PAINLEVE ANALYSIS OF A CLASS OF NONLINEAR DIFFUSION EQUATIONS

We study the Painleve analysis for a class of nonlinear diffusion equations. We find that in some cases it has only the conditional Painleve property and in other cases, just the Painleve property. We also obtained special solutions.


Introduction
In recent years, much attention has been focused on higher order nonlinear partial differential equations, known as evolution equations.Such nonlinear equations often occur in the description of chemical and biological phenomena.Their analytical study has been drawing immense interest.A fundamental question when dealing with nonlinear differential equations is "how can one tell beforehand whether or not they are integrable?"Originally, Ablowitz et al [1] conjectur- ed that a nonlinear partial differential equation is integrable if all its exact reductions to ordinary differential equations have the Painleve property: that is, to have no movable singularities other than poles.This approach poses an obvious operational difficulty in finding all exact reductions.This difficulty was circumvented by Weiss et al [10] by postulating that a partial differential equa- tion has the Painleve property if its solutions are single-valued about a movable singular manifold (z, z2,..., Zn) 0, (1.1) where is an arbitrary function.In other words, a solution u(zi) of a partial differential equation should have a Laurent-like expansion about the movable singular manifold 0: where a is a negative integer.The number of arbitrary functions in expansion (1.2) should be equal to the order of the partial differential equation.Inserting expansion (1.2) into the targeted equation yields a recurrence formula that determines Un(Zi) for all n > 0, except for a finite number of rl, r2, r3,..., rj > 0, called resonances.For some equations, the recurrence formulas at the resonance values may result in constraint equations for the movable singular manifold which implies that it is no longer completely arbitrary.In such cases, one can say that the equation has 78 P. CHANDRASEKARAN and E.K. RAMASAMI the Conditional Painleve property [8].The Painleve property is a sufficient condition for the inte- grability or solvability of equations.Meanwhile, various authors have applied this approach to other nonlinear partial differential equations to decide whether or not these equations are integra- ble.Recent investigations of Cariello and Tabor [3] regarding the Painleve analysis also yield a systematic procedure for obtaining special solutions when an equation possesses only the condition- al Painleve property.Fisher [.4] proposed the nonlinear diffusion equation u Duxx + flu(1 u) (1.3) as a model for the propagation of a mutant gene with an advantageous selection of intensity /.
Roy Choudhury [8] has considered the extended form of equation (1.3) as u fluP(1 uq) + D (umu) (1.4) for Painleve analysis and obtained special solutions for various cases of p, q, and m.
In this paper we consider u uP(1 uq) + #uux + D(umu)z, (1.5) which is a generalization of (1.4) for the Painleve analysis.This equation has several interesting limiting cases which have already been studied: (i) when # rn 0, p 1, and q 0, equation (1.5) is reduced to the generalized Fisher equation.For q 1, equation (1.5) reduces to the Fisher equation and for q 2, (1.5) reduces to the Newell Whitehead equation.
Lemma: For all combinations of integer values of p,q, rn and s, the leading order singularity of equation (1.5) is (i) a movable pole for all combinations with (p + q-m-1) is equal to 1 or 2 for case (i), with (p + q-s-1) being equal to 1 for case (ii), and with s-m being equal to 1 for case (iii); (ii) a rational branch point for all combinations with (p+q-m-1)>2 for case (i), (p + q-s-1) > 1 for case (ii), and s-m > 1 for case (iii).
(2.13) Hence, we have the following theorem.
Here cr is an arbitrary function.Using (3.2) and (3.3) in (3.1) and collecting coefficients of equal powers of , we have: where r dr/dt, and (rtt d2r/dt 2. From (3.8), u 4 is an arbitrary function of this order if and only if the right-hand side of (3.8) is zero: that is, if and only if the singularity manifold is sub- ject to a certain constraint.Therefore, system (3.1)does not have the full Painleve property, but only the Conditional Painleve property.
(3.9)This is the generalized Burgers-Fisher equation with cubic nonlinearity.
3.6 Comparison of the results of equation (1.5) with those of (1.4) In studying the Painleve property for nonlinear differential equations, dominant terms of the equation determine the leading order terms, the Laurent expansion and the resonances.For the case p + q > m >_ s, the dominant terms of equation (1.5) turn out to be the same as those of (1.4).Therefore, in this case, the results of leading order balance for equation (1.5) agree with those of equation (1.4) and hence, we have the same structure of Laurent expansions and the resonances for equation (1.5) and equation (1.4).For (1.5), with the values of p,q,m and s, as above, we find the constraint equations (for example, (3.8)in case (i)and (3.14)in case (ii))for the movable singularity manifold corresponding to the last resonance value.This shows that the singularity manifold is not arbitrary as in the case of (1.4) in [8] for certain values of p, q, and m.Therefore, equation (1.5) for case (i) possesses the Conditional Painleve property for certain values of p, q, m and s.Other cases of (1.5), having either the Painleve property or the Condition- al Painleve property, are not applicable to (1.4).

Special solutions
We are able to find special solutions to some particular cases.