Some Traveling Wave Solutions of a Hyperbolic Model for Chemotaxis in 1-D

Abstract

The traveling wave solution of a hyperbolic model for chemotaxis in one space dimension is studied in this paper. By using some transformations of dependent variables and independent variables, we apply the tanh method and improved tanh method to the model, from which some traveling wave solutions in explicit form are presented.

1. Introduction

Chemotaxis is a widespread phenomenon in biological systems describing the response of individuals to an external chemical together with its gradient. It has attracted significant interest due to its critical role in a wide range of biological phenomena. Chemotaxis was studied mathematically beginning with the early papers of Patlak [1] and Keller and Segel [2, 3]. Since then the mathematical literature on the modeling of chemotaxis has grown rapidly. The parabolic Keller-Segel model of chemotaxis [2] has proved a cornerstone for much of this work. For a review of the recent literature, see Hillen and Painter [4]. However, the parabolic Keller-Segel model allows arbitrarily large speed of the population, it is physically unrealistic for small time. Motivated by the fact, Hillen and Stevens [5] proposed a hyperbolic model for chemotaxis in 1-D, which deal with finite speed. The model is𝑢𝑢^+𝑡_−𝑡+𝛾(𝑆,𝑆_𝑡,𝑆_𝑥)𝑢⁺_𝑥=−𝜇⁺𝑆,𝑆_𝑡,𝑆_𝑥𝑢⁺+𝜇⁻𝑆,𝑆_𝑡,𝑆_𝑥𝑢⁻,−𝛾(𝑆,𝑆𝜏𝑆_𝑡,_𝑡𝑆_𝑥)𝑢⁻_𝑥=𝜇⁺𝑆,𝑆_𝑡,𝑆_𝑥𝑢⁺−𝜇⁻𝑆,𝑆_𝑡,𝑆_𝑥𝑢⁻,=𝐷₀𝑆_𝑥𝑥+𝑓𝑆,𝑢⁺+𝑢⁻,𝜏≥0,(1.1) where 𝑢^±(𝑥,𝑡) denotes the particle densities of right (+)/left (−) moving particles, 𝛾(𝑆,𝑆_𝑡,𝑆_𝑥) denotes the particle speed, and 𝜇^±(𝑆,𝑆_𝑡,𝑆_𝑥) are turning rates (rates of change of direction from + to − and vice versa).𝑆(𝑥,𝑡) denotes the concentration of the chemical external signal. The production and degradation of the external chemical signal is modeled by the reaction term 𝑓.

For the special case with 𝐷=0, constant speed 𝛾 and 𝜇^± depending only on 𝑆 and 𝑆_𝑥,𝑢𝑢^+𝑡_−𝑡+𝛾𝑢^+𝑥=−𝜇⁺𝑆,𝑆_𝑥𝑢⁺+𝜇⁻𝑆,𝑆_𝑥𝑢⁻,−𝛾𝑢^−𝑥=𝑆𝜇⁺_𝑡𝑆,𝑆_𝑥𝑢⁺−𝜇⁻𝑆,𝑆_𝑥𝑢⁻,=𝑓𝑆,𝑢⁺+𝑢⁻.(1.2) Segel in [9] first used it to analyze a very specific scenario. Later Rivero et al. [6] and Ford et al. [7, 8] used it to describe experimental data. In the works [5–9], the issues of local and global in time existence of solutions were considered theoretically and numerically. Hillen and Levine [10] studied finite time blowup of the system (1.2) with𝑓=𝑆𝑢⁺+𝑢⁻,(1.3) and three specific forms of turning rates, one of which is𝜇^±𝑆,𝑆_𝑥=𝛾2𝐷₁𝛾∓𝑆𝛼𝑆^𝑥.(1.4)

Keller and Segel in 1971 [11] considered an experimental setup that results in visible bands of bacteria traveling up a capillary tube filled with a mixture of oxygen and nutrient rich substrate. It raised the question of existence of a traveling wave solution to the Keller-Segel model of chemotaxis. Liu in 2008 [12] constructed some explicit solutions of the system (1.2) with the choice of 𝑓 and 𝜇^± as in (1.3) and (1.4).

