Symmetry analysis of reaction diffusion equation with distributed delay

Accepted Manuscript Symmetry Analysis of Reaction Diffusion Equation with Distributed Delay Zhihong Zhao, Weigao Ge PII: DOI: Reference:

S1007-5704(14)00567-X http://dx.doi.org/10.1016/j.cnsns.2014.12.006 CNSNS 3433

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Communications in Nonlinear Science and Numerical Simulation

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25 July 2014 26 September 2014 12 December 2014

Please cite this article as: Zhao, Z., Ge, W., Symmetry Analysis of Reaction Diffusion Equation with Distributed Delay, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http://dx.doi.org/10.1016/ j.cnsns.2014.12.006

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Symmetry Analysis of Reaction Diffusion Equation with Distributed Delay Zhihong Zhaoa

∗

Weigao Geb

a School of Mathematics Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China b Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China

Abstract: This paper study the reaction-diffusion equation with distributed delay from the Lie group theoretic point of view. we gives at first the evolutionary infinitesimal vector field v and a number of group invariant solutions corresponding to v by general symmetry group theory. Keywords: Symmetry; Evolutionary infinitesimal vector field; Reaction-diffusion equation; Distributed delay; Group invariant solutions.

1

Introduction

Reaction-diffusion equations with time delays appear in problems with delaying links where certain information processing is needed, such as in population dynamics, biology and epidemiology. During the past 30 years, many researchers have paid attention to reaction-diffusion equations with different kinds of time delays. According to the difference of models and the impact of different factors which were take into account in the models, time delays mainly divided into the following form: discrete time delay (see, e.g., [2, 19] and references therein), distributed time delay (see, e.g., [1, 3] and references therein), both time delay and spatially nonlocal effect (see, e.g., [6, 10, 21] and references therein) and spatiotemporal time delay (see, e.g., [4, 17, 20] and references therein). It is noteworthy that the references mentioned above mainly studied the existence of traveling wave solutions. Lie group theory is one of a powerful and direct approach to construct exact solutions of nonlinear differential equations [9, 16]. Around in the middle of nineteenth century, Norwegian mathematician Sophus Lie found the invariance of differential equations under a continuous group of symmetries. The applications of Lie’s continuous symmetry groups, i.e., Lie group, include diverse fields, such as algebraic topology, differential geometry, numerical analysis, bifurcation theory and so on. Roughly speaking, a symmetry group of system of differential equations is a group which transforms solutions of the system to other solutions. For a system of partial differential equations, we can use general symmetry groups to explicitly determine a special type of solutions which are themselves invariant under one-parameter symmetry group of ∗

Corresponding author: Zhihong Zhao, E mail: [email protected]

1

the system. The group-invariant solutions are found by solving a reduced system of differential equations involving fewer independent variables than the original one. We should note that Lie group theory, developed for partial differential equations, can not be directly applied to delay differential equations [13, 14]. The main obstacle is their nonlocality: the nonlocality does not allow using the manifold’s approach to define a Lie group. There have been few investigations of delay differential equations by group approach up to now. It should be mentioned that Linchuk [5] suggested a group approach to research functional differential equations, where delay differential equations are replaced by an underdetermined differential equations for which the classical group analysis is applied. But the extension of equations narrows a set of admitted Lie groups. After then Tanthanuch and Meleshko [13, 14] generalized the group analysis theory to delay differential equation, defining an admitted Lie group and determining equation for delay differential equation. Notice that this definition does not need that the admitted Lie group should transform a solution into a solution. Later, Meleshko and Moyo [8], applying this generalized theory and extended group classification notion to treat delay differential equations, presented the complete group classification of the reactiondiffusion equation with discrete delay. Pue-on et al. [11] applied this generalized theory to second-order delay ordinary differential equations y 00 = f (x, y, yτ , y 0 , yτ0 ), where yτ = y(x − τ ), yτ0 = y 0 (x − τ ) and provided all classes of second-order delay ordinary differential equations admitting a Lie algebra. Recently, Tanthanuch [15] provided a new application of this theory and give a complete group classification of the nonhomogeneous inviscid Burgers equation with delay. We find that the study of delay differential equations by group approach focuses on discrete delay and the main method is group analysis theory. In this paper, we will discuss the symmetry and some corresponding group-invariant solutions of reaction-diffusion equation with distributed delay, which has the following form ut (x, t) = uxx (x, t) + F (t, x, u(x, t), (f ∗ u)(x, t)),

(1.1)

where u is the function of t, x, with x ∈ R being the spatial variable, t is the time variable. And the convolution f ∗ u is defined by Z t (f ∗ u)(x, t) = f (t − s)u(x, s)ds, −∞

where the kernel f : [0, ∞) → [0, ∞) satisfies the following normalization assumption: Z ∞ f (t) ≥ 0, for all t ≥ 0, f (t)dt = 1.

