**3. Logarithmic sensitivity**

The model with logarithmic chemosensitivity function *ϕ*ð Þ� *v* ln *v* is also studied. For the case of *f u*ð Þ¼� , *<sup>v</sup> <sup>v</sup>mu* <sup>þ</sup> <sup>~</sup>*βv*, where <sup>~</sup>*<sup>β</sup>* <sup>¼</sup> *const*, an extensive analysis is performed in [15]. This survey is focused on different aspects of traveling wave solutions. When *m* ¼ 0, this model coincides with Eq. (1) for *b* ¼ 0. When *β* <sup>~</sup> <sup>¼</sup> <sup>0</sup> and *<sup>m</sup>* <sup>¼</sup> 1, the system was studied in [22, 23]. The complete analysis for *<sup>β</sup>*<sup>~</sup> <sup>¼</sup> 0 is performed in [15]. An existence of global solution is established in [24].

Now we consider the system with *<sup>ϕ</sup>*ð Þ¼ *<sup>v</sup>* ln *<sup>v</sup>* and *f u*ð Þ¼ , *<sup>v</sup> <sup>σ</sup>*~*vu* � *<sup>β</sup>*~*v*. Similarly, a replacement of *<sup>t</sup>* ! *<sup>δ</sup>*1*<sup>t</sup>* and *<sup>u</sup>* ! *<sup>σ</sup> <sup>σ</sup>*<sup>~</sup> *δ*1 *<sup>u</sup>* gives *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> 1, *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>1</sup> *δ*1 , *<sup>α</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>2</sup> *δ*1 , *<sup>β</sup>* <sup>¼</sup> *<sup>β</sup>*<sup>~</sup> *δ*1 , and *σ* ¼ �1. Then the model has the form:

$$\begin{cases} u\_t - u\_{\text{xx}} + \eta \left( u \frac{v\_{\text{x}}}{v} \right)\_{\text{x}} = \mathbf{0} \\\\ v\_t - a v\_{\text{xx}} - \sigma v u + \beta v = \mathbf{0}. \end{cases} \tag{21}$$

*ξ*2

solution for *β* ¼ 0 [see Eq. (19)]:

*DOI: http://dx.doi.org/10.5772/intechopen.91214*

biological function.

suitable solution is:

2

**Figure 12.** *v y* ð Þ*; c* ¼ *1.*

**Figure 13.** *σu y* ð Þ*; c* ¼ *1.*

**11**

4

6

8

10

<sup>Ψ</sup>*ξξ* � *<sup>η</sup>*2*<sup>β</sup>*

*y t*ðÞ¼� <sup>2</sup>

*v t*ðÞ¼ <sup>∣</sup>*C*~*ϑ*<sup>∣</sup> 2

*v y* ð Þ¼ *e*

*c* <sup>2</sup>*<sup>η</sup> <sup>y</sup> <sup>C</sup>*� *<sup>e</sup>*

v y

0 50 100 150

*s* u y

5 5 10

10

5

*c*

<sup>2</sup>*c*<sup>2</sup> <sup>Ψ</sup> <sup>þ</sup> *<sup>η</sup>CF*

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis*

ln ð Þ *ϑ*ð Þ*t*

2ð Þ *η* þ 2 *η*

where *ϑ*ð Þ*t* is given in Eq. (18) and *u y* ð Þ may be expressed from Eq. (23). However one may see that *v* ! ∞ as *t* ! �∞, and this solution is unacceptable as a

Another possibility to solve this equation exactly is to put *CF* equal to zero. When *CF* <sup>¼</sup> 0, that means *F y* ð Þ¼ 0, for *<sup>β</sup>* 6¼ 0; Eq. (27) can be linearized by *<sup>ξ</sup>* <sup>¼</sup> *<sup>e</sup><sup>τ</sup>* [21]. Its solution has three forms according to a sign of the expression *<sup>D</sup>* <sup>¼</sup> <sup>2</sup>*η*2*<sup>β</sup>*

> �*<sup>c</sup>* ffiffi *D* p

<sup>4</sup> *<sup>y</sup>* <sup>þ</sup> *<sup>C</sup>*<sup>þ</sup> *<sup>e</sup>*

<sup>4</sup> *<sup>y</sup>* � ��<sup>2</sup>

*c* ffiffi *D* p

*η*

Since *v* should be a nonnegative and bounded function as *cy* ! �∞, the only

Using the result of the symmetry group analysis of Eq. (12), we can write the

*<sup>c</sup>*<sup>2</sup> *<sup>ξ</sup>*<sup>2</sup> <sup>Ψ</sup><sup>4</sup>

*t* 2 þ

� � <sup>1</sup>

2*ηCF c*2

*η*þ2

*<sup>η</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:* (27)

(28)

*<sup>c</sup>*<sup>2</sup> þ 1.

(29)

**0.1; 0.5; 1; 0.5; 2; 0.5; 0.1; 2; 1; 2; 2; 2;**

**0.1; 0.5; 1; 0.5; 2; 0.5; 0.1; 2; 1; 2; 2; 2;**

Let us rewrite the system (21) in terms of the function *υ*ð Þ¼ *x*, *t* ln *v x*ð Þ , *t* :

$$\begin{cases} u\_t - u\_{\text{xx}} + \eta (u \nu\_{\text{x}})\_{\text{x}} = \mathbf{0} \\\\ \nu\_t - a \nu\_{\text{xx}} - a (\nu\_{\text{x}})^2 + \beta - \sigma \mathbf{u} = \mathbf{0}, \end{cases} \tag{22}$$