In this paper, we try to find some exact traveling wave solutions of the system𝑢𝑢^+𝑡_−𝑡+𝛾𝑢^+𝑥=−𝜇⁺𝑆,𝑆_𝑥𝑢⁺+𝜇⁻𝑆,𝑆_𝑥𝑢⁻,−𝛾𝑆𝑢^−𝑥_𝑡=𝜇⁺𝑆,𝑆_𝑥𝑢⁺−𝜇⁻𝑆,𝑆_𝑥𝑢⁻,=𝐷₀𝑆_𝑥𝑥+𝑓𝑆,𝑢⁺+𝑢⁻,(1.5) where 𝐷₀≥0,𝛾>0 are constants, 𝑓(𝑆,𝑢⁺+𝑢⁻) and 𝜇^±(𝑆,𝑆_𝑥) are defined above (1.3) and (1.4). It is clear that system (1.5) is a special case of the system (1.1) and a more generalized case of (1.2).

The system (1.5) can be transformed into an equivalent system for the total particle density, 𝑢=𝑢⁺+𝑢⁻, and particle flux, 𝑣=𝑢⁺−𝑢⁻. The resulting system for (𝑢,𝑣) is (3.1) below. Using tanh method and improved tanh method, we search for traveling wave solutions of the system (3.1). When 𝐷₀≠0, three different kinds of traveling wave solutions are found. To the best of our knowledge, those traveling wave solutions are new and have not appeared in literature. When 𝐷₀=0, the solutions obtained here are different from those known ones in [12].

2. The Two Methods

The tanh method and improved tanh method are powerful techniques to search for traveling wave solutions arising from one dimensional nonlinear wave and evolution equations. Let us first review the main features of two methods. They will be used in this paper. For more details, see [13–17] and reference therein.

For both methods, we first use the wave variable 𝜉=𝑘(𝑥−𝑝𝑡)+𝜉₀ to carry a PDE in two independent variables𝐻𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,…=0(2.1) into an ODE𝑄𝑢,𝑢′,𝑢^,…=0,(2.2) where 𝑘 and 𝑝 are the wave number and the wave speed, respectively.

2.1. The tanh Method

In the standard tanh method, the tanh is used as a new variable and the solution of (2.2) is expressed as a finite series of tanh𝑢(𝜉)=^𝑀_𝑖=0𝑎_𝑖𝑇^𝑖,𝑇=tanh𝜉,(2.3) where 𝑀 is a positive integer that will be determined. Substituting (2.3) into the reduced ODE (2.2) results in an algebraic equation in powers of 𝑇. To determine the parameter 𝑀, we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms. With 𝑀 determined, we collect all coefficients of powers of 𝑇 in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters 𝑘,𝑝, and 𝑎_𝑖. Having determined these parameters, we obtain an analytic solution 𝑢(𝑥,𝑡) in a closed form.

The tanh method is a computerizable method, in which generating an algebraic system and solving it are two key procedures and laborious to do by hand. But they can be implemented on a computer with the help of computer algebra software such as Mathematica.

2.2. The Improved tanh Method

The crucial point of the improved tanh method is to replace the tanh𝜉 in (2.3) by the solution of Riccati equation𝐹^=𝐴+𝐵𝐹+𝐶𝐹²,𝐶≠0,(2.4) where ^∶=𝑑/𝑑𝜉, and 𝐴, 𝐵, 𝐶 are some constants.

