A reaction–diffusion equation and its traveling wave solutions

ARTICLE IN PRESS International Journal of Non-Linear Mechanics 45 (2010) 634–639

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International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm

A reaction–diffusion equation and its traveling wave solutions Zhaosheng Feng a,, Goong Chen b, Qingguo Meng c a b c

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA Department of Mathematics, Texas A&M University, College Station, TX 77843, USA Department of Mechanics, Tianjin University of Technology and Education, Tianjin 300222, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 1 December 2007 Received in revised form 12 February 2010 Accepted 14 March 2010

In the present paper, we study a non-linear reaction–diffusion equation, which can be considered as a generalized Fisher equation. An exact solution and traveling wave solutions to the generalized Fisher equation are obtained by means of the Cole–Hopf transformation and the Lie symmetry method. & 2010 Elsevier Ltd. All rights reserved.

JEL classification: 34C14 35B05 Keywords: Traveling waves Fisher equation Cole–Hopf transformation Infinitesimal generator Prolonged operator Lie symmetry

1. Introduction The problems of the propagation of non-linear waves have fascinated scientists for over two hundred years [1,2]. Modern theories describe non-linear waves and coherent structures in a diverse variety of fields, including general relativity, high energy particle physics, plasmas, atmosphere and oceans, animal dispersal, random media, chemical reactions, biology, non-linear electrical circuits, and non-linear optics. Nowadays it has been universally acknowledged in the physical, chemical and biological communities that the reaction–diffusion equation plays an important role in dissipative dynamical systems. Typical examples are provided by the fact that there are many phenomena in biology where a key element or precursor of a developmental process seems to be the appearance of a traveling wave of chemical concentration (or mechanical deformation). When reaction kinetics and diffusion are coupled, traveling waves of chemical concentration can effect a biochemical change much faster than straight diffusional processes. This usually gives rise to reaction–diffusion equations which in one dimensional space

 Corresponding author. Tel.: + 1 956 292 7483; fax: + 1 956 384 5091.

E-mail address: [email protected] (Z. Feng). 0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2010.03.004

takes the form @u @2 u ¼ k0 2 þ FðuÞ, @t @x

ð1Þ

for a chemical concentration u, where k0 is the diffusion coefficient, and f(u) represents the kinetics. When F(u) is linear, i.e., F(u)¼k2u +k1, where both k1 and k2 are constants, then in many instances Eq. (1) can be solved by the separation of variables methods. However if, as in many of the applications considered in [3], F(u) is non-linear, then the problem is much more intractable. Indeed, it is not usually possible to obtain general exact analytical traveling wave solutions and one must analyze such problems numerically [4]. Despite this, however, under some particular circumstances, many non-linear evolutionary equations have traveling wave solutions of special types, which are of fundamental importance to our understanding of biological phenomena modeled evolutionary equations. The classic and simplest case of the non-linear reaction–diffusion equation is when F(u)¼k3u(1 u), which is the so-called Fisher equation. It was suggested by Fisher as a deterministic version of a stochastic model for the spatial spread of a favored gene in a population [5]. (Although this equation is now referred to as the Fisher equation, the discovery, investigation and analysis of traveling waves in chemical reactions was first presented by Luther at a conference [6]. There, he stated that the wave speed is

ARTICLE IN PRESS Z. Feng et al. / International Journal of Non-Linear Mechanics 45 (2010) 634–639

