Conservation Laws and Exact Solutions for a Reaction-Diffusion Equation with a Variable Coefficient

and Applied Analysis 3 Definition 3. Equation (3) is said to be self-adjoint if the equation obtained from the adjoint equation (4) by the substitutions V = u F ∗ (x, u, V, u (1) , V (1) , . . . , u (s) , V (s) ) = 0 (13) is identical with the original equation (3), in other words, if F ∗ (x, u, V, u (1) , V (1) , . . . , u (s) , V (s) ) 󵄨󵄨󵄨󵄨V=u = φ (x, u, u (1) , . . .) F (x, u, u (1) , . . . , u (s) ) . (14) Definition 4. Equation (3) is said to be quasi-self-adjoint if the equation obtained from the adjoint equation (4) by the substitutions V = φ(u) with a certain function φ(u) such that φ 󸀠 (u) ̸ = 0 F ∗ (x, u, V, u (1) , V (1) , . . . , u (s) , V (s) ) = 0 (15) is identical with the original equation (3). Definition 5. Equation (3) is said to be weak self-adjoint if the equation obtained from the adjoint equation (4) by the substitutions V = φ(x, u) with a certain function φ(x, u) such that φ u ̸ = 0 and φ x ̸ = 0 F ∗ (x, u, V, u (1) , V (1) , . . . , u (s) , V (s) ) = 0 (16) is identical with the original equation (3). Definition 6. Equation (3) is nonlinearly self-adjoint if there exist functions V = φ(x, u) that solve the adjoint equation (4) for all solutions u(x) of (3) and satisfy the conditionφ(x, u) ̸ = 0. Remark 7. Thefirst three definitions were suggested by Ibragimov in [3, 9, 24], respectively, the fourth was introduced by Gandarias in [25], and few time later it was generalized by Ibragimov in [13] in the form of Definition 6. So the nonlinear self-adjointness can be seen the most general concept; namely, the others are all its trivial cases. 2.2. A Theorem on the Order of Local Conservation Laws for a General Class of Second-Order Evolution Equations. The theorem on the order of local conservation laws for a more general class of second-order evolution equations, which covers (1), has been proved by Popovych in [23]. Theorem 8. Any local conservation law of any second-order (1+1)-dimensional quasilinear evolution equation, whose form is u t = S (t, x, u, u x ) u xx + R (t, x, u, u x ) , (17) where S(t, x, u, u x ) ̸ = 0, has the first order and, moreover, there exists its conserved vector with the densityT depending at most on t, x, and u and the fluxX depending at most on t, x, u, and u x . 3. Lie Point Symmetries and Self-Adjointness In this section, we present the most general Lie group of point transformations, which leaves (1) invariant, and study the self-adjointness of (1). First of all, we consider a one-parameter Lie group of infinitesimal transformation: x 󳨀→ x + εξ (x, t, u) , t 󳨀→ t + ετ (x, t, u) , u 󳨀→ u + εφ (x, t, u) , (18) with a small parameter ε ≪ 1.The vector field associatedwith the above group of transformations can be written as