Supposing that 𝐹 is a solution of (2.4), it is easy to verify(1)if 𝐵²−4𝐴𝐶>0, (2.4) possesses a solution 𝐹(𝜉)=−𝐵2𝐶−√𝐵²−4𝐴𝐶2𝐶tanh√𝐵²−4𝐴𝐶2𝜉;(2.5)(2)if 𝐵²−4𝐴𝐶=0, (2.4) possesses a solution 𝐹(𝜉)=−𝐵2𝐶−𝐶1𝜉;(2.6)(3)if 𝐵²−4𝐴𝐶<0, (2.4) possesses a solution 𝐹(𝜉)=−𝐵2𝐶+√4𝐴𝐶−𝐵²2𝐶tan√4𝐴𝐶−𝐵²2𝜉.(2.7)

Based on (2.5), the solution of (2.2) can be expressed as in (2.3), and based on (2.6) and (2.7), the solution of (2.2) can be expressed as𝑢(𝜉)=^𝑀_𝑖=0𝑎_𝑖1𝜉^𝑖𝑢(𝜉)=_𝑀,(2.8)_𝑖=0𝑎_𝑖tan^𝑖𝜉,(2.9) where parameters 𝑀,𝑘,𝑝, and 𝑎_𝑖 are to be determined.

3. Traveling Wave Solutions

A transformation of the dependent variables is necessary, before the just-described methods can be applied to the hyperbolic model (1.5). As in [10], we rewrite system (1.5) as a system for 𝑢=𝑢⁺+𝑢⁻ and 𝑣=𝑢⁺−𝑢⁻:𝑣𝑢_𝑡+𝛾𝑣_𝑥=0,_𝑡+𝛾𝑢_𝑥=𝛾𝐷₁⋅𝛼𝑆^𝑥𝑆𝑢−𝐷𝛾²₁𝑆𝑣,_𝑡=𝐷₀𝑆_𝑥𝑥+𝑆𝑢.(3.1) Eliminating 𝑣 in system (3.1), it is clear that (3.1) is of the form−1𝛾𝑢^𝑡𝑡+𝛾𝑢_𝑥𝑥=𝛼𝛾𝐷₁𝑢𝑆𝑆^𝑥_𝑥+𝛾𝐷₁𝑆_𝑡𝑢_𝑡,=𝐷₀𝑆_𝑥𝑥+𝑆𝑢.(3.2) Taking into account the transformation𝑢=𝑢(𝜉),𝑆=𝑒^𝜑(𝜉),𝜉=𝑘(𝑥−𝑝𝑡)+𝜉₀,(3.3) where 𝑘>0,𝑝>0, and 𝜉₀ is an arbitrary constant, the (3.2) can be put into the format𝑘𝛾−𝑝²𝛾𝑢^=𝛼𝛾𝑘𝐷₁𝑢′𝜑′+𝑢𝜑^−𝐷𝛾𝑝₁−𝑘𝑝𝜑_𝑢′,(3.4)=𝐷₀𝑘²𝜑^2+𝜑^+𝑢,(3.5) where 𝑝²≠𝛾².

3.1. Using tanh Method

Let𝑢(𝜉)=^𝑀_𝑖=0𝑎_𝑖tanh^𝑖𝜉.(3.6) It is noted that ttaannhh₂𝑥𝑑𝑥=ln||cosh𝑥||+𝐶,tanh_𝑘𝑥𝑑𝑥=𝑥−tanh𝑥+𝐶,𝑥𝑑𝑥=−tanh^𝑘−1𝑥𝑘−1+tanh^𝑘−2𝑥𝑑𝑥,(3.7) as a result, we guess the 𝜑 has the form𝜑(𝜉)=𝑏₋₂𝜉+𝑏₋₁ln||cosh𝜉||+^𝑀−2_𝑖=0𝑏_𝑖tanh^𝑖𝜉.(3.8)