(1.2)

0

Two specific cases of delay kernel function f (t), which have been widely used, are weak generic kernel [1]: 1 f (t) = e−t/τ , τ > 0, (1.3) τ and strong generic kernel: t (1.4) f (t) = 2 e−t/τ , τ > 0. τ 2

Here, we assume that f (t) is weak generic kernel. the parameter τ measures the delay, (1.3) implies that the importance of events in the past simply decreases exponentially the further one looks into the past. The existence of solutions of (1.1) with (1.2) can be found in [12, 18]. It is significant to study the symmetries of system (1.1) with (1.2). Utilizing the generalized symmetries theory and undetermined coefficient method of differential equation, we mainly study the symmetry and some corresponding group-invariant solutions of the reaction-diffusion equation with weak distributed delay in this paper. The research results show that different cases of reactions, F (t, x, u, v), possess different symmetries and group-invariant solutions, and we show several common cases of reactions’ symmetry and group-invariant solution. The rest of this chapter is organized as follows. Section 2 gives the theory of generalized symmetries. Section 3 discusses the characteristic of generalized symmetries of reaction-diffusion equations with distributed delay. In the process of solving the equation, only those relatively simple symmetries are useful. We determine some of group-invariant solutions, corresponding to the symmetry of reaction-diffusion equation with distributed delay in section 4.

2

Generalized symmetry method

In this section, we investigate the theory of generalized symmetry [9, 16]. It is useful to introduce some notations. Suppose (x, t) ∈ R2 is independent variable and u = (u1 , u2 ) ∈ R2 is dependent variable. A denotes the space of smooth differential functions P (x, t, u(n) ), depending on x, t, u and derivatives of u up to some finite, but unspecified order n, the range of P is R. Let P [u] := P (x, t, u(n) ). Definition 2.1. A generalized vector field has the form v = ξ[u]

∂ ∂ ∂ ∂ + η[u] + φ1 [u] 1 + φ2 [u] 2 , ∂x ∂t ∂u ∂u

(2.1)

where ξ[u], η[u], φ1 [u] and φ2 [u] ∈ A. Definition 2.2. Given v as above, set Qi [u] = φi [u] − ξ[u]uix − η[u]uit ,

i = 1, 2,

(2.2)

where uix = ∂ui /∂x, /uit = ∂ui /∂t, the 2-tuple Q[u] = (Q1 [u], Q2 [u]) is referred to as the characteristic of the vector field (2.1). Definition 2.3. Let Q[u] = (Q1 [u], Q2 [u]) ∈ A2 . The generalized vector field vQ = Q1 [u]

∂ ∂ + Q2 [u] 2 ∂u1 ∂u

(2.3)

is called an evolutionary vector field. If Q[u] = (Q1 [u], Q2 [u]) of (2.3) take the form (2.2), then generalized vector field vQ is called an evolutionary representative of v (2.1), and Q is called vQ ’s characteristic. Lemma 2.1 ([9]). A generalized vector field v is a symmetry of a system of differential equations if and only if its evolutionary representative vQ is. 3

Consider a system of n order evolution equations ∂u = P [u] ∂t

(2.4)

where P [u] = (P1 [u], P2 [u]) ∈ A2 depends on x ∈ R, u ∈ R2 and x-derivatives of u up to some finite order n. If we want to find generalized symmetries v of above evolution equation, we can simplify the computation by replacing v by its evolutionary representative vQ (2.3), where Q satisfies (2.2), by Lemma 2.1. An evolutionary vector field vQ (2.3) is a symmetry of the system of evolution equation (2.4) if and only if Dt Qi = prvQ (Pi ), i = 1, 2, (2.5) where Dt denotes the total derivative operator and is defined by Dt =

∂ ∂ ∂ ∂ ∂ + u1t 1 + u2t 2 + u1xt 1 + u2xt 2 + · · · , ∂t ∂u ∂u ∂ux ∂ux

and prvQ is the prolongation of the evolutionary vector field vQ given by prvQ (T ) = Tu0 1 Q1 + Tu0 2 Q2 , where T ∈ A, and Tu0 i Qi means Tu0 i Qi =

∂T ∂T ∂T Qi + i Dx Qi + i Dxx Qi + · · · , i ∂u ∂ux ∂uxx

i = 1, 2.

