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Conservation Laws and Exact Solutions for a Reaction-Diffusion Equation with a Variable Coefficient
In this paper a variable-coefficient reaction-diffusion equation is studied. We classify the equation into three kinds by different restraints imposed on the variable coefficient in the process of solving the determining equations of Lie groups. Then, for each kind, the conservation laws corresponding to the symmetries obtained are considered. Finally, some exact solutions are constructed.
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  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 , whereas for a system not arising from a variational principle, such as a single evolution equation, researchers have made various generalizations [3–13] of Noether’s method to construct conservation laws.
Among these generalizations, after suggesting some concepts, such as adjoint equation, strict self-adjointness, quasi-self-adjointness, and nonlinear self-adjointness, Ibragimov  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 corresponding to any Lie point, Lie-Bcklund, or nonlocal symmetry. But involves a “nonphysical” variable . If the equation considered is (nonlinearly) self-adjoint, one can eliminate via a certain substitution to obtain the conservation law of the original equation. However, if the equation does not have any self-adjoint 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.
In this paper, we consider a variable-coefficient reaction-diffusion equation  where represents the population density, 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 , Cantrell and Cosner determined how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled.
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  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  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.
We assume that all functions are smooth, and the summation over the repeated indices is understood.
2.1. 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 (here denote .) and a dependent variable , where denote the sets of the partial derivatives of the first, second, and so forth, orders, ,.
Definition 1. The adjoint equation to (3) is
where 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 , Ibragimov proved that (8) inherits all symmetries of (3) and, using Noether’s identity , obtained a formula of conservation law corresponding to every symmetry of (3).
Theorem 2. Any Lie point, Lie-Bäcklund, or nonlocal symmetry of (3) provides a conservation law for (8). The conserved vector is given by where and are defined as follows: For a second-order equation, (10) becomes
Sometimes the multiplier can be removed from the conservation law provided that (3) has a property of self-adjointness.
Definition 4. Equation (3) is said to be quasi-self-adjoint if the equation obtained from the adjoint equation (4) by the substitutions 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 with a certain function such that and is identical with the original equation (3).
Remark 7. The first three definitions were suggested by Ibragimov in [3, 9, 24], respectively, the fourth was introduced by Gandarias in , and few time later it was generalized by Ibragimov in  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
Theorem 8. Any local conservation law of any second-order -dimensional quasilinear evolution equation, whose form is where , 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 .
3. Lie Point Symmetries and Self-Adjointness
First of all, we consider a one-parameter Lie group of infinitesimal transformation: with a small parameter . The vector field associated with the above group of transformations can be written as
Applying the second prolongation to (1), we find that the coefficient functions , , and must satisfy the symmetry condition where , are the coefficients in and where , are the total derivatives with respect to and , respectively.
Substituting , into the symmetry condition, replacing by whenever it occurs, and equating the coefficients of the various monomials in the first- and second-order partial derivatives of , we obtain the following system of determining equations:
By substituting , , and obtained in (28) and (29), respectively, we obtain As is uncertain, we split (34) into two parts: the part contains and and the remainder; that is, We rearrange (35) as Then it splits into Equation (36) splits into We obtain a simultaneous system Equation (33) and Equations (38)–(41). Equation (39) yields ; then . Equation (41) yields . Then, by (40), , that is, is a constant. So (33) reduces to the form whence , , and hence where are arbitrary constants. Ultimately, we arrive at and the general solution of the system of determining equations: And, invoking (35), the restriction on the variable-coefficient is
According to , we can classify (1) into the following three kinds.
Case 2. When ,, the restriction on is ; namely, , where is an integral constant, and the infinitesimal generator is where ,, denote ,,, respectively. So we conclude that the partial differential equation has the infinitesimal generators ,. The commutation table of the Lie algebra generated by the infinite symmetries is
Case 3. When ,, the restriction on is ; namely, , and the infinitesimal generator is So we conclude that the partial differential equation has the infinitesimal generators ,. The commutation table of the Lie algebra generated by the infinite symmetries is
Now we study the self-adjoint property of (1).
Setting , we study where The reckoning shows that Equation (59) should be satisfied identically in , , and . Therefore we nullify the coefficient of and obtain . According to Definition 6, we claim that for , (1) is not nonlinearly self-adjoint.
Hence, for , (1) is neither quasi-self-adjoint nor nonlinearly self-adjoint.
4. Conservation Laws
For Case 1 we have and (12) yields the conservation law with This vector involves an arbitrary solution 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 . We have and (12) yields the nontrivial conservation law with This vector involves an arbitrary solution 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 . We have and (12) yields the nontrivial conservation law with This vector involves an arbitrary solution 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 For , according to (71), we can assume then so we assume Now Substitution of and obtained into (72) yields We can set as they only contribute to the trivial part of the conservation law. Taking into account the remainder of the later expression and splitting it with respect to the powers of , we obtain the system of PDEs on It is clear that the only solution of this system is . It means that (1) does not have any local conservation law.
5. Some Exact Solutions
5.1. Scale-Invariant Solution of (48)
For (48), we consider which corresponds to the scaling group On the half space , 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 . The usual substitution reduces (84) to the first-order equation For (86) the odeadvisor command returns to ; that is, (86) is a class A of second kind of Abel’s equation, whose general form is where , are arbitrary functions. According to , there is as yet no general solution for this ordinary differential equation.
5.2. Traveling Wave Solutions of (51)
Remark 9. The solutions (89) have been verified by the pdetest command of Maple.
If , for (51), we consider the translation group generated by , in which is a fixed constant and determines the speed of the waves. Global invariants of this group are so that a group-invariant solution takes the familiar form 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 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 with infinitional generator . We can construct canonical variables by solving the equations and we obtain Then so (93) becomes which is a first-order equation for : that is Setting , we obtain With the aid of the substitution , (102) becomes When , according to , one can get the solution of (103) by the elliptic Weierstrase function. That is, if , the solution of (103) in the parametric form is where , is the classical elliptic Weierstrase function , and , where is a constant; if , the solution in the parametric form is with .
6. 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 . Each solution of (84) leads to a scale-invariant solution of (48). For (51), we have obtained some exact solutions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the referee for his/her valuable comments and suggestions which have led to an improvement of the presentation. This research is supported by the National Natural Science Foundation of China (no. 11061016).
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Copyright © 2014 Zhijie Cao and Yiping Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.