a simple consequence of the differential equations. This recently re-discovered paper has been translated into English by Arnold et al. [7] and Luther’s remarkable discovery and analysis of chemical waves has been put in a modern context by Showalter and Tyson [8].) In the 20th century, the Fisher equation has became the basis for a variety of models for spatial spread. The typical examples are that Aoki discussed gene-culture waves of advance [9] and Ammerman and Cavali-Sforza, in an interesting direct application of the model, applied it to the spread of early farming in Europe [10,11]. Meanwhile, the qualitative analysis in the phase plane and traveling wave solutions of the Fisher equation have been widely investigated. The seminal and now classical references are that by Kolmogorov, Petrovsky and Piscunov [12], Albowitz and Zeppetella [13], Fife [14] and Britten [15]. In [12], Kolmogorov et al. showed that any initial concentration which is one for large negative spatial variable x and vanishes for largepx, ffiffiffiffiffievolves to a traveling wavefront with minimal velocity v ¼ 2 k0 . Different initial values propagate with different traveling waves, depending on the behavior at x- 71. The first explicit analytic form of a cline solution for the Fisher equation was obtained by Albowitz and Zeppetella making use of the Painleve´ analysis [13]. A full discussion of this equation and an extensive bibliography can be seen in [14,15]. The singular ¨ property, auto-Backlund transformation and analytic solutions including some heteroclinic and homoclinic solutions of the Fisher equation were obtained by Guo and Chen via the expanded Painleve´ for carrier flow equation in semiconductor devices [16,17]. A discrete singular convolution algorithm was introduced to solve Fisher’s equation and predicted long-time traveling wave behavior by Zhao and Wei [18]. In the present work, we consider Eq. (1) with FðuÞ ¼ uðm þ bugu2 Þ, namely @u @2 u ¼ a 2 þ uðm þ bugu2 Þ, @t @x

ð2Þ

where a, b, m and g are real constants. This equation can be regarded as a generalization of the Fisher equation, which is used as a density-dependent diffusion model, in the one-dimensional situation, for studying insect and animal dispersal with growth dynamics [3], and as a genetic model arising from the classical theory of population genetics and combustion [19,20]. During the past decade, considerable attention has been received to exact solutions and traveling wave solutions of some special cases of Eq. (2). When b ¼ 0, exact solutions were obtained by Clarkson and Mansfield [21] using the non-classical method. When both b and g are non-zero, exact solutions of Eq. (2) have been found by Chen and Guo using a truncated Painleve´ expansion [22], by Chowdhury [23] and Este´vez and Gordoa [24] using a complete Painleve´ test, and by Clarkson and Mansfield [21]. More profound results have been established by Clarkson and Mansfield making use of the non-classical reductions method [25]. Herrera et al. [26] obtained traveling wave solutions of (2) when p¼ 2 in their equations. Kudryashov [27] derived an exact solitary wave solution to Eq. (2) by utilizing the Riccati equation and the Jacobi elliptic function. The study of the properties of the traveling waves and their applications were undertook by de Pablo and Sanchez [28]. Note that since two non-linearities occur in Eq. (2), in general it is not integrable. Therefore, to seek exact solutions of Eq. (2), qualitative analysis together with ingenious mathematical techniques for treating such non-linear system appears to be more powerful and important. Recently, qualitative results for some physical and biological systems have been studied extensively [29–32] and some innovative mathematical methods, such as the Lie group analysis and symmetry method have been developed and widely applied to many non-linear systems [33–35].

635

Our goal in this paper is to find exact solutions and traveling wave solutions to Eq. (2) under certain parametric conditions. The rest of the paper is organized as follows. In Section 2, we consider a special case of Eq. (2) where m ¼ 0. A traveling wave solution is found by utilizing the Cole–Hopf transformation, and an exact solution is presented by means of the Lie symmetry method. In Section 3, we focus on traveling wave solutions in terms of elliptic functions for Eq. (2). Section 4 is a brief conclusion.

2. Exact solution in the case l ¼ 0 In this section we mainly study exact solutions for Eq. (2) when m ¼ 0. Make a Cole–Hopf transformation: u ¼ RðlnpÞx ,

ð3Þ

where R is a real constant, and p is a function of x and t to be determined. Substitution of (3) into Eq. (2) yields 3px pxx p2 px pt p2 pxxx p3 2ðpx Þ3 p þ pxt p3 Rbðpx Þ2 p2 þ R2 gðpx Þ3 p ¼ 0: To reduce (4) to a bilinear equation as follows, we set R ¼ 7 3px pxx px pt pxxx p þpxt pRbðpx Þ2 ¼ 0:

ð4Þ pffiffiffiffiffiffiffiffi 2=g ð5Þ

Assume that Eq. (5) admits the solution of the form pðx,tÞ ¼ k1 eðm1 x þ n1 tÞ þk2  eðm2 x þ n2 tÞ þ k3  eðm3 x þ n3 tÞ ,