Then in terms of the traveling wave variable *y* ¼ *x* � *ct*, where *c* ¼ *const*, Eq. (22) has the form:

$$\begin{cases} u\_\mathcal{\mathcal{V}} + cu - \eta u \nu\_\mathcal{\mathcal{V}} + \lambda = \mathbf{0} \\\\ a \nu\_\mathcal{\mathcal{W}} + a \left( \nu\_\mathcal{\mathcal{V}} \right)^2 + c \nu\_\mathcal{\mathcal{V}} - \beta + \sigma u = \mathbf{0}, \end{cases} \tag{23}$$

where *u* ¼ *u y* ð Þ, *υ* ¼ *υ*ð Þ*y* , and *λ* is an integration constant. To integrate Eq. (23), we tested this system on the Painlevé ODE test. One can show that for *η*>0, it passes this test only if *<sup>α</sup>* <sup>¼</sup> 2 with the additional condition *<sup>λ</sup>* ¼ �*σc<sup>β</sup>* <sup>1</sup> <sup>þ</sup> *<sup>η</sup>* 2 � � [25]. If we express *u y* ð Þ as *υ*ð Þ*y* from Eq. (23), we obtain an equation only for *υ*ð Þ*y* ; for *α* ¼ 2, it has the form:

$$2\eta\_{\eta\eta} + 3c\eta\_{\eta\eta} + \left(c^2 + \eta\beta\right)\nu\_{\eta} + 2(2-\eta)\nu\_{\eta}\nu\_{\eta\eta} + 2(2-\eta)\left(\nu\_{\eta}\right)^2 - 2\eta\left(\nu\_{\eta}\right)^3 - c\beta - \sigma\lambda = 0. \tag{24}$$

For *<sup>λ</sup>* ¼ �*σc<sup>β</sup>* <sup>1</sup> <sup>þ</sup> *<sup>η</sup>* 2 � �, this equation can be linearized. It becomes equivalent to the following linear equation for *F*:

$$F\_{\mathcal{Y}} + cF = \mathbf{0}, \text{ where } F(\mathcal{Y}) = \epsilon^{2\nu} \left( 2\nu\_{\mathcal{Y}} + c\nu\_{\mathcal{Y}} - \eta \left( \nu\_{\mathcal{Y}} \right)^2 + \frac{\eta\beta}{2} \right) \tag{25}$$

that gives the equation for *υ*ð Þ*y* :

$$
\Delta n\_{\mathcal{Y}} + \varepsilon n\_{\mathcal{Y}} - \eta \left( n\_{\mathcal{Y}} \right)^2 + \frac{\eta \beta}{2} = C\_F e^{-2\nu - c\eta}, \tag{26}
$$

where *CF* <sup>¼</sup> *const*. If we rewrite Eq. (26) in terms of the variable *<sup>ξ</sup>* <sup>¼</sup> *<sup>e</sup>*�*cy* <sup>2</sup> for the function <sup>Ψ</sup>ð Þ¼ *<sup>ξ</sup> <sup>e</sup>*�*<sup>η</sup>* <sup>2</sup>*<sup>υ</sup>*, we obtain an equation similar to Eq. (12) with zero right-hand side:

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis DOI: http://dx.doi.org/10.5772/intechopen.91214*

$$
\xi^2 \Psi\_{\xi\xi} - \frac{\eta^2 \beta}{2c^2} \Psi + \frac{\eta C\_F}{c^2} \,\xi^2 \Psi^{\frac{4}{\eta} + 1} = \mathbf{0}.\tag{27}
$$

Using the result of the symmetry group analysis of Eq. (12), we can write the solution for *β* ¼ 0 [see Eq. (19)]:

$$\begin{split} y(t) &= -\frac{2}{c} \ln \left( \theta(t) \right) \\ v(t) &= \frac{|\tilde{\mathbf{C}}\_{\theta}|}{2} \left( \frac{2(\eta + 2)}{\eta} t^{2} + \frac{2\eta \mathbf{C}\_{F}}{c^{2}} \right)^{\frac{1}{\eta + 2}} \end{split} \tag{28}$$

where *ϑ*ð Þ*t* is given in Eq. (18) and *u y* ð Þ may be expressed from Eq. (23). However one may see that *v* ! ∞ as *t* ! �∞, and this solution is unacceptable as a biological function.

Another possibility to solve this equation exactly is to put *CF* equal to zero. When *CF* <sup>¼</sup> 0, that means *F y* ð Þ¼ 0, for *<sup>β</sup>* 6¼ 0; Eq. (27) can be linearized by *<sup>ξ</sup>* <sup>¼</sup> *<sup>e</sup><sup>τ</sup>* [21]. Its solution has three forms according to a sign of the expression *<sup>D</sup>* <sup>¼</sup> <sup>2</sup>*η*2*<sup>β</sup> <sup>c</sup>*<sup>2</sup> þ 1. Since *v* should be a nonnegative and bounded function as *cy* ! �∞, the only suitable solution is:

**1; 0.5; 2; 0.5; 0.1; 2; 1; 2; 2; 2;**

5 5 10

10

5

**3. Logarithmic sensitivity**

a replacement of *<sup>t</sup>* ! *<sup>δ</sup>*1*<sup>t</sup>* and *<sup>u</sup>* ! *<sup>σ</sup> <sup>σ</sup>*<sup>~</sup>

*σ* ¼ �1. Then the model has the form:

has the form:

*α* ¼ 2, it has the form:

For *<sup>λ</sup>* ¼ �*σc<sup>β</sup>* <sup>1</sup> <sup>þ</sup> *<sup>η</sup>*

the function <sup>Ψ</sup>ð Þ¼ *<sup>ξ</sup> <sup>e</sup>*�*<sup>η</sup>*

right-hand side:

**10**

2

*Fy* þ *cF* ¼ 0, where *F y* ð Þ¼ *e*

2*υyy* þ *cυ<sup>y</sup>* � *η υ<sup>y</sup>*

the following linear equation for *F*:

that gives the equation for *υ*ð Þ*y* :

2*υyyy* þ 3*cυyy* þ *c*

The model with logarithmic chemosensitivity function *ϕ*ð Þ� *v* ln *v* is also studied. For the case of *f u*ð Þ¼� , *<sup>v</sup> <sup>v</sup>mu* <sup>þ</sup> <sup>~</sup>*βv*, where <sup>~</sup>*<sup>β</sup>* <sup>¼</sup> *const*, an extensive analysis is performed in [15]. This survey is focused on different aspects of traveling wave solutions. When *m* ¼ 0, this model coincides with Eq. (1) for *b* ¼ 0. When *β*

and *<sup>m</sup>* <sup>¼</sup> 1, the system was studied in [22, 23]. The complete analysis for *<sup>β</sup>*<sup>~</sup> <sup>¼</sup> 0 is