Introduction
Conservation law is an important concept in physics.It describes a quantity that is conserved; that is, the total amount is the same before and after something occurs.In mathematics, conservation law provides one of the basic principles in formulating and investigating models.For instance, sometimes, the existence of a large amount of conservation laws of a partial differential equation (PDE) is a strong indication of its integrability.
For an Euler-Lagrange equation, which arises out of a variational principle, the remarkable Noether's theorem [1] constructed a one-to-one correspondence between a nontrivial generalized variational symmetry of some functionals and a nontrivial conservation law.By this correspondence, one can establish conservation laws for an Euler-Lagrange equation, as illustrated in [2], whereas for a system not arising from a variational principle, such as a single evolution equation, researchers have made various generalizations [3][4][5][6][7][8][9][10][11][12][13] of Noether's method to construct conservation laws.
Among these generalizations, after suggesting some concepts, such as adjoint equation, strict self-adjointness, quasiself-adjointness, and nonlinear self-adjointness, Ibragimov [3] derived a convenient formula to establish conservation laws for the simultaneous system of the target equation together with its adjoint equation.By this formula, one can construct a conservation law for the combined system through a formal Lagrangian L corresponding to any Lie point, Lie-B ä cklund, or nonlocal symmetry.But L involves a "nonphysical" variable V.If the equation considered is (nonlinearly) self-adjoint, one can eliminate V via a certain substitution to obtain the conservation law of the original equation.However, if the equation does not have any selfadjoint property, the conservation laws for it corresponding to each symmetry can be considered as local conservation laws of the simultaneous system but reflect the symmetry property of the original equation.
Following Ibragimov, many researchers have been studying this interesting area and there are a lot of works in the literature, such as [14][15][16][17][18][19] and the references therein.
In this paper, we consider a variable-coefficient reactiondiffusion equation [20] where  = (, ) represents the population density,  ( ̸ = 0) is a constant of the diffusion rate, and the variable-coefficient () describes the growth rate of the living being.As a diffusive logistic equation, (1) describes the dynamics of a population inhabiting a strongly hetergeneous environment.The growth rate () is positive on favourable habitats and negative on unfavorable ones.In [20], Cantrell and Cosner determined how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled.
In [22], Ide and Okada designed numerical schemes that preserve energy property and showed numerical experiments for (1).
We start our work with solving the system of determining equations to obtain the infinitesimal generator and meantime classifying (1) into three kinds.To determine the form of the conservation law corresponding to any symmetry of every kind, we then study the self-adjointness of (1).By the theorem [23] on the order of conservation laws for a general class of second-order evolution equations, which covers (1), we illustrate that the local conservation law of (1) does not exist.With the aid of Ibragimov's formula, we construct conservation laws corresponding to every symmetry of each kind.Finally, using the symmetry group obtained and another method, we establish some exact solutions of the last two kinds of (1).
The contents of this paper are as follows.In Section 2, we will give some preliminaries, namely, some definitions, Ibragimov's theorem, and a theorem on the order of local conservation laws for a general class of second-order evolution equations which covers (1).Next we subject (1) to the effective computational procedure [2] for finding the Lie point symmetries and study its self-adjointness in Section 3. Section 4 is devoted to discussing the conservation laws of (1).Some exact solutions of (1) are obtained in Section 5. Finally, concluding remarks are given in Section 6.

Preliminaries
We assume that all functions are smooth, and the summation over the repeated indices is understood.

Ibragimov's Theory.
In this subsection, we recall Ibragimov's procedure of constructing conservation laws corresponding to the given symmetries of any system of PDEs, provided that the number of equations in the system is equal to the number of dependent variables.
For convenience, we consider a scalar evolution equation with independent variables  = ( with where V = V( 1 ,  2 ) is a multiplier and denotes the variational derivatives (the Euler-Lagrange operator) and are the total differentiations.We now extend (3) to a system In [3], Ibragimov proved that (8) inherits all symmetries of (3) and, using Noether's identity [13], obtained a formula of conservation law corresponding to every symmetry of (3).
of (3) provides a conservation law   (  ) = 0 for (8).The conserved vector is given by where  and L are defined as follows: For a second-order equation, (10) becomes According to Theorem 2, we know that every symmetry of (3) can provide a conservation law.To express these conservation laws one depends on not only the original variables but the multiplier V.
Sometimes the multiplier V can be removed from the conservation law provided that (3) has a property of selfadjointness.
Definition 3. Equation ( 3) is said to be self-adjoint if the equation obtained from the adjoint equation ( 4) by the substitutions is identical with the original equation ( 3), in other words, if , . ..)  (, ,  (1) , . . .,  () ) . ( Definition 4. Equation ( 3) is said to be quasi-self-adjoint if the equation obtained from the adjoint equation ( 4) by the substitutions V = () with a certain function () such that is identical with the original equation (3).
Definition 5. Equation ( 3) is said to be weak self-adjoint if the equation obtained from the adjoint equation ( 4) by the substitutions V = (, ) with a certain function (, ) such that   ̸ = 0 and is identical with the original equation ( 3).Definition 6. Equation ( 3) is nonlinearly self-adjoint if there exist functions V = (, ) that solve the adjoint equation ( 4) for all solutions () of ( 3) and satisfy the condition (, ) ̸ = 0.
Remark 7. The first three definitions were suggested by Ibragimov in [3,9,24], respectively, the fourth was introduced by Gandarias in [25], and few time later it was generalized by Ibragimov in [13] in the form of Definition 6.So the nonlinear self-adjointness can be seen the most general concept; namely, the others are all its trivial cases.

A Theorem on the Order of Local Conservation Laws for a General Class of Second-Order Evolution Equations.
The theorem on the order of local conservation laws for a more general class of second-order evolution equations, which covers (1), has been proved by Popovych in [23].

