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

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.

MSC:

35C05 Solutions to PDEs in closed form
35K55 Nonlinear parabolic equations
35Q51 Soliton equations
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[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).; · Zbl 0187.03502
[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).;
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