Now we consider the system with *<sup>ϕ</sup>*ð Þ¼ *<sup>v</sup>* ln *<sup>v</sup>* and *f u*ð Þ¼ , *<sup>v</sup> <sup>σ</sup>*~*vu* � *<sup>β</sup>*~*v*. Similarly,

*<sup>u</sup>* gives *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> 1, *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>1</sup>

*<sup>x</sup>* ¼ 0

*v* � �

*vt* � *αvxx* � *σvu* þ *βv* ¼ 0*:*

*<sup>υ</sup><sup>t</sup>* � *αυxx* � *α υ*ð Þ*<sup>x</sup>* <sup>2</sup> <sup>þ</sup> *<sup>β</sup>* � *<sup>σ</sup><sup>u</sup>* <sup>¼</sup> 0,

Then in terms of the traveling wave variable *y* ¼ *x* � *ct*, where *c* ¼ *const*, Eq. (22)

where *u* ¼ *u y* ð Þ, *υ* ¼ *υ*ð Þ*y* , and *λ* is an integration constant. To integrate Eq. (23),

we tested this system on the Painlevé ODE test. One can show that for *η*>0, it passes this test only if *<sup>α</sup>* <sup>¼</sup> 2 with the additional condition *<sup>λ</sup>* ¼ �*σc<sup>β</sup>* <sup>1</sup> <sup>þ</sup> *<sup>η</sup>*

we express *u y* ð Þ as *υ*ð Þ*y* from Eq. (23), we obtain an equation only for *υ*ð Þ*y* ; for

<sup>2</sup> <sup>þ</sup> *ηβ* � �*υ<sup>y</sup>* <sup>þ</sup> 2 2ð Þ � *<sup>η</sup> <sup>υ</sup>yυyy* <sup>þ</sup> 2 2ð Þ � *<sup>η</sup> <sup>υ</sup><sup>y</sup>*

� �<sup>2</sup> <sup>þ</sup> *<sup>c</sup>υ<sup>y</sup>* � *<sup>β</sup>* <sup>þ</sup> *<sup>σ</sup><sup>u</sup>* <sup>¼</sup> 0,

� �, this equation can be linearized. It becomes equivalent to

<sup>2</sup>*<sup>υ</sup>* <sup>2</sup>*υyy* <sup>þ</sup> *<sup>c</sup>υ<sup>y</sup>* � *η υ<sup>y</sup>*

<sup>2</sup> <sup>¼</sup> *CFe*

<sup>2</sup>*<sup>υ</sup>*, we obtain an equation similar to Eq. (12) with zero

� �<sup>2</sup> <sup>þ</sup> *ηβ*

where *CF* <sup>¼</sup> *const*. If we rewrite Eq. (26) in terms of the variable *<sup>ξ</sup>* <sup>¼</sup> *<sup>e</sup>*�*cy*

� �

Let us rewrite the system (21) in terms of the function *υ*ð Þ¼ *x*, *t* ln *v x*ð Þ , *t* :

*δ*1 , *<sup>α</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>2</sup> *δ*1 , *<sup>β</sup>* <sup>¼</sup> *<sup>β</sup>*<sup>~</sup> *δ*1 , and

� �<sup>2</sup> � <sup>2</sup>*η υ<sup>y</sup>*

� �<sup>2</sup> <sup>þ</sup> *ηβ* 2

�2*υ*�*cy*, (26)

performed in [15]. An existence of global solution is established in [24].

*Mathematical Theorems - Boundary Value Problems and Approximations*

*δ*1

(

(

8 < : *ut* � *uxx* <sup>þ</sup> *<sup>η</sup> <sup>u</sup> vx*

*ut* � *uxx* þ *η*ð Þ *uυ<sup>x</sup> <sup>x</sup>* ¼ 0

*uy* þ *cu* � *ηuυ<sup>y</sup>* þ *λ* ¼ 0

*αυyy* þ *α υ<sup>y</sup>*

<sup>~</sup> <sup>¼</sup> <sup>0</sup>

(21)

(22)

(23)

(24)

(25)

<sup>2</sup> for

2 � � [25]. If

� �<sup>3</sup> � *<sup>c</sup><sup>β</sup>* � *σλ* <sup>¼</sup> <sup>0</sup>*:*

where *C*� are positive constants and *β* > 0. Unfortunately, the corresponding solution for *u y* ð Þ is alternating and has the form:

$$\begin{split} u(\mathbf{y}) &= -\frac{\sigma c^2(\eta+2)}{2\eta^2} \left( \mathbf{C}\_-^2 \left( \mathbf{1} + \sqrt{D} \right) e^{-\frac{\sqrt{D}}{4}\eta} + \mathbf{C}\_+^2 \left( \mathbf{1} - \sqrt{D} \right) e^{\frac{\sqrt{D}}{4}\eta} \right. \\ &\left. - \frac{4\eta^2 \beta}{c^2} \mathbf{C}\_- \mathbf{C}\_+ \right) \left( \mathbf{C}\_- e^{-\frac{c\sqrt{D}}{4}\eta} + \mathbf{C}\_+ e^{\frac{c\sqrt{D}}{4}\eta} \right)^{-\frac{2}{\eta}}. \end{split} \tag{30}$$

The function *Zν*ð Þ*z* satisfies the modified Bessel's equation and can be present as

*<sup>v</sup>ν*ð Þ*<sup>z</sup>* ! 0; *<sup>u</sup>ν*ð Þ!*<sup>z</sup>* <sup>0</sup>*:* (35)

2 ;

1 2 ;

� �, (37)

4 � *λ c*3 ,

1 2 ;

(36)

(38)

<sup>2</sup> and

, when *Zν*ð Þ*z*

(39)

*u*1 2 ð Þ*z*

<sup>2</sup> as functions of *z*.