Thus, (2.5) can be reformed as Q1 Q2

!

t

=

0 P1u 1 0 P2u 1

0 P1u 2 0 P2u 2

!

Q1 Q2

!

.

(2.6)

Lemma 2.2 ([16]). If evolutionary vector field vQ (2.3) is a symmetry of the system of evolution equation (2.4), then u is the group-invariant solution of (2.4) corresponding vQ if and only if u satisfies   ∂u = P [u], ∂t (2.7)  Q[u] = 0.

where Q[u] is the characteristic of the vector field vQ .

3

Symmetries for reaction-diffusion equation with distributed delay

In this section, we consider the generalized symmetries of (1.1) with weak delay kernel (1.3). We do this by showing that the governing equation can be recast as a system of ordinary differential equations without delay of order two, and then applying the generalized symmetry method we has introduced above to this latter system. Thus, define v by Z t 1 t−s v(x, t) = (f ∗ u)(x, t) = e τ u(x, s)ds, −∞ τ 4

Differentiating with respect to t, we can have ∂v 1 = (u − v). ∂t τ Hence, system (1.1) with weak delay kernel (1.3) can be replaced by the system   ut = uxx + F (t, x, u, v),  vt = 1 (u − v). τ

(3.1)

Theorem 3.1. The characteristic Q of evolutionary vector field vQ , which is a symmetry of system (3.1), has the following form: ! ! ! ! ! ! ux ut Q1 τ vt − u + v 0 m =µ +β +α +θ + , (3.2) 0 τ vt − u + v n vx vt Q2 where β, µ are arbitrary constants, α, θ are arbitrary functions in x, t, u, v and n, m are arbitrary functions in x, t and they satisfy   mt = mxx + mFu + nFv − µFx − βFt , (3.3)  nt = 1 (n − m). τ

Proof. we mainly use the method of undetermined coefficients [16]. By (3.1) and (2.4), we have P1 = uxx + F (u, v),

P2 =

1 (u − v) τ

and 0 = D2 + F , P 0 = F , P1u v u x 1v 1 0 0 P2v = − τ1 , P2u = τ ,

∂F ∂F d2 , Fu = and Fv = . where Dx2 = dx2 ∂u ∂v For easy the convenience of calculation, we assume the characteristic of evolutionary vector field has the entries Q1 = a1 ux + a2 vx + b1 ut + b2 vt + c1 u + c2 v + m, Q2 = e1 ux + e2 vx + f1 ut + f2 vt + g1 u + g2 v + n, where ai , bi , ci , ei , fi , gi (i = 1, 2) are undetermined functions of x, t, u, v, and m, n are undetermined functions of x, t. Thus Q1t = (a1t + a1u ut + a1v vt )ux + a1 uxt + (a2t + a2u ut + a2v vt )vx + a2 vxt +(b1t + b1u ut + b1v vt )ut + b1 utt + (b2t + b2u ut + b2v vt )vt + b2 vtt +(c1t + c1u ut + c1v vt )u + c1 ut + (c2t + c2u ut + c2v vt )v + c2 vt + mt , 0 P1u Q1

= (a1xx + 2a1xu ux + 2a1xv vx + 2a1uv ux vx + a1uu u2x + a1vv vx2 +a1u uxx + a1v vxx )ux + 2(a1x + a1u ux + a1v vx )uxx + a1 uxxx +(a2xx + 2a2xu ux + 2a2xv vx + 2a2uv ux vx + a2uu u2x + a2vv vx2 5

+a2u uxx + a2v vxx )vx + 2(a2x + a2u ux + a2v vx )vxx + a2 vxxx +(b1xx + 2b1xu ux + 2b1xv vx + 2b1uv ux vx + b1uu u2x + b1vv vx2 +b1u uxx + b1v vxx )ut + 2(b1x + b1u ux + b1v vx )uxt + b1 uxxt +(b2xx + 2b2xu ux + 2b2xv vx + 2b2uv ux vx + b2uu u2x + b2vv vx2 +b2u uxx + b2v vxx )vt + 2(b2x + b2u ux + b2v vx )vxt + b2 vxxt +(c1xx + 2c1xu ux + 2c1xv vx + 2c1uv ux vx + c1uu u2x + c1vv vx2 +c1u uxx + c1v vxx )u + 2(c1x + c1u ux + c1v vx )ux + c1 uxx +(c2xx + 2c2xu ux + 2c2xv vx + 2c2uv ux vx + c2uu u2x + c2vv vx2 +c2u uxx + c2v vxx )v + 2(c2x + c2u ux + c2v vx )vx + c2 vxx +mxx + Fu (a1 ux + a2 vx + b1 ut + b2 vt + c1 u + c2 v + m), 0 P1v Q2

= Fv (e1 ux + e2 vx + f1 ut + f2 vt + g1 u + g2 v + n).