Substituting 𝑢(𝜉) and 𝜑(𝜉) into (3.4)-(3.5), we obtain 𝑀=2, and the solutions of system (3.4)-(3.5) is of the form𝑢=𝑎₀+𝑎₁tanh𝜉+𝑎₂tanh²𝜉,𝜑=𝑏₋₂𝜉+𝑏₋₁ln||cosh𝜉||.(3.9) With the help of the symbolic software Mathematica, substituting (3.9) into (3.4) and (3.5)and setting the coefficients of tanh^𝑖𝜉 to be zero, we obtain−2𝑘𝑘𝐷𝐷₁₁𝛾²−𝑝²𝑎₂−𝛼𝛾²𝑘𝑎₁𝑏₋₂+𝑎₀𝑏₋₁+𝛾²𝑝𝑎₁=0,𝛾²−−3𝑘𝑝²2𝐷𝑎₁₁−𝛼𝛾²𝑘𝑎₂𝑏₋₂+𝑎₁𝑏₋₁+𝛾²𝑝𝑎₂=0,𝑘𝑝𝑏₋₂𝛾²−𝑝²+𝛼𝛾²𝑏₋₁𝑎₂=0,𝑘𝑝𝑏+₋₁𝐷₀𝑘²𝑏²⁻²+𝑏₋₁+𝑎₀=0,𝐷+2𝐷₀𝑘²𝑏₋₂𝑏₋₁+𝑎₁=0,₀𝑘²𝑏²⁻¹−𝑏₋₁+𝑎₂=0.(3.10) If 𝐷₀≠0 and 𝑝 satisfiesΔ=+𝛾2²−𝛾₂𝑝²²𝐷²¹4𝐷²⁰+4𝛼𝐷₀+5𝛼²>0,−𝑝²𝐷₁𝛼³𝛾²−𝛼²𝛾⁴𝐷₀𝛼+𝐷₀(3.11) solving the algebraic equations (3.10) with the aid of Mathematica, we arrive at𝑘=𝛾²𝑝√Δ𝑏₋₁2𝐷₀𝐷₁||4𝐷₁𝛾²−𝑝²+𝛼𝛾²𝛾²−𝑝²||,=−2𝐷¹𝛾²−𝑝²𝛼𝛾²𝑏₋₂,=1𝛼𝑘𝐷₀⋅2𝑝𝐷⁰𝐷₁𝛾²−𝑝²+𝑝𝐷₀𝛼𝛾²−𝑝𝐷₁𝛼𝛾²−𝑝²4𝐷₁𝛾²−𝑝²+𝛼𝛾²𝑎₀,=2𝑘²𝐷₀𝐷₁𝛾²−𝑝²𝛼𝛾²−𝑝𝑝𝛼𝛾²𝐷₀−𝑝𝛼𝐷₁𝛾²−𝑝²+2𝑝𝐷₀𝐷₁𝛾²−𝑝²−𝐷𝛼𝐷₀𝛼𝛾²+4𝐷₁𝛾²−𝑝²₀𝑝𝛼𝛾²𝐷₀−𝑝𝛼𝐷₁(𝛾²−𝑝²)+2𝑝𝐷₀𝐷₁(𝛾²−𝑝²)²𝛼²𝐷²⁰𝛼𝛾²+4𝐷₁(𝛾²−𝑝²)²𝑎₁,=×2𝑘𝑝𝐷𝛼₁₂𝛾²−𝑝²𝛾²+2𝑝𝛼𝛾²𝐷₀+2𝑝𝛼𝐷₁𝛾²−𝑝²+4𝑝𝐷₀𝐷₁𝛾²−𝑝²𝑎𝛼²𝛾²𝛼𝛾²+4𝐷₁𝛾²−𝑝²₂,=−2𝑘²𝐷₀𝐷₁𝛾²−𝑝²𝛼𝛾²+2𝐷₁𝛾²−𝑝²𝛼𝛾²²,(3.12) which leads to the solutions of system (3.1)𝑢𝑆(𝑥(𝑥,𝑡,)𝑡)==𝑎||₀c+os𝑎h₁t𝑘an(h𝑥−𝑘𝑝(𝑥𝑡)−+𝑝𝜉𝑡₀)+𝜉₀+𝑎₂tanh²𝑘(𝑥−𝑝𝑡)+𝜉₀,𝑣(𝑥,𝑡)=||𝑝^𝑏−1⋅exp𝑏₋₂𝑘(𝑥−𝑝𝑡)+𝜉₀,𝛾𝑢(𝑥,𝑡).(3.13) If 𝐷₀=0, we found 𝑎₂=0, and 𝑏₋₁=−𝐷¹𝛾²−𝑝²𝛼𝛾²,𝑏₋₂=𝑝𝑎2𝛼𝑘₀,=−𝑝²2𝛼,𝑎¹=𝑘𝑝𝐷¹𝛾²−𝑝²𝛼𝛾².(3.14) Consequently, we obtain the following solutions of system (3.1)𝑢(𝑥,𝑡)=−𝑝²2𝑎+𝐷¹𝑘𝑝𝛾²−𝑝²𝛼𝛾²𝑆(𝑥,𝑡)=cosh𝑘(𝑥−𝑝𝑡)+𝜉₀tanh𝑘(𝑥−𝑝𝑡)+𝜉₀,𝑣(𝑥,𝑡)=−𝑝₃^{−𝐷1(𝛾2−𝑝2)/𝛼𝛾2}⋅exp2𝑎𝑝𝑘𝑘(𝑥−𝑝𝑡)+𝜉⁰,2𝑎𝛾+𝐷¹𝑘𝑝²𝛾²−𝑝²𝛼𝛾³tanh𝑘(𝑥−𝑝𝑡)+𝜉₀,(3.15) which is different from the solutions obtained in [12].