2

∞, 0≤ *ν*<

ð Þ *<sup>π</sup>* <sup>þ</sup> <sup>2</sup> <sup>2</sup> , *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup>

8*π*

0, *ν*>

þ *BK<sup>ν</sup>*

<sup>2</sup> , its form coincides with the well-known Korteweg-de Vries

2 ð Þ*z* :

*<sup>z</sup>* , *<sup>n</sup>* <sup>¼</sup> <sup>2</sup>*<sup>k</sup>* <sup>þ</sup> 1; *<sup>k</sup>* <sup>¼</sup> 0, 1 … ;

*vn*þ<sup>1</sup>

. It is interesting to present the expressions for *e*

<sup>2</sup> and *un*þ<sup>1</sup>

ð Þ *z* þ *ζ* (40)

ð Þ *z* þ *ζ* , (41)

*κ* ∣*c*∣ *e* �*cy* 2

<sup>2</sup> *<sup>ν</sup>* � <sup>1</sup> 2

<sup>2</sup> agrees with Eq. (39).

� � � � <sup>2</sup> " #

asymptotic behavior [36], one may obtain the following asymptotic forms for *evν*ð Þ*<sup>z</sup>*

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis*

*κ* ∣*c*∣*C*<sup>2</sup>

Using the series expansion of the Infeld's function, as well as theirs

8 >>>>>><

>>>>>>:

*uν*ð Þ!*z c*

*κ* ∣*c*∣ *e* �*cy* 2 � �

�*cy* <sup>þ</sup> *λ c* � �, where *<sup>ν</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

Consider now the class of solutions with half-integer index *<sup>ν</sup>* <sup>¼</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

*zdz* � �*<sup>n</sup>* cosh ð Þ *<sup>z</sup>* <sup>þ</sup> *<sup>ζ</sup>*

*zdz* � �*<sup>n</sup>* sinh ð Þ *<sup>z</sup>* <sup>þ</sup> *<sup>ζ</sup>*

*<sup>π</sup>* , *<sup>C</sup>* <sup>¼</sup> *const:*

where *κ* >0, *A*, and *B* are arbitrary constants and the functions *I<sup>ν</sup>* and *K<sup>ν</sup>* are Infeld's and Macdonald's functions, respectively. This solution is not satisfactory from the biological point of view, since *v y* ð Þ is an alternating function for any *ν*. However it seems interesting because of the following: in the case of *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup>

can be expressed in hyperbolic functions. The requirement of absence of divergence

*<sup>z</sup>* , *<sup>n</sup>* <sup>¼</sup> <sup>2</sup>*k*,

a linear combination of Infeld's and Macdonald's functions.

*z* ! ∞ : *e*

*<sup>v</sup>ν*ð Þ*<sup>z</sup>* !

*z* ! 0 : *e*

*DOI: http://dx.doi.org/10.5772/intechopen.91214*

So, the exact solution obtained has the form:

�*cy* <sup>2</sup> *A*<sup>2</sup> *I<sup>ν</sup>*

*u* ! �∞ for finite *z* leads to the following form for *Zn*þ<sup>1</sup>

At first let us consider the solutions obtained for *e*

2

*e v*1 2 ð Þ*<sup>z</sup>* <sup>¼</sup> *<sup>κ</sup> C*2 ∣*c*∣ *sech*<sup>2</sup>

*u*1 2 ð Þ¼ *<sup>z</sup> <sup>z</sup>*<sup>2</sup> *c* <sup>2</sup> *sech*<sup>2</sup>

*Cz<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> *d*

8 >>>>>><

>>>>>>:

We begin with *<sup>n</sup>* <sup>¼</sup> 0 or *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup>

*Cz<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> *d*

*<sup>ζ</sup>* <sup>¼</sup> <sup>1</sup> 2 ln <sup>2</sup>

� �<sup>2</sup> � *<sup>κ</sup>*<sup>2</sup> *<sup>e</sup>*

*v* ¼ � ln *e*

*u* ¼ �*σ vy*

where the expression for *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup>

and *uν*ð Þ*z* :

*<sup>B</sup>* <sup>¼</sup> <sup>2</sup>þ*<sup>π</sup>*

soliton.

and *u*<sup>1</sup> 2 ð Þ*z* :

**13**

*Zn*þ<sup>1</sup> 2 ð Þ¼ *z*

<sup>2</sup>*<sup>π</sup>* in terms of *<sup>e</sup>*�*cy*

It is easy to see that *<sup>σ</sup>u y* ð Þ! *<sup>c</sup>*2ð Þ *<sup>η</sup>*þ<sup>2</sup> <sup>2</sup>*η*<sup>2</sup> �<sup>1</sup> � ffiffiffiffi *<sup>D</sup>* � � <sup>p</sup> as *cy* ! �∞. These functions are presented in **Figures 12** and **13**.

## **4. Linear sensitivity**

Let us consider the system with linear function *ϕ*ð Þ� *v v*. When *f u*ð Þ¼ , *v u* � *v*, the system is called the minimal chemotaxis model following the nomenclature of [26]. This model is often considered with *f u*ð Þ¼ , *<sup>v</sup> <sup>σ</sup>*~*<sup>u</sup>* � *<sup>β</sup>*~*<sup>v</sup>* (*σ*<sup>~</sup> and *<sup>β</sup>*<sup>~</sup> are constants), and it is studied in many papers. As was proved in [27, 28], the solutions of this system are global and bounded in time for one space dimension. The case of positive *σ*~ and nonnegative *β*~ is studied in [29–33]. As we noted earlier, a sign of *σ*~ may effect on the mathematical properties of the system, which changes its solvability conditions [34].

