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Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities

Published online by Cambridge University Press:  14 November 2011

Victor A. Galaktionov
Victor A. Galaktionov
Affiliation:
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Square 4, 125047 Moscow, Russia
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Abstract

We present new explicit solutions to some classes of quasilinear evolution equations arising in different applications, including equations of the Boussinesq type:

and quasilinear heat equations:

The method is based on construction of finite-dimensional linear functional subspaces which are invariant with respect to spatial operators having quadratic nonlinearities. The corresponding nonlinear evolution equations on invariant subspaces are shown to be equivalent to finite-dimensional dynamical systems. Examples of two-, three- and five- dimensional invariant subspaces are given. Some generalisations to N-dimensional quadratic operators are also considered.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 2 , 1995 , pp. 225 - 246
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Bertch, M., Kersner, R. and Peletier, L. A.. Positivity versus localization in degenerate diffusion equations. Nonlinear Anal. 9 (1985), 987–1008.CrossRefGoogle Scholar
2Budd, C., Dold, J. W. and Stuart, A.. Blow-up in a parabolic system with convection and constrain first integral (in preparation).Google Scholar
3Clarkson, P. A. and Kruskal, M. D.. New similarity reductions of the Boussinesq equation. J. Math. Phys. 30(1989), 2201–2213.CrossRefGoogle Scholar
4Dorodnitsyn, V. A.. On invariant solutions of a nonlinear heat equation with source. Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), 1393–1400 (in Russian).Google Scholar
5Galaktionov, V. A.. On new exact blow-up solutions for nonlinear heat conduction equations with source and applications. Differential Integral Equations 3 (1990), 863–874.CrossRefGoogle Scholar
6Galaktionov, V. A., Dorodnitsyn, V. A., Elenin, G. G., Kurdyumov, S. P. and Samarskii, A. A.. Quasilinear heat conduction equation with source: blow-up, localization, symmetry, exact solutions, asymptotics, structures. In Modern Mathematical Problems. New Achievements, Vol. 28, pp. 95–206 (Moscow: VINITI AN USSR, 1986 (in Russian); English translation: New York: Plenum Press, 1987).Google Scholar
7Galaktionov, V. A. and Posashkov, S. A.. Exact solutions of parabolic equations with quadratic nonlinearities (Preprint No. 115, Keldysh Inst. Appl. Math. Acad. Sci. USSR, Moscow, 1988 (in Russian)).Google Scholar
8Hocking, L. M., Stewartson, K. and Stuart, J. T.. A non-linear instability burst in plane parallel flow. J. Fluid Mech. 51 (1972), 705–735.CrossRefGoogle Scholar
9Kersner, R.. On some properties of weak solutions of quasilinear degenerate parabolic equations. Ada Math. Hungar. 32 (1978), 301–330 (in Russian).Google Scholar
10King, J. R.. Exact multidimensional solutions to some non-linear diffusion equations (preprint).Google Scholar
11Martinson, L. K.. Propagation of a heat wave in a nonlinear medium with absorption. Prikl. Mekh.Techn. Fiz. 4 (1979), 36–39 (in Russian).Google Scholar
12Olver, P. J.. Symmetry and explicit solutions of partial differential equations (to appear).Google Scholar
13Olver, P. J. and Rosenau, P.. The construction of special solutions to partial differential equations. Phys. Lett. A 114 (1986), 107–112.CrossRefGoogle Scholar
14Ovsiannikov, L. V.. Group properties of a nonlinear heat conduction equation. Dokl. Akad. Nauk SSSR, Ser. Math. 125 (1959), 492–495 (in Russian).Google Scholar