3.2. Using Improved tanh Method

Case 1. We seek solutions in the form of (2.8) 𝑢(𝜉)=∑^𝑀𝑖=0𝑎_𝑖/𝜉^𝑖. Motivated by the expression of (3.5), we turn to the assumption 𝜑(𝜉)=𝑏₋₂𝜉+𝑏₋₁ln|𝜉|+∑^{𝑀−2𝑗=0}𝑏_𝑗/𝜉^𝑗 and also imply that 𝑀=2. Therefore, we have 𝑢(𝜉)=𝑎₀+𝑎₁1𝜉+𝑎²1𝜉²,𝜑(𝜉)=𝑏₋₂𝜉+𝑏₋₁ln||𝜉||.(3.16)
By substituting (3.16) into (3.4) and (3.5), and setting the coefficients of 1/𝜉^𝑗(𝑗=0,1,2) to be zero, we set 2𝑘𝐷₁𝛼𝛾𝑘𝑎₁𝑏₋₂+𝑎₀𝑏₋₁−𝛾𝑝𝑎₁=0,𝛾²−6𝑘𝑝𝐷²₁𝑎₁+2𝛼𝛾²𝑘𝑎₁𝑏₋₁+𝑎₂𝑏₋₂−2𝛾²𝑝𝑎₂=0,𝑘𝑝𝛾𝑏²−₋₂𝑝²𝑎₂+3𝛼𝛾²𝑘𝑎₂𝑏₋₁=0,𝑘𝑝𝑏₋₁+𝐷₀𝑘²𝑏²⁻²+𝑎₀=0,𝐷+2𝐷₀𝑘²𝑏₋₂𝑏₋₁+𝑎₁=0,₀𝑘²𝑏²⁻¹−𝑏₋₁+𝑎₂=0.(3.17) If 𝐷₀≠0 and Δ=1+𝛼³±𝛼2𝐷₀+𝛼𝐷²⁰+𝛼𝐷₀+𝛼²5𝛼²𝐷₁+4𝐷²⁰𝐷₁+4𝛼𝐷₀𝐷₁>0,(3.18) solving algebraic equations (3.17) gives 𝑏₋₁=−2𝐷¹𝛾²−𝑝²𝛼𝛾²𝑏₋₂,=1𝛼𝑘𝐷₀⋅2𝑝𝐷⁰𝐷₁𝛾²−𝑝²+𝑝𝐷₀𝛼𝛾²−𝑝𝐷₁𝛼𝛾²−𝑝²4𝐷₁𝛾²−𝑝²+𝛼𝛾²𝑎₀,=−𝑝²2𝐷₀𝐷₁−𝐷₁𝛼𝛾²−𝑝²+𝛼𝛾²𝐷₀𝛼²𝐷₀4𝐷₁(𝛾²−𝑝²)+𝛼𝛾²²𝑎₁×3𝛼𝐷₁+2𝐷₀𝐷₁𝛾²−𝑝²+𝛼²𝛾²+𝛼𝛾²𝐷₀,=2𝑘𝑝𝐷¹𝛾²−𝑝²×𝛼²𝛾𝛼²𝛼𝛾²+4𝐷₁𝛾²−𝑝²₂𝑎𝛾²+2𝛼𝛾²𝐷₀+2𝛼𝐷₁𝛾²−𝑝²+4𝐷₀𝐷₁𝛾²−𝑝²₂,=−2𝑘²𝐷₀𝐷₁𝛾²−𝑝²𝛼𝛾²²𝑝=𝛾√𝛼𝛾²+2𝐷₁𝛾²−𝑝²,Δ,(3.19) where Δ is specified by (3.18).
This in turn gives the solutions of system (3.1) 𝑢(𝑥,𝑡)=𝑎₀+𝑎¹𝑘(𝑥−𝑝𝑡)+𝜉₀+𝑎²𝑘(𝑥−𝑝𝑡)+𝜉₀²𝑆(𝑥,𝑡)=||𝑘(𝑥−𝑝𝑡)+𝜉₀,𝑣(𝑥,𝑡)=𝑝||^𝑏−1⋅exp𝑏₋₂𝑘(𝑥−𝑝𝑡)+𝜉₀,𝛾𝑎⁰+𝑎¹𝑘(𝑥−𝑝𝑡)+𝜉₀+𝑎²𝑘(𝑥−𝑝𝑡)+𝜉₀²,(3.20) where 𝑘 is an arbitrary constant and 𝑎_𝑖(𝑖=0,1,2), 𝑏_−𝑗(𝑗=1,2),𝑝 are given in (3.19).