Now we consider the linear chemosensitivity function *ϕ*ð Þ¼ *v v* and *f u*ð Þ¼ , *v <sup>σ</sup>*~*<sup>u</sup>* � *<sup>β</sup>*~*v*. The replacement of *<sup>t</sup>* ! *<sup>δ</sup>*1*t*, *<sup>v</sup>* ! *<sup>η</sup>*<sup>1</sup> *δ*1 *<sup>v</sup>*, and *<sup>u</sup>* ! *<sup>σ</sup> ση*<sup>~</sup> <sup>1</sup> *δ*2 1 *u* leads to *δ*<sup>1</sup> ¼ *η*<sup>1</sup> ¼ 1, *<sup>α</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>2</sup> *δ*1 , *<sup>β</sup>* <sup>¼</sup> *<sup>β</sup>*<sup>~</sup> *δ*1 , and *σ* ¼ �1. Then the system has the form:

$$\begin{cases} u\_t - u\_{\text{xx}} + (uv\_x)\_x = \mathbf{0} \\ v\_t - av\_{\text{xx}} + \beta v - \sigma u = \mathbf{0}. \end{cases} \tag{31}$$

This system reduces to the system of ODEs in terms of traveling wave variable *y* ¼ *x* � *ct*, where *c* ¼ *const*:

$$\begin{cases} u\_{\mathcal{Y}} + cu - \mu v\_{\mathcal{Y}} + \lambda = \mathbf{0} \\ av\_{\mathcal{Y}} + cv\_{\mathcal{Y}} - \beta v + \sigma u = \mathbf{0}, \end{cases} \tag{32}$$

where *u* ¼ *u y* ð Þ, *v* ¼ *v y* ð Þ, and *λ* is an integration constant. As shown in [35], this system passes the Painlevé ODE test only if *α* ¼ 2 and *β* ¼ 0. Let us focus on this case.

It is convenient to solve Eq. (32) in terms of variable:

$$z = \frac{\kappa}{|c|} \ e^{-\frac{\sigma}{2}},\tag{33}$$

where *κ* >0 is an arbitrary constant. Then for *v* and *u*, we obtain the solutions in the form:

$$v = -\ln\left[\frac{|c|}{\kappa} z Z\_{\nu}^{2}(z)\right]$$

$$u = c^{2} z^{2} \left(1 - \frac{1}{4} (v\_{x})^{2}\right) - \frac{\lambda}{c}, \text{where } \nu^{2} = \frac{1}{4} - \frac{\lambda}{c^{3}}.\tag{34}$$

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis DOI: http://dx.doi.org/10.5772/intechopen.91214*

The function *Zν*ð Þ*z* satisfies the modified Bessel's equation and can be present as a linear combination of Infeld's and Macdonald's functions.

Using the series expansion of the Infeld's function, as well as theirs asymptotic behavior [36], one may obtain the following asymptotic forms for *evν*ð Þ*<sup>z</sup>* and *uν*ð Þ*z* :

$$z \to \infty : \quad e^{v\_{\nu}(x)} \to 0; \ u\_{\nu}(z) \to 0. \tag{35}$$

$$z \to 0: \quad e^{\nu\_{\nu}(z)} \to \begin{cases} \infty, & 0 \le \nu < \frac{1}{2}; \\ \frac{\kappa}{|c| \cdot \mathbb{C}^2} \frac{8\pi}{\left(\pi + 2\right)^2}, & \nu = \frac{1}{2}; \\ 0, & \nu > \frac{1}{2}; \end{cases} \tag{36}$$

$$u\_{\nu}(z) \to c^2 \left(\nu - \frac{1}{2}\right), \tag{37}$$

where the expression for *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> agrees with Eq. (39). So, the exact solution obtained has the form:

$$\begin{split} v &= -\ln \left[ e^{-\frac{\alpha}{2}} A^2 \left( I\_{\nu} \left( \frac{\kappa}{|c|} e^{-\frac{\alpha}{2}} \right) + B K\_{\nu} \left( \frac{\kappa}{|c|} e^{-\frac{\alpha}{2}} \right) \right)^2 \right] \\ u &= -\sigma \left( \left( v\_{\mathcal{Y}} \right)^2 - \kappa^2 e^{-c\mathcal{Y}} + \frac{\lambda}{c} \right), \text{ where } \nu^2 = \frac{1}{4} - \frac{\lambda}{c^3}, \end{split} \tag{38}$$

where *κ* >0, *A*, and *B* are arbitrary constants and the functions *I<sup>ν</sup>* and *K<sup>ν</sup>* are Infeld's and Macdonald's functions, respectively. This solution is not satisfactory from the biological point of view, since *v y* ð Þ is an alternating function for any *ν*. However it seems interesting because of the following: in the case of *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> and *<sup>B</sup>* <sup>¼</sup> <sup>2</sup>þ*<sup>π</sup>* <sup>2</sup>*<sup>π</sup>* in terms of *<sup>e</sup>*�*cy* <sup>2</sup> , its form coincides with the well-known Korteweg-de Vries soliton.

Consider now the class of solutions with half-integer index *<sup>ν</sup>* <sup>¼</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2 , when *Zν*ð Þ*z* can be expressed in hyperbolic functions. The requirement of absence of divergence *u* ! �∞ for finite *z* leads to the following form for *Zn*þ<sup>1</sup> 2 ð Þ*z* :

$$Z\_{n+\frac{1}{2}}(z) = \begin{cases} Cz^{n+\frac{1}{2}} (\frac{d}{dx})^n \frac{\cosh\left(z+\zeta\right)}{z}, & n=2k, \\ Cz^{n+\frac{1}{2}} (\frac{d}{dx})^n \frac{\sinh\left(z+\zeta\right)}{z}, & n=2k+1; \ k=0,1\ldots; \\ \zeta = \frac{1}{2} \ln\frac{2}{\pi}, & C=const. \end{cases} \tag{39}$$

At first let us consider the solutions obtained for *e vn*þ<sup>1</sup> <sup>2</sup> and *un*þ<sup>1</sup> <sup>2</sup> as functions of *z*. We begin with *<sup>n</sup>* <sup>¼</sup> 0 or *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup> 2 . It is interesting to present the expressions for *e u*1 2 ð Þ*z* and *u*<sup>1</sup> 2 ð Þ*z* :

$$e^{\frac{\nu\_{\frac{1}{2}}(x)}{2}} = \frac{\kappa}{C^2|c|} \operatorname{sech}^2(z + \zeta) \tag{40}$$

$$
\mu\_{\frac{1}{2}}(z) = z^2 c^2 \operatorname{sech}^2(z + \zeta), \tag{41}
$$

where *C*� are positive constants and *β* > 0. Unfortunately, the corresponding

<sup>4</sup> *<sup>y</sup>* <sup>þ</sup> *<sup>C</sup>*<sup>þ</sup> *<sup>e</sup>*

