Solitary wave and other solutions for nonlinear heat equations. (English) Zbl 1116.35035

Nikitin, Anatoly G.; Barannyk, Tetyana A.

Cent. Eur. J. Math. 2, No. 5, 840-858 (2004).

The authors describe a class of nonlinear heat equations with a polynomial nonlinearity for which explicit solutions can be found by a certain ansatz. The method allows to find infinitely many solutions in terms of e.g. Jacobi elliptic functions or (in the case of Fisher’s equation) the Weierstrass \(\wp\)-function. This includes solutions which had already been found by symmetry methods but also some new exact solutions. Vice versa, the authors describe a modification of Fisher’s equation which possesses solitary wave solutions of any given propagation velocity.

Reviewer: Jörg Härterich (Berlin)

MSC:

35C05	Solutions to PDEs in closed form
35K55	Nonlinear parabolic equations
35Q51	Soliton equations

Keywords:

Fisher equation; Newell-Whitehead equation; KPP equation; polynomial nonlinearity; Jacobi elliptic functions

References:

[1]	J.D. Murray: Mathematical Biology, Springer, 1991. · Zbl 0704.92001
[2]	A.N. Kolmogorov, I.G. Petrovskii and N.S. Piskunov: “A study of the diffusion equation with increase in the quantity of matter and its application to a biological problem”, Bull. Moscow Univ. Sér. Int. A, Vol. 1(1), (1937).
[3]	R. Fitzhugh: “Impulses and physiological states in models of nerve membrane“, Biophys. J., Vol. 1(445), 1961; J.S. Nagumo, S. Arimoto and S. Yoshizawa: “An active pulse transmission line simulating nerve axon”, Proc. IRE, Vol. 50(2061), (1962).
[4]	A.C. Newell and J.A. Whitehead: “Finite bandwidth, finite amplitude convection”, J. Fluid Mech., Vol. 38(279), (1969). · Zbl 0187.25102
[5]	V.A. Dorodnitsyn: “On invariant solutions of nonlinear heat equation with source”, Comp. Meth. Phys., Vol. 22(115), (1982). · Zbl 0535.35040
[6]	S. Lie: Transformationgruppen, Leipzig, 1883.
[7]	P. Olver: Application of Lie groups to differential equations, Springer, Berlin, 1986. · Zbl 0588.22001
[8]	A.G. Nikitin and R. Wiltshire: “Symmetries of Systems of Nonlinear Reaction-Diffusion Equations“, In: A.M. Samoilenko (Ed.): Symmetries in Nonlinear Mathematical Physics, Proc. of the Third Int. Conf., Kiev, July 12-18, 1999, Inst. of Mathematics of Nat. Acad. Sci. of Ukraine, Kiev, 2000; R. Cherniha and J. King: “Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I”, J. Phys. A, Vol. 33(257), (2000); A.G. Nikitin and R. Wiltshire: “Systems of Reaction Diffusion Equations and their symmetry properties”, J. Math. Phys., Vol. 42(1666), (2001).
[9]	G.W. Bluman and G.D. Cole: “The general similarity solution of the heat equation”, J. Math. Mech., Vol. 18(1025), (1969).
[10]	W.I. Fushchych and A.G. Nikitin: Symmetries of Maxwell’s equations, Reidel, Dordrecht, 1987; W.I. Fushchych: “Conditional symmetry of mathematical physics equations”, Ukr. Math. Zh., Vol. 43(1456), 1991.
[11]	D. Levi and P. Winternitz: “Non-classical symmetry reduction: example of the Boussinesq equation”, J. Phys. A, Vol. 22(2915), (1989). · Zbl 0694.35159
[12]	W.I. Fushchich and M.I. Serov: “Conditional invariance and reduction of the nonlinear heat equation”, Dokl. Akad. Nauk Ukr. SSR, Ser. A, Vol. 4(24), (1990). · Zbl 0727.35066
[13]	P.A. Clarkson and E.L. Mansfield: “Symmetry reductions and exact solutions of a class of nonlinear heat equations”, Physica D, Vol. 70(250), (1993). · Zbl 0788.35067
[14]	E. Fan: “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method”, J. Phys. A, Vol. 35(6853), (2002). · Zbl 1039.35029
[15]	R. Hirota and J. Satsuma: “Soliton solutions of a coupled Korteweg-de Vries equation”, Phys. Lett. A, Vol. 85(07), (1981).
[16]	M.J. Ablowitz and A. Zeppetella: “Explicit solution of Fisher’s equation for a special wave speed”, Bull. Math. Biol., Vol. 41(835), (1979). · Zbl 0423.35079
[17]	A.S. Fokas and Q.M. Liu: “Generalized Conditional Symmetries and Exact Solutions of Non Integrable Equations”, Theor. Math. Phys., Vol. 99(371), (1994). · Zbl 0850.35097
[18]	R.Z. Zhdanov and V.I. Lahno: “Conditional symmetry of a porous medium equation”, Physica D, Vol. 122(178), (1998). · Zbl 0952.76087
[19]	D.J. Needham and A.C. King: “The evolution of travelling waves in the weakly hyperbolic generalized Fisher model“, Proc. Roy. Soc. (London) Vol. 458(1055), (2002). P.S. Bindu, M. Santhivalavan and M. Lakshmanan: “Singularity structure, symmetries and integrability of generalized Fisher-type nonlinear diffusion equation”, J. Phys. A, Vol. 34(l689), (2001).