Case 2. We seek solutions in the form of (2.9). Let 𝑢(𝜉)=∑^𝑀𝑖=0𝑎_𝑖tan^𝑖𝜉 and 𝜑(𝜉)=𝑏₋₂𝜉+𝑏₋₁ln|cos𝜉|+∑^{𝑀−2𝑗=0}𝑏_𝑗tan^𝑗𝜉. Substituting 𝑢(𝜉),𝜑(𝜉) into (3.4) and (3.5), balancing the linear terms of highest order with the highest order nonlinear terms, directly results in 𝑀=2 and 𝑢=𝑎₀+𝑎₁tan𝜉+𝑎₂tan²𝜉,𝜑=𝑏₋₂𝜉+𝑏₋₁ln||cos𝜉||.(3.21) To progress further substituting (3.21) into (3.4) and (3.5) and setting the coefficients of tan^𝑖𝜉(𝑖=0,1,2) in the resulting equations to be zero, we obtain a set of algebraic equations 𝑘2𝐷𝑘𝐷₁₁𝛾²−𝑝²𝑎₂−𝛼𝛾²𝑘𝑎₁𝑏₋₂−𝑎₀𝑏₋₁+𝛾²𝑝𝑎₁=0,𝛾²−6𝑎𝑝²₂𝑎₁−𝛼𝛾²𝑘−𝑎₁𝑏₋₁+𝑎₂𝑏₋₂+𝛾²𝑝𝑎₂=0,−𝑘𝑘𝑝𝐷𝑏₁₋₂𝛾²−𝑝²+3𝛼𝛾²𝑘𝑎₂𝑏₋₁=0,𝑘𝑝𝑏₋₁−𝐷₀𝑘²𝑏²⁻²−𝑏₋₁=𝑎₀,−𝐷₀+2𝐷₀𝑘²𝑏₋₁𝑏₋₂=𝑎₁,𝑘²𝑏²⁻¹−𝑏₋₁=𝑎₂.(3.22) If 𝐷₀≠0 and 𝑝 satisfies Δ=−+𝑝⁴𝑝₂5𝐷²¹𝛼²+4𝐷₀𝐷²¹𝛼+4𝐷²⁰𝐷²¹+𝛾₄𝛾²2𝐷₁𝛼³+10𝛼²𝐷²¹+8𝛼𝐷₀𝐷²¹+8𝐷²⁰𝐷²¹>0,𝛼³𝐷₀+𝛼²𝐷²⁰−2𝛼³𝐷₁−5𝛼²𝐷²¹−4𝛼𝐷₀𝐷²¹−4𝐷²⁰𝐷²¹(3.23) we know 𝑏₋₁=−2𝐷¹𝛾²−𝑝²𝛼𝛾²𝑏,(3.24)₋₂=𝑝𝛼𝑘𝐷₀⋅2𝐷⁰𝐷₁𝛾²−𝑝²+𝛼𝛾²𝐷₀−𝐷₁𝛼𝛾²−𝑝²4𝐷₁𝛾²−𝑝²+𝛼𝛾²𝑎,(3.25)₀=−𝑝²𝛼𝐷₀⋅2𝐷⁰𝐷₁𝛾²−𝑝²+𝛼𝛾²𝐷₀−𝛼𝐷₁𝛾²−𝑝²4𝐷₁𝛾²−𝑝²+𝛼𝛾²−𝑝²𝛼²𝐷₀⋅2𝐷⁰𝐷₁(𝛾²−𝑝²)+𝛼𝛾²𝐷₀−𝛼𝐷₁(𝛾²−𝑝²)4𝐷₁(𝛾²−𝑝²)+𝛼𝛾²²−2𝑘²𝐷₀𝐷₁𝛾²−𝑝²𝛼𝛾²𝑎,(3.26)₁=−2𝑘𝑝𝐷¹𝛾²−𝑝²𝛼+2𝐷₀𝛼𝛾²+2𝐷₁𝛾²−𝑝²𝑎𝛼²𝛾²𝛼𝛾²+4𝐷₁𝛾²−𝑝²,(3.27)₂=−2𝑘²𝐷₀𝐷₁𝛾²−𝑝²𝛼𝛾²+2𝐷₁𝛾²−𝑝²𝛼𝛾²²𝑘=𝑝𝛾₂,(3.28)2Δ𝐷²⁰𝐷²¹𝛾²−𝑝²²4𝐷₁𝛾²−𝑝²+𝛼𝛾²²,(3.29) where Δ is given by (3.23). To this end, we therefore derive the solutions of system (3.1) as follows: 