<sup>2</sup>*η*<sup>2</sup> �<sup>1</sup> � ffiffiffiffi

Let us consider the system with linear function *ϕ*ð Þ� *v v*. When *f u*ð Þ¼ , *v u* � *v*, the system is called the minimal chemotaxis model following the nomenclature of [26]. This model is often considered with *f u*ð Þ¼ , *<sup>v</sup> <sup>σ</sup>*~*<sup>u</sup>* � *<sup>β</sup>*~*<sup>v</sup>* (*σ*<sup>~</sup> and *<sup>β</sup>*<sup>~</sup> are constants), and it is studied in many papers. As was proved in [27, 28], the solutions of this system are global and bounded in time for one space dimension. The case of positive *σ*~ and nonnegative *β*~ is studied in [29–33]. As we noted earlier, a sign of *σ*~ may effect on the mathematical properties of the system, which changes its solvability

Now we consider the linear chemosensitivity function *ϕ*ð Þ¼ *v v* and *f u*ð Þ¼ , *v*

*ut* � *uxx* þ ð Þ *uvx <sup>x</sup>* ¼ 0 *vt* � *αvxx* þ *βv* � *σu* ¼ 0*:*

This system reduces to the system of ODEs in terms of traveling wave variable

where *u* ¼ *u y* ð Þ, *v* ¼ *v y* ð Þ, and *λ* is an integration constant. As shown in [35], this system passes the Painlevé ODE test only if *α* ¼ 2 and *β* ¼ 0. Let us focus on this

where *κ* >0 is an arbitrary constant. Then for *v* and *u*, we obtain the solutions in

*κ zZ*<sup>2</sup> *<sup>ν</sup>*ð Þ*z* � �

� *λ c* , where *<sup>ν</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

*uy* þ *cu* � *uvy* þ *λ* ¼ 0 *αvyy* þ *cvy* � *βv* þ *σu* ¼ 0,

> *<sup>z</sup>* <sup>¼</sup> *<sup>κ</sup>* ∣*c*∣ *e* �*cy*

*<sup>v</sup>* ¼ � ln <sup>∣</sup>*c*<sup>∣</sup>

<sup>4</sup> ð Þ *vz* <sup>2</sup> � �

, and *σ* ¼ �1. Then the system has the form:

�

�

It is convenient to solve Eq. (32) in terms of variable:

*<sup>z</sup>*<sup>2</sup> <sup>1</sup> � <sup>1</sup>

*u* ¼ *c* 2 *δ*1

*<sup>v</sup>*, and *<sup>u</sup>* ! *<sup>σ</sup> ση*<sup>~</sup> <sup>1</sup>

*δ*2 1

<sup>2</sup> , (33)

<sup>4</sup> � *<sup>λ</sup>* c3 *:*

*u* leads to *δ*<sup>1</sup> ¼ *η*<sup>1</sup> ¼ 1,

(31)

(32)

(34)

<sup>4</sup> *<sup>y</sup>* � ��<sup>2</sup>

�*<sup>c</sup>* ffiffi *D* p <sup>4</sup> *<sup>y</sup>* <sup>þ</sup> *<sup>C</sup>*<sup>2</sup>

<sup>4</sup> *<sup>y</sup>* �

*c* ffiffi *D* p

*η :*

<sup>þ</sup> <sup>1</sup> � ffiffiffiffi *D* � � <sup>p</sup> *<sup>e</sup>*

*<sup>D</sup>* � � <sup>p</sup> as *cy* ! �∞. These functions are

*c* ffiffi *D* p

(30)

� <sup>1</sup> <sup>þ</sup> ffiffiffiffi *D* � � <sup>p</sup> *<sup>e</sup>*

*Mathematical Theorems - Boundary Value Problems and Approximations*

�*<sup>c</sup>* ffiffi *D* p

solution for *u y* ð Þ is alternating and has the form:

<sup>2</sup>*η*<sup>2</sup> *<sup>C</sup>*<sup>2</sup>

*<sup>c</sup>*<sup>2</sup> *<sup>C</sup>*�*C*þÞ *<sup>C</sup>*� *<sup>e</sup>*

*u y* ð Þ¼� *<sup>σ</sup>c*<sup>2</sup> ð Þ *<sup>η</sup>* <sup>þ</sup> <sup>2</sup>

� <sup>4</sup>*η*2*<sup>β</sup>*

presented in **Figures 12** and **13**.

**4. Linear sensitivity**

conditions [34].

*y* ¼ *x* � *ct*, where *c* ¼ *const*:

*<sup>α</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>2</sup> *δ*1 , *<sup>β</sup>* <sup>¼</sup> *<sup>β</sup>*<sup>~</sup> *δ*1

case.

the form:

**12**

It is easy to see that *<sup>σ</sup>u y* ð Þ! *<sup>c</sup>*2ð Þ *<sup>η</sup>*þ<sup>2</sup>

*<sup>σ</sup>*~*<sup>u</sup>* � *<sup>β</sup>*~*v*. The replacement of *<sup>t</sup>* ! *<sup>δ</sup>*1*t*, *<sup>v</sup>* ! *<sup>η</sup>*<sup>1</sup>

where Eq. (40) appears the one-soliton solution exactly the same as the wellknown one of the Korteweg-de Vries equation. Returning to the variable *y*:

$$e^{\nu\left(\epsilon^{-\frac{\gamma}{2}}\right)} = \frac{\kappa}{C^2|c|} \operatorname{sech}^2\left(\frac{\kappa}{|c|} \operatorname{e}^{-\frac{\gamma}{2}} + \frac{1}{2} \ln \frac{2}{\pi}\right)$$

$$u(y) = \frac{\sigma(\pi B - 1)\kappa^2 \epsilon^{-\gamma}}{\left(\sinh\left(\frac{\kappa}{|c|} \operatorname{e}^{-\frac{\gamma}{2}}\right) + \frac{\pi}{2} B e^{-\frac{\kappa}{|c|} \epsilon^{-\frac{\gamma}{2}}}\right)^2}. \tag{42}$$

