New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model

One-dimensional parabolic-parabolic Keller-Segel (PP-KS) model of chemotaxis is considered. By using the generalized tanh function method, ( G′ /G)-expansion method and variable-separating method, plenty of new explicit exact solutions, including travelling wave solutions and non-travelling wave solutions, are obtained for the PP-KS model. Compared to the existing results, more new exact solutions are derived and the obtained solutions all have explicit expressions.


Introduction
Investigating exact solutions of nonlinear evolution equations plays an important role in nonlinear science. For example, the wave phenomena observed in fluid dynamics, plasma and elastic media are often modelled by bell-shaped sech solutions and kink-shaped tanh solutions. The effort to find these solutions is significant for the understanding of many phenomena in physics, chemistry and biology, because they may give more quantitative information. In the past several decades, many effective methods for obtaining exact solutions of nonlinear partial differential equations (NLPDEs) have been presented such as Hirota's bilinear method [1], inverse scattering method [2], Backlund transformation method [3], Painleve expansion method [4], Jacobi elliptic function expansion method, the new generalized algebraic method and so on [5][6][7][8][9][10].
The prototypical chemotaxis model was proposed by Keller and Segel in the 1970s to describe the aggregation of cellular slime molds Dictyostelium discoideum in response to the chemical cyclic adenosine monophosphate [11][12]. In its general form, Keller-Segel model reads ( , ) f u v is a function characterizing the chemical growth and degradation. When ( ) ln , ( , ) , Keller and Segel [13] performed theoretical analysis of the one-dimensional form of (1) to interpret the propagating travelling bands of bacterial chemotaxis experimentally observed in [14,15]. Since then, the study of travelling wave solutions to (1) has received extensive attentions [16][17] and the references quoted therein. The readers are referred to [18] and [19,20] for more detail about biological motivation and mathematical introduction of Eq. (1).
In this paper, exact solutions of the one-dimensional parabolic-parabolic Keller-Segel (PP-KS) model of chemotaxis are considered. The model is made of two parabolic equations as follows It is a special case of Eq.
(1) when 1, ( ) , ( , ) , is an integral constant. In [21], the authors have shown that (3) is Painleve integrable when 2 α = and soliton solutions for the particular integrable case are investigated. In this paper, more explicit exact solutions of (2) will be given. The rest of the paper is organized as follows.
In section 2, exact solutions of (2) are derived by the generalized tanh function method. In section 3, exact solutions of (2) are studied by the ( G′ /G)-expansion method. In section 4, two variable-separating methods are used to get rational solutions of (2). Conclusions will be finally presented.

Generalized Tanh Function Method
Let , y f v = (5) can be simplified to 2 2 0.
From above analysis, u and v can be obtained by solving (6) , . yy y v fdy According to the main steps of the generalized tanh function method in [22], (6) is assumed to has solutions of the form Here, , A B and N are constants, solutions of (9) have been found in [22][23]. Balancing yy f with y ff in (6) gives 1 M = .

Variable-Separating Method
Variable-separating method is a classical method to solve partial differential equations. In [26], the author proposed a new variable-separating method. In the following (2) will be solved by the two variable-separating methods.
Form the second equation of (2), one can easily get Substituting (18) into the first equation of (2), one can obtain 2 0.

New Variable-Separating Method
According to the new variable-separating method, one may assume where 11 G and 12 G are constants, 11 12 The solutions (29), (31) and (33) are satisfied for all values of α in (2). Although the expressions of (29), (31) and (33) are simple, they all reflect different phenomenon of chemotaxis. For example, (31) describes a particular case when the cell density u is unchanging, the corresponding concentration of the chemical substance v is a linear function of time .
t Remark 4 Compared with the work in [21], more new exact solutions for the particular integrable case when 2 α = are given. Furthermore, exact solutions for the general case 0 α > are obtained. In addition, all the obtained solutions have explicit expressions, so they are easier to use.

Conclusion
A mathematical model of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) named as parabolic -parabolic Keller-Segel (PP-KS) equation is considered in this paper. By using the generalized tanh function method and ( G′ /G)-expansion method, plenty of new travelling solutions are obtained for the particular integrable case ( 2 α = ) of the PP-KS model (2). These solutions contain hyperbolic function solutions, triangular periodic solutions and rational function solutions. By using classical variable-separating method and new variable-separating method, a lot of algebraically explicit analytical solutions are obtained for the general case ( 0 α > ) of the PP-KS model (2). Compared with the results in [21], more new exact solutions for the PP-KS model have been derived whether it is Painleve integrable or not, and the obtained solutions in this paper all have explicit expressions. They can be used in numerical simulation and help one to understand the mechanism reflected by PP-KS model. In the future, exact solutions of the generalizations of the KS model will be studied since they play critical roles in a wide range of biological phenomena.