ð6Þ

where ki (i¼1, 2, 3), and mj and nj (i¼1, 2, 3) are constants. Substituting (6) into (5) and equating the corresponding coefficients of the resulting exponential functions, we get an algebraic system 8 > 3mi mj ðmi þ mj Þðmi Þ3 ðmj Þ3 2Rbmi mj > < þ ðni nj Þðmi mj Þ ¼ 0 ð1r i oj r3Þ, ð7Þ > > : 2ðmq Þ3 Rbðmq Þ2 ¼ 0 ð1r q r3Þ: System (7) can be solved with the aid of Maple. Changing to our original variables, we obtain that Eq. (2) has exact solutions as ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b k2  e ðb =2agÞx þ ðb =2gÞt ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi u1 ðx,tÞ ¼  g k2  e ðb2 =2agÞx þ ðb2 =2gÞt þ ðk1 þ k3 Þ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b e ðb =2agÞx þ ðb =2gÞt ffi ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffi , ð8Þ g e ðb2 =2agÞx þ ðb2 =2gÞt þC1 and pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 e ðb =2agÞx þ ðb =2gÞt ffi , u2 ðx,tÞ ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffi g e ðb2 =2agÞx þ ðb2 =2gÞt þ C2

b

ð9Þ

where C1 and C2 are arbitrary. Note that (8) and (9) are actually traveling wave solutions. When both C1 and C2 are positive, u1(x, t) and u2(x, t) each describe a kind-profile traveling wave (see Fig. 1). These two solutions are monotone with respect to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ 7 ðb2 =2agÞx þ ðb2 =2gÞt. They are analytic on the whole (x, t)-plane, but blow up at infinite points of (x, t) when both C1 and C2 are negative. Now let us briefly state the general idea of the Lie symmetry method for partial differential equations (PDEs) [33–35]. Here we only consider PDEs with one dependent variable u and two independent variables x and t. A point transformation is a diffeomorphism ~ ~ ~ T : ðx,t,uÞ/ðxðx,t,uÞ, tðx,t,uÞ, uðx,t,uÞÞ, which maps the surface u ¼u(x, t) to the surface ~ x~ ¼ xðx,t,uðx,tÞÞ,

~ t~ ¼ tðx,t,uðx,tÞÞ,

~ u~ ¼ uðx,t,uðx,tÞÞ:

ð10Þ

ARTICLE IN PRESS 636

Z. Feng et al. / International Journal of Non-Linear Mechanics 45 (2010) 634–639

1.2

1

1

0.8

0.8

0.6

0.6

u

u

1.2

-10

-5

0.4

0.4

0.2

0.2

0 -0.2

0

5 xi

10

-10

-5

0 -0.2

0

5 xi

10

Fig. 1. The left figure: traveling wave solution u1 ðx,tÞ ¼ u1 ðxÞ when a ¼ b ¼ g ¼ 1 and C1 is positive. The right figure: traveling wave solution u2 ðx,tÞ ¼ u2 ðxÞ when a ¼ b ¼ g ¼ 1 and C2 is positive.

In order to get the prolongation of a given transformation, we need to differentiate (10) with respect to each of the parameters x and t (for details, see [34,35]). First, we need to calculate two total derivatives: Dx ¼ @x þ ux @u þuxx @ux þuxt @ut þ    , Dt ¼ @t þut @u þuxt @ux þ utt @ut þ    :

XD ¼ 0

Consider PDEs of the form

D ¼ up Fðx,t,u,ux ,ut , . . .Þ ¼ 0,

ð11Þ

where up is one of the m th order derivatives of u and F is independent of up. The point transformation T is a point symmetry of (11) if ~ u, ~ t, ~ u~ x~ , u~ t~ , . . .Þ ¼ 0: u~ p Fðx,

ð12Þ

Typically, the symmetry condition (12) is extremely complicated, so one does not need to try to solve it. However, it is usually possible to carry out a systematic search for one-parameter Lie group of point symmetries. Namely, we try to find point symmetries of the form x~ ¼ x þ exðx,t,uÞ þOðe2 Þ, t~ ¼ t þ etðx,t,uÞ þ Oðe2 Þ, u~ ¼ u þ eZðx,t,uÞ þ Oðe2 Þ: Just as for Lie group point transformations of the plane, each oneparameter (local) Lie group of point transformations is obtained by exponentiating its infinitesimal generator, which is X ¼ x@x þ t@t þ Z@u :