Substitution (3.1) into above three formulas, replacing ut , vt by uxx +F , τ1 (u−v) respectively, whenever they occur, then a1v (u − v)]ux + a1 (uxxx + Fx + Fu ux + Fv vx ) τ a2 a2v (u − v)]vx + (ux − vx ) +[a2t + a2u (uxx + F ) + τ τ b1v +[b1t + b1u (uxx + F ) + (u − v)](uxx + F ) + b1 [uxxxx + Fuu u2x τ +Fu uxx + Fvv vx2 + Fv vxx + Fxx + 2Fxu ux + 2Fxv vx + 2Fuv ux vx Fv (u − v)] +Ft + Fu (uxx + F ) + τ 1 b2 b2v 1 + [b2t + b2u (uxx + F ) + (u − v)](u − v) + [uxx + F − (u − v)] τ τ τ τ c1v +[c1t + c1u (uxx + F ) + (u − v)]u + c1 (uxx + F ) τ c2 c2v (u − v)]v + (u − v) + mt , +[c2t + c2u (uxx + F ) + τ τ = (a1xx + 2a1xu ux + 2a1xv vx + 2a1uv ux vx + a1uu u2x + a1vv vx2

Q1t = [a1t + a1u (uxx + F ) +

0 Q1 P1u

+a1u uxx + a1v vxx )ux + 2(a1x + a1u ux + a1v vx )uxx + a1 uxxx +(a2xx + 2a2xu ux + 2a2xv vx + 2a2uv ux vx + a2uu u2x + a2vv vx2 +a2u uxx + a2v vxx )vx + 2(a2x + a2u ux + a2v vx )vxx + a2 vxxx +(b1xx + 2b1xu ux + 2b1xv vx + 2b1uv ux vx + b1uu u2x + b1vv vx2 +b1u uxx + b1v vxx )(uxx + F ) + 2(b1x + b1u ux + b1v vx )(uxxx + Fu ux +Fv vx + Fx ) + b1 (uxxxx + Fuu u2x + Fu uxx + Fvv vx2 + Fv vxx 1 +Fxx + 2Fxu ux + 2Fxv vx + 2Fuv ux vx ) + (b2xx + 2b2xu ux + 2b2xv vx τ +2b2uv ux vx + b2uu u2x + b2vv vx2 + b2u uxx + b2v vxx )(u − v) 2 b2 + (b2x + b2u ux + b2v vx )(ux − vx ) + (uxx − vxx ) τ τ +(c1xx + 2c1xu ux + 2c1xv vx + 2c1uv ux vx + c1uu u2x + c1vv vx2 +c1u uxx + c1v vxx )u + 2(c1x + c1u ux + c1v vx )ux + c1 uxx 6

+(c2xx + 2c2xu ux + 2c2xv vx + 2c2uv ux vx + c2uu u2x + c2vv vx2

0 Q2 P1v

+c2u uxx + c2v vxx )v + 2(c2x + c2u ux + c2v vx )vx + c2 vxx + mxx 1 +Fu [a1 ux + a2 vx + b1 (uxx + F ) + b2 (u − v) + c1 u + c2 v + m], τ 1 = Fv [e1 ux + e2 vx + f1 (uxx + F ) + f2 (u − v) + g1 u + g2 v + n]. τ

Similarly, we can get e1v (u − v)]ux + e1 (uxxx + Fu ux + Fv vx + Fx ) τ e2v e2 +[e2t + e2u (uxx + F ) + (u − v)]vx + (ux − vx ) τ τ f1v (u − v)](uxx + F ) +[f1t + f1u (uxx + F ) + τ +f1 [uxxxx + Fuu u2x + Fu uxx + Fvv vx2 + Fv vxx + Fxx Fv +2Fxu ux + 2Fxv vx + 2Fuv ux vx + Ft + Fu (uxx + F ) + (u − v)] τ 1 f2v f2 1 + [f2t + f2u (uxx + F ) + (u − v)](u − v) + [uxx + F − (u − v)] τ τ τ τ g1v (u − v)]u + g1 (uxx + F ) +[g1t + g1u (uxx + F ) + τ g2v g2 +[g2t + g2u (uxx + F ) + (u − v)]v + (u − v) + nt , τ τ 1 1 = [a1 ux + a2 vx + b1 (uxx + F ) + b2 (u − v) + c1 u + c2 v + m], τ τ 1 1 = − [e1 ux + e2 vx + f1 (uxx + F ) + f2 (u − v) + g1 u + g2 v + n]. τ τ

Q2t = [e1t + e1u (uxx + F ) +

0 P2u Q1 0 P2v Q2

By (2.6), we have 0 0 Q1t = P1u Q1 + P1v Q2 .