𝑆𝑢((𝑥𝑥,,𝑡𝑡))==𝑎||c₀o+s𝑎𝑘₁(ta𝑥n−𝑘𝑝𝑡()𝑥+−𝜉𝑝₀𝑡)+𝜉₀+𝑎₂tan²𝑘(𝑥−𝑝𝑡)+𝜉₀,𝑣(𝑥,𝑡)=𝑎₀||^𝑏−1𝑒^{𝑏−2(𝑘(𝑥−𝑝𝑡)+𝜉0)},+𝑎₁tan𝑘(𝑥−𝑝𝑡)+𝜉₀+𝑎₂tan²𝑘(𝑥−𝑝𝑡)+𝜉₀𝑝𝛾,(3.30) where 𝑏_−𝑗(𝑗=1,2) and 𝑎_𝑖(𝑖=0,1,2) and 𝑘 are given by (3.24)–(3.29), respectively.

4. Conclusion

In the present paper, we have considered the exact traveling wave solutions of a one-space dimensional hyperbolic model for chemotaxis (1.5). Using a transformation of dependent variables and eliminating the dependent variable 𝑣 in the resultant system (3.1), the system (1.5) is reduced to a couple system of two equations (3.2). Applying the transformation (3.3) to unite the independent variables 𝑥 and 𝑡, the (3.2) is convert to a coupled ordinary differential system. Then, the tanh method and improved tanh method are used, with the help of symbolic software MATHMATICA, to search for exact traveling wave solutions of system (3.1). Finally, three kinds of traveling wave solutions including soliton, rational, and triangular solutions are found, all of which are different from those known ones in the literature.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11001050). The authors gratefully acknowledge valuable discussions with Professor J. B. Chen. Also, the authors thank the three anonymous editors whose suggestions helped improve the paper.