A note on the FKPP equation approached with the hyperbolic scaling
aa r X i v : . [ m a t h - ph ] D ec A note on the FKPP equation approached with the hyperbolicscaling
M.A. Reyes ∗ Departamento de F´ısica, DCI Campus Le´on, Universidad de Guanajuato,Apdo. Postal E143, 37150 Le´on, Gto., Mexico.
H. C. Rosu † IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica,Apdo Postal 3-74 Tangamanga, 78231 San Luis Potos´ı, S.L.P., Mexico. (Dated: November 8, 2018)
Abstract
We consider the hyperbolic scaling of the FKPP equation and introduce two solutions of theaction functional type in the limit of zero hyperbolic parameter. Furthermore, we show that theaction functional of the Ablowitz-Zepetella kink is a special case of one of those solutions.
PACS numbers: – Ordinary differential equations (ODE), partial differential equations (PDE), integrodifferentialmodels – Chemical kinetics and reactions: special regimes and techniques – Population dynamics and ecological pattern formation ∗ Electronic address: marco@fisica.ugto.mx † Electronic address: [email protected] u t = ∇ u for example, the parabolically-scaled solution u ( γx, γ x ) also solves the equationfor any positive γ parameter. In other words, the standard solution u ( x, t ) is the special case γ = 1 of the parabolically-scaled family of solutions describing the long-time large-distanceasymptotics of more microscopic transport phenomena. As pointed out by Fedotov [1],the propagating fronts of reaction diffusion equations should be approached with the more‘democratic’ hyperbolic scaling t → tǫ , x → xǫ ( ǫ ≪
1) if one wants to include their micro-scopic origin. Following Fedotov, we consider the Fisher-Kolmogorov-Petrovskii-Piskunovequation for a scalar field ρ ( x, t ) [1] dρdt = D d ρdx + U ρ (1 − ρ ) , (1)where D is the diffusion constant and U is the constant reaction rate. The logistic term de-scribes the spreading of the scalar field in an environment, originally applied to the spreadingof an advantageous gene in a population. Using the hyperbolically-rescaled field ρ (cid:18) xǫ , tǫ (cid:19) = ρ ǫ ( x, t ) , (2)one gets ǫ dρ ǫ dt = ǫ D d ρ ǫ dx + U ρ ǫ (1 − ρ ǫ ) . (3)We also notice that if in this scaled equation we substitute the imaginary scaling parameter ǫ = i ~ , we obtain formally a nonlinear cubic Schr¨odinger equation, where D is identifiedwith the inverse of the mass of a quantum entity, whereas the reaction parameter is now thestrength of the nonlinear interaction. However, this is not our concern in the following.If we make the change of dependent variable ρ ǫ = e − G ( x,t ) ǫ the following equation isobtained ∂G∂t = ǫD ∂ G∂x − D (cid:18) ∂G∂x (cid:19) − U (cid:16) − e − Gǫ (cid:17) . (4)In addition, in the limit ǫ → ∂G∂t + D (cid:18) ∂G∂x (cid:19) + U = 0 . (5)The latter equation is the Hamilton-Jacobi equation ∂G∂t + H (cid:0) ∂G∂x (cid:1) = 0 for the Hamiltonian H ( p ) = Dp + U , where G plays the role of the action functional. To solve this Hamilton-Jacobi equation we propose the ansatz G ( x, t ) = cx a t b − αt , which gives bcx a t b − − α + Dc a x a − t b + U = 0 . (6)2hus, α = U , a = 2, and b = −
1, which provide c = 1 / D and the solution: G ( x, t ) = x Dt − U t . (7)The reaction front can be found from the conditions G ( t, x ( t )) = 0 and x = vt , implyingthat the reaction velocity is v = √ DU . Solution (7) is also given by Fedotov but here weobtained it by other means.We here write down another solution, which comes out from the similarity with theharmonic oscillator solution of the HJ equation G osc = W ( x ) − βt : G ( x, t ) = r β − UD x − β q β − UD t . (8)This solution has not been given in the literature, although it is easily obtainable throughthe HJ route. Indeed, since ∂G∂x = dWdx , equation (5) reduces to the following form − β + D (cid:18) dWdx (cid:19) + U = 0 , (9)which immediately leads to the solution (8) if the initial condition x ( t = 0) = 0 is used forthe propagation front.Finally, a third form of the solution can be obtained if it is asked to be directly of travelingtype, i.e., G ( x, t ) = G ( z ), z = x − vt . Equation (5) takes the form D (cid:18) dG dz (cid:19) − v dG dz + U = 0 (10)and the solution will be G ( x, t ) = v ± √ v − DU D ( x − vt ) . (11)One can see that if v − DU = 0, then p = √ DU D , thus again the mass of the equivalentmicroscopic entity is m = D .We want also to discuss in this context the well-known Ablowitz-Zeppetella solution ofthe FKKP equation [2], following the results of a paper by Rosu and Cornejo-P´erez [3], whoobtained it through a factorization of equation (3). Writing the equation in the form ∂ρ ǫ ∂t = ǫD ∂ ρ ǫ ∂x + Uǫ ρ ǫ (1 − ρ ǫ ) , (12)3ntroducing the new variables ˜ t = Uǫ t and ˜ x = ǫ (cid:0) UD (cid:1) / x and passing to the travelingcoordinate ˜ z = ˜ x − ˜ v ˜ t , one gets the ordinary differential equation ρ ′′ + ˜ vρ ′ + ρ (1 − ρ ) = 0 , (13)where the prime denotes the derivative d/d ˜ z . According to [3], for ˜ v = √ equation (13)can be factorized in the way they proposed which leads to a first order differential equationeasy to integrate. The solution is precisely the Ablowitz-Zepetella solution, i.e., ρ (˜ z ) = 14 (cid:18) − tanh ˜ z √ (cid:19) . (14)In the asymptotic limit ˜ z → ∞ , we get ρ ∼ e − √ ˜ z . This leads to the following actionfunctional G AZ ( x, t ) = r U D (cid:16) x − √ DU t (cid:17) . (15)It is worth noting that G AZ is identical to G in the special case β = q U . The latter valueimplies a negative reaction rate, i.e., a rate of disappearance of the reactant.In conclusion, we obtained two novel solutions of traveling type of the Hamilton-Jacobiequation corresponding to the FKPP equation in the limit of zero hyperbolic parameter. Inaddition, we have shown that the action functional of the Ablowitz-Zepetella kink solutionof the FKPP equation corresponds to a special case of one of those solutions. [1] S. Fedotov, Front propagation into an unstable state of reaction-transport systems , Phys. Rev.Lett. 86 (2000) 926.[2] M. Ablowitz, A. Zeppetella,
Explicit solutions of Fisher’s equation for a special wave speed ,Bull. Math. Biol. 41 (1979) 835.[3] H.C. Rosu, O. Cornejo-P´erez,
Supersymmetric pairing of kinks for polynomial nonlinearities ,Phys. Rev. E 71 (2005) 046607.,Phys. Rev. E 71 (2005) 046607.