ð13Þ

dx~ ~ t~ , uÞ, ~ ¼ xðx, de

when D ¼ 0:

@u @ 2 u ¼ a 2 þ nu ½u ðu Þ2 , @t @x

du~ ~ uÞ, ~ t, ~ ¼ Zðx, de

2

where n ¼ b =g. It is also convenient at the outset to re-scale (17) by writing  n 1=2 n t ¼ t, x ¼ x: 2 2a Omitting the asterisks for notational simplicity and assuming that the resulting equation has a two-parameter Lie group of (classical) point symmetries, generated by X1 ¼ @x ,

X2 ¼ @t ,

then one can obtain that the invariant surface condition becomes

Zxx Zt ¼ 3u2 Z2uZ,

~ uÞj ~ t, ~ e ¼ 0 ¼ ðx,t,uÞ: ðx,

where

A surface u¼ u(x, t) is mapped to itself by the group of transformations generated by X if

Zx ¼ Dx Zux Dx xut Dx t, Zt ¼ Dt Zux Dt xut Dt t:

when u ¼ uðx,tÞ:

ð14Þ

The above equation can be rewritten neatly by using the characteristic of the group. That is G ¼ Zxux tut : From (14), the surface u¼u(x,t) is invariant if it holds that G¼0

when u ¼ uðx,tÞ:

ð17Þ

ð18Þ

From (16), we have the linearized symmetry condition for these non-classical symmetries

subject to the initial condition:

Xðuuðx,tÞÞ ¼ 0

ð16Þ

Eq. (11) enables us to eliminate up from (16); then we split the remaining terms (according to their dependence on derivatives of u) to obtain a linear system of determining equations for x, t and Z. Solving this resulting system and writing in terms of the original variables, we can obtain the exact solution to the PDE (11). Next, let us revert to Eq. (2). Note that (2) is a particular case of (11) with the highest order 2 of derivative. In order to simplify our calculations, we may convert Eq. (2) with m ¼ 0 to the following form by assuming u ¼ ðb=gÞu

ut Z þ xux ¼ 0:

~ uÞ ~ t, ~ can be obtained by solving Similarly, ðx, dt~ ~ uÞ, ~ t, ~ ¼ tðx, de

Eq. (15) is the so-called invariant surface condition, which usually plays a central role to some of the main techniques for finding exact solutions of PDEs. Lie point symmetries are obtained by differentiating the symmetry condition (12) with respect to e at e ¼ 0. So the linearized symmetry method condition is

ð15Þ

ð19Þ

ð20Þ

To get the Lie point symmetries, using (19) and (20) we have

Z ¼ Zt xt ux þ Zu ut tt ut xu ux ut tu ðut Þ2 , Zxx ¼ txx ut þ Zxx þ 2Zxu ux xxx ux þ Zuu ðux Þ2 2xxu ðux Þ2 2txu ux ut xuu ðux Þ3 tuu ðux Þ2 ut þ Zu uxx 2xx uxx 2tx uxt 3xu ux uxx tu ut uxx 2tx ux uxt : t

Substituting the above two equations into (19), writing (19) out in full, using (17) and (18) to eliminate uxx and ut, and then splitting the resulting equation by equating powers of ux, we obtain the

ARTICLE IN PRESS Z. Feng et al. / International Journal of Non-Linear Mechanics 45 (2010) 634–639

determining equations for x, t and Z as follows:

637

then Eq. (28) becomes

xuu ¼ 0, Zuu 2xxu þ2xxu ¼ 0, 2Zxu xxx 2Zxu 6u3 xu þ 6u2 xu þ2xxx þ xt ¼ 0, Zxx 2Zxx þ 2u3 Zu 4u3 xx 2u2 Zu þ 4u2 xx Zt þ 4uZ6u2 Z ¼ 0:

r 2 t2

d2 u 2 au3 bu du ¼ 0: dt2

ð30Þ

Take the coordinate transformation as ð21Þ

q ¼ tk ,

rðqÞ ¼ u  t1=2ðk1Þ ,

ð31Þ

then Eq. (30) reduces to a simple form Although (21) contains some non-linear equations and looks complicated, it is easily solved since it happens to be in a triangle form. The general solution of the first equation is

d2 r

x ¼ Hðx,tÞu þVðx,tÞ,

4d : r2 Consider the set of transformations (  q ¼ f0 ðq, r, eÞ, q je ¼ 0 ¼ q, Le ¼ r ¼ c0 ðq, r, eÞ, r je ¼ 0 ¼ r,