Thus, equating the coefficients of the various monomials in third, second and first order partial derivatives of u and v, the coefficient functions of characteristic of evolutionary vector field of equation (3.1) satisfy the following partial differential equations: Monomial

Coefficient

uxxx uxxx ux uxxx vx vxxx uxx ux uxx vx uxx vxx vxx u vx ux u2x

0 = b1x 0 = b1u 0 = b1v 0 = a2 0 = a1u 0 = a1v b1t = 2a1x + f1 Fv 0 = − τ1 b2 + c2 0 = τ1 b2v + c1v a1 Fv = e2 Fv a1t = a1xx + τ2 b2x + 2c1x + Fv e1 0 = τ1 b2u + c1u 7

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

v u 1

1 1 τ b1 Fv = ( τ f2 − g2 )Fv 1 1 1 1 τ (b1 Fv + b2t ) + c1t = τ b2xx + c1xx + Fu ( τ b2 + c1 ) + Fv ( τ f2 + g1 ) a1 Fx + b1t F + b1 Ft + ( τ1 b2 + c1 )F + mt = mxx + mFu + f1 Fv F +

nFv

(13) (14) (15)

Following 0 0 Q2t = P2u Q1 + P2v Q2 ,

we have other partial differential equations Monomial

Coefficient

uxxxx uxxx uxx v u 1

0 = f1 0 = e1 1 1 τ f2 + g1 = τ b1 f2t − τ g2t = τ1 b2 − c2 f2t + τ g1t + g2 = τ1 b2 + c1 − g1 ( τ1 f2 + g1 )F + nt = τ1 (b1 F + m + n)

(16) (17) (18) (19) (20) (21)

From the above two tables, we have a1 = µ,

a2 = 0,

b1 = β,

b2 = τ α,

c1 = −α,

e1 = 0,

e2 = µ,

f1 = 0,

f2 = β + τ θ,

c2 = α;

g1 = −θ,

g2 = θ;

where µ, β are arbitrary constants, α, θ arbitrary functions in x, t, u, v. And n, m satisfy   mt = mxx + mFu + nFv − µFx − βFt ,  nt = 1 (n − m). τ Thus, we obtain

Q1 Q2

!

=

µux + βut + τ αvt − αu + αv + m µvx + (β + τ θ)vt − θu + θv + n

!

,

that is (3.2). By Theorem 3.1 and Definition 2.3, we have Theorem 3.2. The generalized vector field, which characteristic (3.2) correspond, is a symmetry of system (3.1) and has the following form v=µ

∂ ∂ ∂ ∂ + β + [α(τ vt − u + v) + m] + [θ(τ vt − u + v) + n] , ∂x ∂t ∂u ∂v

where µ, β are arbitrary constants, α, θ arbitrary functions in x, t, u, v. And n, m satisfy   mt = mxx + mFu + nFv − µFx − βFt ,  nt = 1 (n − m). τ 8

(3.4)

4

Some group-invariant solutions

In the process of solving the equations, only those relatively simple symmetry is useful. Therefore, we will only calculate some of group-invariant solutions, which, by Lemma 2.2 and Theorem 3.1 in this section, are corresponding to simple evolutionary symmetry vQ of (1.1). The group-invariant solutions are found by solving a reduced system of differential equations involving fewer independent variables than the original one. Let the coefficient function θ of (3.2) be identically zero, since we mainly calculate the group-invariant solutions of (1.1). Case 1. Let µ = 1, β = α = θ = m = n = 0, then the characteristic (3.2) has the form ! ! Q1 ux = . Q2 vx From (3.3), we find F satisfies Fx = 0, that is, F = F (t, u, v).