The explicit form of our solution in terms of the variable *y* can be obtained by

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis*

formulae are complicated and slightly difficult for analytic analysis; it seems to be

the values of velocity *c* and arbitrary constant *κ*. One may see that these curves become higher and shift to the right with different rates for the rising *κ*. The *u*<sup>1</sup>

values of *c* and *κ*; the curves become higher and more tight, and they shift to the

solution has the negative section converging to zero for *cy* ! �∞ (**Figures 18**–**21**).

is the positive function whose altitude and sharpness of peak depend on *c*

*<sup>c</sup>* ¼ �*c*2*n n*ð Þ <sup>þ</sup> <sup>1</sup> . The resulting

ð Þ*<sup>y</sup>* is retained for different

ð Þ*y* , the obtained

c 2, k 1 c 1, k 1 c 1 2, k 1 c 2, k 5 c 1, k 5 c 1 2, k 5

> c 2, k 1 c 1, k 1 c 1 2, k 1 c 2, k 50 c 1, k 50 c 1 2, k 50

ð Þ*y* are presented

2

2 ð Þ*y*

, we have the "step" whose altitude depends on

*vn*þ<sup>1</sup> 2

ð Þ*y* has to be positive (nonnegative), we see that these

2

direct substitution of Eq. (33) into Eq. (39), where *<sup>λ</sup>*

*v*1 2 ð Þ*y*

For *n*≥1 we can see that the solitonic behavior of *e*

2

20

0.5

1.0

1.5

2.0

2.5

40

60

80

100

right also with an increase of *<sup>c</sup>* and *<sup>κ</sup>*. For the cell density *un*þ<sup>1</sup>

The curves for the concentration of the chemical substance *vn*þ<sup>1</sup>

functions do not satisfy this requirement in all domains of definition.

e^v y , 1 C^2 10

10 10 20 30 40

u y

5 0 5 10 15 20 25

more convenient to present the plots. For *n* ¼ 0 in the function *e*

*DOI: http://dx.doi.org/10.5772/intechopen.91214*

(see **Figures 16** and **17**).

in **Figure 22**. Since *vn*þ<sup>1</sup>

Out[62]=

**Figure 16.**

Out[75]=

**Figure 17.** *un*þ*<sup>1</sup> 2*

**15**

ð Þ*y ; n* ¼ *0.*

*e vn*þ*<sup>1</sup> 2* ð Þ*<sup>y</sup> ; n* <sup>¼</sup> *0.*

One can see that for *σ* ¼ 1 (an increase of a chemical substance), the cell density *u y* ð Þ≥0 for *<sup>B</sup>* <sup>≥</sup> <sup>1</sup> *<sup>π</sup>* and that for *B*>0 *u y* ð Þ is the solitary continuous solution vanishing as *y* ! �∞, whereas for *B*< 0 *u y* ð Þ has a point of discontinuity. One can say that when *B*<0, we obtain "blow-up" solution in the sense that it goes to infinity for finite *y*, and this is true for different *ν*.

The expressions for *n*≥ 1 become more complicated, and one can see the solitonic behavior of *e vn*þ<sup>1</sup> 2 ð Þ*<sup>z</sup>* and the curves for *un*þ<sup>1</sup> 2 ð Þ*z* in **Figures 14** and **15**.

**Figure 14.** *e vn*þ*<sup>1</sup> 2* ð Þ*<sup>z</sup> ; n* <sup>¼</sup> *<sup>0</sup>*, *:::6; c* <sup>¼</sup> *1.*

**Figure 15.** *un*þ*<sup>1</sup> 2* ð Þ*z ; n* ¼ *0*, *:::5; c* ¼ *1.*

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis DOI: http://dx.doi.org/10.5772/intechopen.91214*

The explicit form of our solution in terms of the variable *y* can be obtained by direct substitution of Eq. (33) into Eq. (39), where *<sup>λ</sup> <sup>c</sup>* ¼ �*c*2*n n*ð Þ <sup>þ</sup> <sup>1</sup> . The resulting formulae are complicated and slightly difficult for analytic analysis; it seems to be more convenient to present the plots.

For *n* ¼ 0 in the function *e v*1 2 ð Þ*y* , we have the "step" whose altitude depends on the values of velocity *c* and arbitrary constant *κ*. One may see that these curves become higher and shift to the right with different rates for the rising *κ*. The *u*<sup>1</sup> 2 ð Þ*y* is the positive function whose altitude and sharpness of peak depend on *c* (see **Figures 16** and **17**).

For *n*≥1 we can see that the solitonic behavior of *e vn*þ<sup>1</sup> 2 ð Þ*<sup>y</sup>* is retained for different values of *c* and *κ*; the curves become higher and more tight, and they shift to the right also with an increase of *<sup>c</sup>* and *<sup>κ</sup>*. For the cell density *un*þ<sup>1</sup> 2 ð Þ*y* , the obtained solution has the negative section converging to zero for *cy* ! �∞ (**Figures 18**–**21**).

The curves for the concentration of the chemical substance *vn*þ<sup>1</sup> 2 ð Þ*y* are presented in **Figure 22**. Since *vn*þ<sup>1</sup> 2 ð Þ*y* has to be positive (nonnegative), we see that these functions do not satisfy this requirement in all domains of definition.

5 0 5 10 15 20 25

**Figure 17.** *un*þ*<sup>1</sup> 2* ð Þ*y ; n* ¼ *0.*

where Eq. (40) appears the one-soliton solution exactly the same as the well-

(42)

þ *π* <sup>2</sup> *B e*� *<sup>κ</sup>* <sup>∣</sup>*c*<sup>∣</sup> *e* �*cy* 2

2

ð Þ*z* in **Figures 14** and **15**.