ð22Þ

Z ¼ 13H2 ðx,tÞu3 þ Hx ðx,tÞu2 Hðx,tÞVðx,tÞu2 þ Aðx,tÞu þBðx,tÞ:

ð23Þ

Substituting (22) and (23) into system (21) and equating powers of u, we obtain either Vðx,tÞ ¼ 1,

Vðx,tÞ ¼ 81,

Aðx,tÞ ¼ Bðx,tÞ ¼ 0:

x ¼ 1, Z ¼ 0 or x ¼ 7 3u 8 1, Z ¼ 3u3 þ 3u2 :

ð24Þ

From (13), the first solution in (24) corresponds to the classical symmetry generator X1 +X2, but the second solution in (24) corresponds to the non-classical symmetry generators 3

2

X ¼ 7 ð3u1Þ@x þ@t þð3u þ 3u Þ@u :

ð25Þ

Combining (15) and (25) gives the invariant surface condition for the non-classical symmetries as

which yields an exact solution to the generalized Fisher equation (2) as follows: uðx,tÞ ¼

g

2



b2



g

1 þ c1  eðb !1=2

t7

b2

2ag

, 2

x þ c1  eðb

xðq, rÞ ¼

 @f0 ðq, r, eÞ  @e

2

1=2

=2gÞt 7 ðb =2agÞ

ð26Þ

x þc 2

3. Traveling waves in terms of elliptic functions

ð27Þ

where v is the wave velocity and v A R. Substituting (27) into Eq. (2) gives 2

ð28Þ

where r ¼ v=a, a ¼ g=a, b ¼ b=a and d ¼ m=a. When all coefficients of (28) are non-zero, in the general case, we know that Eq. (4) does not pass the Painleve´ test, and is not integrable either. In order to avoid complicated calculations, we make the natural logarithmic transformation 1 x ¼ lnt, r

ð35Þ

Zðq, rÞ ¼

,

 @c0 ðq, r, eÞ  @e

:

e¼0

For the infinitesimal transformation (35), the associated infinitesimal generator and prolonged operator are given as X ¼ xðq, rÞ

@ @ , þ Zðq, rÞ @q @r

and

DX ¼ xðq, rÞ

2 X @ @ @ þ z ðq, r, r0 , rqðiÞ Þ ðiÞ , þ Zðq, rÞ @q @r i ¼ 1 k @rq

respectively, where @rðkÞ , z1 ¼ Zq þðZr xq Þr0q xr ðr0q Þ2 ¼ Dq ðZÞr0q Dq ðxÞ, @q þ 1Þ zk þ 1 ¼ Dq ðzk Þrðk Dq ðxÞ, q þ 1Þ rðk ¼ q

here Dq ¼ @q þ r0q @r þ    is the operator of total derivative with respect to q. If we assume that Eq. (32) admits the Lie group (34), from the invariance condition ¼ Fðq, rÞ

¼ 0,

we have

Assume that Eq. (2) has a traveling wave solution of the form

u00 ðxÞru0 ðxÞau3 bu du ¼ 0,

r ¼ r þ eZðq, rÞ þ Oðe2 Þ,

e¼0

DXðr00 Fðq, rÞÞjr00

x ¼ xvt,

ð34Þ

2

=2gÞt 7 ðb =2agÞ1=2 x

where c1 and c2 are arbitrary constants. Formula (26) appears more general than formulas (8) and (9) obtained by using the Cole–Hopf transformation.

uðx,tÞ ¼ uðxÞ,

ð33Þ

here

ut 7 ð3u1Þ@x þ ð3u3 3u2 Þ ¼ 0,

b

ð32Þ

where k is given by

q ¼ q þ exðq, rÞ þ Oðe2 Þ,

Thus, the two solutions of the determining equations are either

a ð13kÞ=k 3 b q r þ 2 2 qð15kÞ=2k r2 ¼ Fðq, rÞ, r 2 k2 r k

where f0 and c0 are smooth functions of their arguments and e is a real parameter. The set Le is called a continuous one-parameter Lie group of point transformations if the relation Le1 þ e2 ¼ Le1 3Le2 holds for any real e1 and e2 . Taylor’s expansion of q* and r* in (34) with respect to the parameter e at e ¼ 0 yields

