed using the non-standard Hirota's bilinearization method. Three families of analytical solitary wave solutions of generalized coupled cubic-quintic Ginzburg-Landau equations are obtained in [27]. Even though kink solitary solutions are the simplest type of solitons, their construction and analysis is far from being trivial [21] . Kinks and bell-type soliton solutions to a differential equation describing the dynamics of microtubules are constructed in [28] . The interaction of kink-type solutions of harmonic map equations is studied in [7] . The kink solutions to the negative-order KdV equation are constructed using the Lax pair in [18] . Kink solutions to models of transport phenomena and mathematical biology are considered in [25] . The main objective of this paper is to seek kink solitary solutions in a hepatitis C virus infection model [20] that explicitly includes proliferation of infected and uninfected hepatocytes. The mathematical equations of the model are:
doi:10.3934/dcdsb.2020106 fatcat:rmu5axfwtna6tkbdjxaw67tzie