Theorem 8. Any local conservation law of any second-order (1+1)-dimensional quasilinear evolution equation, whose form is
where (, , ,   ) ̸ = 0, has the first order and, moreover, there exists its conserved vector with the density  depending at most on , , and  and the flux  depending at most on , , , and   .

Lie Point Symmetries and Self-Adjointness
In this section, we present the most general Lie group of point transformations, which leaves (1) invariant, and study the self-adjointness of (1).
For Case 1 we have  = −  and ( 12) yields the conservation law with This vector involves an arbitrary solution V of the adjoint equation ( 53) and hence provides an infinite number of conservation laws.With the aid of Mathematica, we find For Case 2, we find the conservation law provided by  2 .We have  = −−(1/2)  −  and ( 12) yields the nontrivial conservation law with This vector involves an arbitrary solution V of the adjoint equation ( 53) and hence provides an infinite number of conservation laws.With the aid of Mathematica, we find For Case 3, we find the conservation law provided by the symmetry  3 .We have  = −  and (12) yields the nontrivial conservation law with This vector involves an arbitrary solution V of the adjoint equation ( 53) and hence provides an infinite number of conservation laws.With the aid of Mathematica, we find According to the definition of conservation law, we say that we have obtained local conservation laws for the simultaneous system of (1) and its adjoint equation (53); however, they are nonlocal for the single equation (1).Then we will illustrate that there is no local conservation law for (1).
In view of Theorem 8, for (1) we have the density  = (, , ) and the flux  = (, , ,   ).We substitute the expression of   deduced from (1) into and we obtain The coefficient of   gives therefore Taking into account (69) and splitting the rest of (67) with respect to the powers of   , we obtain the system of PDEs on the functions (, , ) and X(, , ) It is clear that the only solution of this system is (, ) = 0.It means that (1) does not have any local conservation law.

Some Exact Solutions
In this section, we will construct scale-invariant solution for (48) and traveling wave solutions for (51).
5.1.Scale-Invariant Solution of (48).For (48), we consider which corresponds to the scaling group On the half space {(, , ),  > 0}, global invariants of this one-parameter group are provided by the functions then Substitutions into (48) yield As guaranteed by the general theory, this equation is equivalent to one in which the parametric variable  does not occur; namely, which forms the reduced equation for the scale-invariant solutions.
If we get a solution of this equation, we can construct a scale-invariant solution of (48).
When we used the odeadvisor command of Maple for classifying (84) according to standard text books, it returns to [NONE].The usual substitution reduces (84) to the first-order equation For (86) the odeadvisor command returns to [rational, [Abel, 2nd type, class A]]; that is, (86) is a class A of second kind of Abel's equation, whose general form is where  2 (),  1 () are arbitrary functions.According to [26], there is as yet no general solution for this ordinary differential equation.
so that a group-invariant solution V = ℎ() takes the familiar form  = ℎ( − ) determining a wave of unchanging profile moving at the constant velocity .Solving for the derivatives of  with respect to  and  in terms of those of V with respect to  we find Substituting these expressions into (51), we find the reduced ordinary differential equation for the traveling wave solution to be This is a nonlinear, constant coefficient equation.If we get a solution of this equation, we can construct a traveling wave solution of (51).This is clearly invariant that under the group of translations in the -direction (, V)  → ( + , V) , When / 2 = ±6/25, according to [28], one can get the solution of (103) by the elliptic Weierstrase function.That is, if / 2 = 6/25, the solution of (103) in the parametric form is where  = −(125 2 /6 3 ),  = ∫(/ ± √4 3 − 1) −  2 ,  is the classical elliptic Weierstrase function  = ( +  2 , 0, 1), and  = √±(4 3 − 1) + 2, where  2 is a constant; if / 2 = −6/25, the solution in the parametric form is with  2 =  2  ∓ 1.

Concluding Remarks
Some exact solutions and a discussion on local conservation laws for the variable-coefficient reaction-diffusion equation were presented.Firstly, by solving the determining equations in Lie symmetry analysis, we classified the equation into three kinds.For every kind, we obtained the Lie symmetries.
Corresponding to these Lie point symmetries, we have expressed the conservation laws, respectively.For (48), we studied its scale-invariant solution by global invariants of one-parameter group  2 .Each solution of (84) leads to a scale-invariant solution of (48).For (51), we have obtained some exact solutions.
+   , in which  is a fixed constant and determines the speed of the waves.Global invariants of this group are