(4.1)

Actually, Q[u] = 0 is equivalent to ux = 0,

vx = 0,

that is, u = g(t), v = h(t). Substituting these solutions into system (3.1), we reduce it to be the following ordinary differential equations (ODEs):  0  g = F (t, g, h) (4.2)  h0 = 1 (g − h). τ

Assume (g0 (t), h0 (t)) is a solution of (4.2), then u = g0 (t) is an x-translation invariant solution of (1.1) with (4.1), and corresponds to the symmetry generator v = −∂x by (2.1) and (2.2). Example 1. If F (t, k, k) = 0 (4.3)

for some constant k, then u = k is a x-translation invariant solution of (1.1), where reaction term satisfies (4.1) and (4.3). Example 2. If F (t, u, v) = A1 (t)u + A2 (t)v + A3 (t), (4.4) where Ai (t) are continuous functions, i = 1, 2, 3. System (4.2) has non-zero solution (g0 (t), h0 (t)) corresponding to any non-zero initial condition. Thus, u = g0 (t) is an x-translation invariant solution of (1.1), where reaction term satisfies (4.4). Case 2. Let µ = 1, β = α = θ = 0 and m = n = k1 x + k2 , where k1 , k2 are constants and satisfy k12 + k22 6= 0, then the characteristic becomes ! ! Q1 ux + k1 x + k2 . = Q2 vx + k1 x + k2

9

By (3.3), we obtain that F should satisfies (k1 x + k2 )Fu + (k1 x + k2 )Fv − Fx = 0, that is, F = Ψ1 (t, u − v, u +

k1 2 x + k2 x), 2

(4.5)

where Ψ1 (t, ξ1 , ξ2 ) is a continuously differentiable function in ξ1 , ξ2 . In fact, Q[u] = 0 is equivalent to ux + k1 x + k2 = 0, vx + k1 x + k2 = 0, that is, u = g(t) − k21 x2 − k2 x, v = h(t) − k21 x2 − k2 x. Substituting this solution into system (3.1), we get the following ODEs  0  g = Ψ1 (t, g − h, g) − k1 (4.6)  h0 = 1 (g − h). τ

If (g1 (t), h1 (t)) is a solution of (4.6), then u = g1 (t) − k21 x2 − k2 x is a group-invariant solution of (1.1) with (4.5), and corresponds to the symmetry generator v = −∂x + (k1 x + k2 )∂u by (2.1) and (2.2). Example 3. Assume F = A4 (t)(u − v) + A5 (t), (4.7)

where Ai (t) are continuous functions, i = 4, 5. System (4.6) has non-zero solution (g1 (t), h1 (t)) for any given non-zero initial condition. Therefore, u = g1 (t)− k21 x2 −k2 x is an invariant solution of (1.1), with the reaction term satisfying (4.7). Case 3. Let β = 1, µ = α = θ = m = n = 0, the characteristic changes to ! ! ut Q1 . = vt Q2 From (3.3), F satisfies Ft = 0, that is, F = F (x, u, v).

(4.8)

In fact, Q[u] = 0 is equivalent to ut = 0,

vt = 0,

that is, u = g(x), v = h(x). Substituting this solution into system (3.1) one has the following   0 = g 00 + F (x, g, h) (4.9)  0 = 1 (g − h). τ In this case, (4.9) satisfies the second-order ODE

g 00 = −F (x, g, g).

(4.10)

If u = g2 (x) is a solution of ordinary equation (4.10), then (u, v) = (g2 (x), g2 (x)) is a t-translation invariant solution of (3.1). Therefore, u = g2 (x) is a t-translation invariant solution of (1.1) with the reaction term satisfying (4.8), and corresponds to the symmetry v = −∂t by (2.1) and (2.2). 10

Case 4. Let β = 1, µ = α = θ = 0 and m = n = k, where k 6= 0 is a constant, the characteristic turns to ! ! Q1 ut + k = . vt + k Q2 By (3.3), F satisfies kFu + kFv − Ft = 0, that is, F = Ψ2 (x, u − v, u + kt).

(4.11)

where Ψ2 (x, ξ1 , ξ2 ) is a continuously differentiable function in ξ1 , ξ2 . Actually, Q[u] = 0 is equivalent to ut + k = 0, vt + k = 0, that is, u = g(x) − kt, v = h(x) − kt. Substituting these solutions into system (3.1), we have  00  g + Ψ2 (x, g − h, g) + k = 0 (4.12)  1 (g − h) + k = 0. τ In this case, (4.12) satisfies the second-order ODE

g 00 + k + Ψ2 (x, −kτ, g) = 0.