<sup>2</sup> *:*

*sech*<sup>2</sup> *<sup>κ</sup>* ∣*c*∣ *e* �*cy* <sup>2</sup> þ 1 2 ln <sup>2</sup> *π*

*u y* ð Þ¼ *σ π*ð Þ *<sup>B</sup>* � <sup>1</sup> *<sup>κ</sup>*<sup>2</sup> *<sup>e</sup>*�*cy* sinh *<sup>κ</sup>*

<sup>∣</sup>*c*<sup>∣</sup> *<sup>e</sup>*�*cy* 2 

One can see that for *σ* ¼ 1 (an increase of a chemical substance), the cell density

vanishing as *y* ! �∞, whereas for *B*< 0 *u y* ð Þ has a point of discontinuity. One can say that when *B*<0, we obtain "blow-up" solution in the sense that it goes to

The expressions for *n*≥ 1 become more complicated, and one can see the

n=1 n=3 n=5 k/C^2 =10

0 2 4 6 8

u z

5 10 15 20 25

ð Þ*<sup>z</sup>* and the curves for *un*þ<sup>1</sup>

*<sup>π</sup>* and that for *B*>0 *u y* ð Þ is the solitary continuous solution

known one of the Korteweg-de Vries equation. Returning to the variable *y*:

<sup>¼</sup> *<sup>κ</sup> C*2 ∣*c*∣

*Mathematical Theorems - Boundary Value Problems and Approximations*

*e v e*�*cy* 2 

infinity for finite *y*, and this is true for different *ν*.

n=0 n=2 n=4 n=6

n=5

n=4

n=3 n=2 n=1 n=0

*vn*þ<sup>1</sup> 2

*u y* ð Þ≥0 for *<sup>B</sup>* <sup>≥</sup> <sup>1</sup>

solitonic behavior of *e*

10

ð Þ*<sup>z</sup> ; n* <sup>¼</sup> *<sup>0</sup>*, *:::6; c* <sup>¼</sup> *1.*

10

ð Þ*z ; n* ¼ *0*, *:::5; c* ¼ *1.*

20

30

40

**Figure 14.**

**Figure 15.** *un*þ*<sup>1</sup> 2*

**14**

*e vn*þ*<sup>1</sup> 2* 20

30

40

50

60

*Mathematical Theorems - Boundary Value Problems and Approximations*

k=17, c=2 n=6 n=4 n=2

6 4 2 2 4

v y

50

n=0

nu = 7 nu = 45 nu = 1 5 nu = 8 17

5 5 10

100

150

200

250 u y

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis*

k=1, c=2

n=5 n=3 n=1

e^v y , k C^2=10

0 20 40 60

2 4

k=1, c=1

*DOI: http://dx.doi.org/10.5772/intechopen.91214*

n=6 n=4 n=2

**Figure 21.** *un*þ*<sup>1</sup> 2*

**Figure 22.** *vn*þ*<sup>1</sup> 2*

**Figure 23.**

**17**

*ev<sup>ν</sup>* ð Þ*<sup>y</sup> ; <sup>ν</sup>* <sup>¼</sup> *<sup>1</sup>=5*; *<sup>8</sup>=17*; *<sup>7</sup>*; *45.*

ð Þ*y ; n* ¼ *0*, *1*, *:::6.*

Out[30]=

0.02

0.04

0.06

0.08

ð Þ*y ; n* ¼ *2*; *4*; *6.*

**Figure 18.** *vn*þ*<sup>1</sup>*

**Figure 19.** *e vn*þ*<sup>1</sup> 2* ð Þ*<sup>y</sup> ; n* <sup>¼</sup> *<sup>2</sup>*; *<sup>4</sup>*; *6.*

**Figure 20.** *un*þ*<sup>1</sup> 2* ð Þ*y ; n* ¼ *1*; *3*; *5.*

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis DOI: http://dx.doi.org/10.5772/intechopen.91214*

**Figure 21.** *un*þ*<sup>1</sup> 2* ð Þ*y ; n* ¼ *2*; *4*; *6.*

n=5 n=3 n=1

k=1, c=2

k=1, c=2

n=5 n=3 n=1

**Figure 18.**

**Figure 19.**

**Figure 20.** *un*þ*<sup>1</sup> 2*

**16**

ð Þ*y ; n* ¼ *1*; *3*; *5.*

ð Þ*<sup>y</sup> ; n* <sup>¼</sup> *<sup>2</sup>*; *<sup>4</sup>*; *6.*

*e vn*þ*<sup>1</sup> 2* ð Þ*<sup>y</sup> ; n* <sup>¼</sup> *<sup>1</sup>*; *<sup>3</sup>*; *5.*

*e vn*þ*<sup>1</sup> 2* k=1, c=1

50

n=6 n=4 n=2 k=10, c=2

k=1, c=1

k=1, c=2

k=1, c=1

100

150

200

250

300

*Mathematical Theorems - Boundary Value Problems and Approximations*

e^v y , 1 C^2 10

4 2 0 2 4 6

4 2 2 4

6 4 2 2 4 6

50

100

150

u y

k=17, c=2

10

20

30

40

50

e^v y , 1 C^2 10

k=10, c=2

**Figure 22.** *vn*þ*<sup>1</sup> 2* ð Þ*y ; n* ¼ *0*, *1*, *:::6.*

**Figure 23.** *ev<sup>ν</sup>* ð Þ*<sup>y</sup> ; <sup>ν</sup>* <sup>¼</sup> *<sup>1</sup>=5*; *<sup>8</sup>=17*; *<sup>7</sup>*; *45.*

*uν*ð Þ*y ; ν* ¼ *1=5*; *8=17.*

In conclusion it seems interesting to present the plots for *evν*ð Þ*<sup>y</sup>* and *<sup>u</sup>ν*ð Þ*<sup>y</sup>* for different values of *ν* (**Figures 23**–**25**). It is interesting to see that there are irregular solutions for *evν*ð Þ*<sup>y</sup>* ; however, the corresponding solutions for *<sup>u</sup>ν*ð Þ*<sup>y</sup>* are regular [see Eqs. (35)–(37)].