Aðx,tÞ ¼ Bðx,tÞ ¼ 0,

or Hðx,tÞ ¼ 73,

¼

k2 ¼ 1þ

which leads to the solution of the second equation as

Hðx,tÞ ¼ 0,

2

dq

ð29Þ

Zqq þ ð2Zqr xqq Þr0q þðZrr 2xqr Þðr0q Þ2 xrr ðr0q Þ3 ¼ ð2xq Zr þ 3xr r0q ÞFðq, rÞ þ xFq ðq, rÞ þ ZFr ðq, rÞ:

ð36Þ

This is a second-order partial differential equation for two unknown functions xðq, rÞ and Zðq, rÞ. Since the unknown functions do not depend on the derivative r0q , after setting the coefficients of the powers ðr0q Þi ði ¼ 0,1,2,3Þ in (36) to zero, one can get the determining equations which can be split and represented as the system

xrr ¼ 0,

ð37Þ

2xqr Zrr ¼ 0,

ð38Þ

2Zqr xqq 3xr Fðq, rÞ ¼ 0,

ð39Þ

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Zqq þ ðZr 2xq ÞFðq, rÞxFq ðq, rÞZFr ðq, rÞ ¼ 0:

ð40Þ

From Eqs. (37) and (38), we get

xðq, rÞ ¼ gðqÞr þf ðqÞ,

Zðq, rÞ ¼ g 0 ðqÞr2 þ fðqÞr þ cðqÞ,

ð41Þ

where g(q), f(q), fðqÞ and cðqÞ are functions to be determined later. Substituting (41) into Eqs. (39) and (40), respectively, we have 0

3g 00 ðqÞr þ 2f ðqÞf 00 ðqÞ3gðqÞFðq, rÞ ¼ 0,

ð42Þ

g 000 ðqÞr2 þ f00 ðqÞr þ c00 ðqÞ þ ½fðqÞ2f 0 ðqÞFðq, rÞ ½gðqÞr þ f ðqÞFq ðq, rÞ½g 0 ðqÞr2 þ fðqÞr þ cðqÞFr ðq, rÞ ¼ 0: ð43Þ Choosing g(q)¼ 0 and fðqÞ ¼ 12 f 0 ðqÞ þ c0 , where c0 is an arbitrary constant, and substituting g(q) and fðqÞ into Eq. (43), we have   c00 þ 12f 000 ðqÞrf ðqÞFq ðq, rÞ þ c0 32f 0 ðqÞ Fðq, rÞ    c0 þ 12f 0 ðqÞ r þ cðqÞ Fr ðq, rÞ ¼ 0: ð44Þ When k a 7 1 and 71/3, setting c00 and the coefficients of r and r2 in (44) to zero, we have

where a1, b1 and c1 (j ¼0, 1, 2, 4) are constants. Substituting cðqÞ and f(q) into Eq. (44), and equating corresponding coefficients of powers of q while setting the coefficient of r3 to zero, we obtain an algebraic system which, to within non-zero coefficients, gives k¼5/3 or  5/3. It thus yields two cases of prolongation of the admissible group: Case 1: when k ¼ 53, we obtain that Eq. (32) admits the infinitesimal generator @ @ , þ Zðq, rÞ @q @r 15ab ð4=5Þ 3ab xðq, rÞ ¼ q , Zðq, rÞ ¼ 2 qð1=5Þ r þ a: 2r 2 r X1 ¼ xðq, rÞ

@ @ , þ Zðq, rÞ X2 ¼ xðq, rÞ @q @r 15ab 6=5 9ab 1=5 xðq, rÞ ¼ q , Zðq, rÞ ¼ q raq: 2r 2 2r 2 Consequently, we obtain that (I) when k ¼ 53, we have ð45Þ 3 1=5 1 2 a2 b2 k1 ,

where g=m ¼ 8 b=m ¼ 7 and j is the elliptic function [36] defined by Z j ¼ ð1 7 o4 Þ1=2 do þ C3 ðwhere the modulus k1 ¼ 7 1Þ:

(II) when k

¼  53,

q ¼ a2 C15 j5 ,

where x0 is an arbitrary constant, r and q are given by the parametric form (45). The other solution is pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi u2 ðx,tÞ ¼ e 8 ð2 am=aÞðx 8 ð3=2Þ amt þ x0 Þ  rðqÞ, where r and q are given by the parametric form (47) and o is the same as (46). It is remarkable that when g ¼ 0, choosing k ¼ 15 in Eq. (32), we can obtain the same result as that in [13]. When b ¼ 0, using the closely similar arguments, we obtain that when k ¼  13, namely, the velocity v satisfies v2 ¼ 92 am, Eq. (2) has a traveling wave solution as pffiffiffiffiffiffiffi pffiffi pffiffiffiffiffi u3 ðx,tÞ ¼ e 8 ð 2am=aÞðx 8 ð3 2=2Þ amt þ x0 Þ  rðqÞ,

Z

ð1 þ j41 Þ1=2 dj1 þC2

1

,

where g=m ¼ 7 a43 =b23 , and j1 is defined by Z dt C2 : j1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ð4t3 1Þ Here the upper sign in this formula corresponds to the classical elliptic Weierstrass function t ¼ tðj1 þC2 ,0,1Þ [36]. Note that when b ¼ 0, if k ¼ 13, (32) becomes an autonomous system, thus Eq. (2) has a traveling wave solution as pffiffiffiffiffiffiffi pffiffi pffiffiffiffiffi u4 ðx,tÞ ¼ e 8 ð 2am=2aÞðx 8 ð3 2=2Þ amt þ x0 Þ  rðqÞ, where x0 is an arbitrary constant, r and q satisfy Z dr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 : q¼ ðg=mÞr4 7 1

Case 2: when k ¼  53, Eq. (32) admits the infinitesimal generator

1 2=5 2 2 a2 b2 ,

Note that when k ¼ 7 53, formula (33) gives v2 ¼  94 am. Using (27) as well as the inverse transformations of (29) and (31), we obtain that: Suppose that the velocity v satisfies v2 ¼  94 am. Then Eq. (2) has two traveling wave solutions. One is pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi u1 ðx,tÞ ¼ e 7 am=2aðx 8 ð3=2Þ amt þ x0 Þ  rðqÞ,

r ¼ b3 C14 j1

þ 3aða1 q þ b1 Þqð1=k3Þ ¼ 0,

r ¼ b2 C1 jðojk1 Þ,

4=5

where x0 is an arbitrary constant, r and q are given by the parametric form

Z 1 ð1 þ j41 Þ1=2 dj1 þ C2 q ¼ a3 C12 ,

32a1 bk qð1=2k þ 3=2Þ r 2 ð1kÞð1 þ kÞð1 þ 3kÞ 32b1 bk c1 þ 2 qð1=2k þ 1=2Þ þ x2 þ c2 x 2 r ð13kÞð1kÞð1 þ kÞ



 5 0 1 5 1=2k5=2 þ c3 , f ðqÞ þc0 bq þb  f ðqÞqð1=2k7=2Þ 2 2k 2

cðqÞ ¼ a1 q þb1 , f ðqÞ ¼

q ¼ a2 C15 j5 ,

8=5

3 1 where g=m ¼ 8 12 a2 b2 2 , b=m ¼ 7 2 a2 b2 k1 , and j is the same as (46).

4. Conclusion Many physical, chemical and biological phenomena can be described by non-linear reaction–diffusion models. Typical examples are given by the Fisher equation after Fisher who proposed the one-dimensional version as a model for the spread of an advantageous gene in a population. In this work, we are concerned with a non-linear reaction–diffusion equation, which can be considered as a generalized Fisher equation. By making use of the Cole–Hopf transformation and the Lie symmetry method, we obtain an exact solution and a class of traveling wave solutions. At the present stage, the chemical and biological explanation is being sought for traveling wave solutions of this special type.

ð46Þ

Acknowledgments

ð47Þ

The work is supported by UTPA Faculty Research Council Grant 119100 and NSF Grant CCF-0514768.

we have

r ¼ b2 C14 j4 ðojk1 Þ,

ARTICLE IN PRESS Z. Feng et al. / International Journal of Non-Linear Mechanics 45 (2010) 634–639

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