(4.13)

If g3 (x) is a solution of (4.13), then u = g3 (x) − kt is a group-invariant solution of (1.1) with the reaction term satisfying (4.11), and corresponds to the symmetry v = −∂t + k∂u by (2.1) and (2.2). Case 5. Let µ = c, β = 1 and α = θ = m = n = 0, where c 6= 0 is a constant, then the characteristic reduces to ! ! cux + ut Q1 . = cvx + vt Q2 From (3.3), F satisfies cFx + Ft = 0, that is, F = F (x − ct, u, v).

(4.14)

where F (ξ, u, v) is a continuously differentiable function in ξ. In fact, Q[u] = 0 is equivalent to cux + ut = 0,

cvx + vt = 0,

that is, u = g(ξ), v = h(ξ), ξ = x − ct. Substituting this solution into system (3.1), we yield the following ODEs  00  g + cg 0 + F (ξ, g, h) = 0 (4.15)  ch0 + 1 (g − h) = 0. τ The solution u = g4 (x + ct), v = h4 (x + ct) of (4.15) is a traveling wave solution of (3.1). At the beginning of this paper, we have introduced lots of papers studying the existence of traveling wave for (3.1). Therefore, u = g4 (x − ct) is a traveling wave solution of (1.1) with the reaction term satisfying (4.14), and corresponds to the symmetry v = −c∂x − ∂t by (2.1) and (2.2).

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Case 6. Let µ = c, β = 1, α = θ = 0 and m = n = k1 x + k2 , where c 6= 0, k1 , k2 are constants and satisfying k12 + k22 6= 0, then the characteristic becomes ! ! cux + ut + k1 x + k2 Q1 . = cvx + vt + k1 x + k2 Q2 By (3.3), F satisfies (k1 x + k2 )Fu + (k1 x + k2 )Fv − cFx − Ft = 0. Then F = Ψ3 (x − ct, u − v, cu +

k1 2 x + k2 x). 2

(4.16)

where Ψ3 (ξ1 , ξ2 , ξ3 ) is a continuously differentiable function in ξ1 , ξ2 , ξ3 . In fact Q[u] = 0 is equivalent to cux + ut + k1 x + k2 = 0, cvx + vt + k1 x + k2 = 0, which has a special kind of solutions u = g(x − ct) − 1c ( k21 x2 + k2 x), v = h(x − ct) − 1c ( k21 x2 + k2 x). Let ξ = x − ct and substitute this solution into system (3.1). Then we get ODEs  k   g 00 + cg 0 + Ψ3 (ξ, g − h, cg) − 1 = 0 c (4.17) 1   ch0 + (g − h) = 0. τ

where 0 denote differentiation with respect to ξ. If g = g5 (ξ), h = h5 (ξ) a the solution of (4.17), then u = g5 (x − ct) − 1c ( k21 x2 + k2 x) is a group-invariant solution of (1.1) where F satisfies (4.16), and corresponds to the symmetry v = −c∂x − ∂t + (k1 x + k2 )∂u by (2.1) and (2.2). Case 7. Let µ = α = 1 and β = θ = m = n = 0, then the characteristic becomes to ! ! Q1 ux + (τ vt − u + v) . = vx Q2 From (3.3), F satisfies (4.1). In fact Q[u] = 0 is equivalent to ux + (τ vt − u + v) = 0,

vx = 0,

which yields u = cex + τ ht (t) + h(t), v = h(t), where c is a constant to be chosen. Then system (3.1) turn to   τ htt + ht = cex + F (t, cex + τ ht + h, h) (4.18)  ht = 1 (cex + τ ht ). τ Thus c = 0. By (4.18), we have second-order linear ordinary differential equation τ htt + ht = F (t, τ ht + h, h).

(4.19)

If h(t) is a solution of the above equation, then u = τ ht + h is a group-invariant solution of (1.1) where F satisfies (4.1), and corresponds to the symmetry v = −∂x by (2.1) and (2.2). Example 4. Let F (t, u, v) satisfy (4.4). Then system (4.19) has non-zero solution h6 (t) for any non-zero initial condition, and u = τ h6t + h6 is a group-invariant solution of (1.1). Case 8. Let β = α = 1 and µ = θ = m = n = 0, then the characteristic reduces to ! ! ut + (τ vt − u + v) Q1 = . Q2 vt 12

From (3.3), F satisfies (4.8). In fact Q[u] = 0 is equivalent to ut + (τ vt − u + v) = 0,

vt = 0,

which gives u = cet + h(x), v = h(x), where c is a constant to be chosen. Then system (3.1) turns to  t  ce = hxx + F (x, cet + h, h) (4.20)  0 = 1 cet . τ Thus c = 0. By (4.20), u = v = h(t) satisfy second-order ODE (4.10). Therefore, we can’t find new group-invariant solution of (1.1) by this characteristic. Remark 4.1. If the characteristic has the form τ vt − u + v 0

!

,

(4.21)

by Lemma 2.2, Q[u] = 0 is equivalent to τ vt − u + v = 0. Then vt = τ1 (u − v). We can’t get a reduced ODE system. And similar to the discussion of Case 7 and Case 8, if the characteristic (3.2) contains(4.21), then there is no new group-invariant solution of (1.1).

5

Conclusion

The present paper deals with the reaction-diffusion equation with a distributed delay, mainly using the weak generic kernel. The characteristic of this equation is developed. And we calculate several group-invariant solutions, corresponding to evolutionary symmetry of this equation. For the strong generic delay case (1.4), the general approach is still applicable. Tanthanuch and Meleshko [13, 14] generalized the group analysis theory to delay differential equation, defining an admitted Lie group and determining equation for delay differential equation. Notice that this admitted Lie group can’t transform a solution into a solution in the definition. But in this paper, we mainly utilize the classical generalized symmetries of differential equation theory, therefore the solutions, corresponding to the symmetry, are group-invariant solutions.

6

Acknowledgements

This work is supported by Beijing Higher Education Young Elite Teacher Project, and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP), Grant No. 20120006120007.

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[2] J. Huang, X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Contin. Dyn. Syst., 9(2003), 925-936. [3] D. Liang, J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13(2003), 289-310. [4] Guo Lin, Wan-Tong Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244(2008), 487-513. [5] L.V. Linchuk, On group analysis of functional differential equations, In: Proceedings of the International Conference MOGRAN 2000, Modern Group Analysis for the New Millennium, USATU Publishers, Ufa, (2000), 111-115. [6] Shiwang Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation, J. Differential Equations, 237(2007), 259-277. [7] R.H. Martin, H.L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413(1991), 1-35. [8] S.V. Meleshko, S. Moyo, On the complete group classification of the reaction-diffusion equation with a delay, J. Math. Anal. Appl., 338(2008), 448-466. [9] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, (1993). [10] Chunhua Ou, Jianhong Wu, Persistence of wavefronts in delayed nonlocal reactiondiffusion equations, J. Differential Equations, 235(2007), 219-261. [11] P. Pue-on, S.V. Meleshko, Group classification of second-order delay ordinary differential equation, Commun. Nonlinear Sci. Numer. Simul., 15(2010), 1444-1453. [12] Shigui Ruan, Jianhong Wu, Reaction-diffusion equations with infinite delay, Can. Appl. Math. Q., 2(1994), 485-550. [13] J. Tanthanuch, S.V. Meleshko, On definition of an admitted Lie group for functional differential equations, Commun. Nonlinear Sci. Numer. Simul., 9 (2004), 117-125. [14] J. Tanthanuch, S.V. Meleshko, Application of group analysis to delay differential equations, Proceedings of ISNA-16, nonlinear acoustics at the beginning of the 21st century. Moscow: Moscow State University,(2002), 607-610. [15] J. Tanthanuch, Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4978-4987. [16] C. Tian, Lie Groups and its Applications to Differential Equations, Science Press, Beijing, (2001) (in Chinese). [17] Zhi-Cheng Wang, Wan-Tong Li, Shigui Ruan, Travelling wave fronts in reaction-diffusion systems with spatiotemporal delays, J. Differential Equations, 222(2006), 185-232. 14

[18] J.H. Wu, Theory and Applications of partial functional differential equations, Applied Mathematical Sciences, Wol.119, Springe-Werlag, Berlin, New York, (1996). [19] J. Wu, X. Zou, Travelling wave fronts of reaction-diffusion systems with delay, J. Dynamical Differential Equations, 13(2001), 651-687. [20] Zhihong Zhao, Erhua Rong, Reaction diffusion equation with spatio-temporal delay, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2252-2261. [21] X. Zou, Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comput. Appl. Math., 146 (2002), 309-321.

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Highlights 1. There have been few investigations of the symmetry and group invariant solutions of delay differential equations till now. This research gives at first the evolutionary infinitesimal vector field of reaction-diffusion equation with distributed delay by classical Lie group theory. 2. Tanthanuch and Meleshko [2002, 2004] generalized the group analysis theory to delay differential equation, defining an admitted Lie group and determining equation for delay differential equation. Notice that this admitted Lie group can't transform a solution into a solution in the definition. But in this paper, we mainly utilize the classical generalized symmetries of differential equation theory, therefore the solutions, corresponding to the symmetry, are group-invariant solutions.