Similarity reductions from extended Painlevé expansions for nonintegrable evolution equations. (English) Zbl 0745.35046
Summary: The use of singular manifold expansions to find exact solutions to nonintegrable evolution equations is extended to include arbitrary (resonance) coefficients in such a way as to make the resulting infinite series exactly resummable. The technique involves the use of a rescaling ansatz analogous to that used to analyze the psi-series of nonintegrable ordinary differential equations. The result is a similarity reduction of the equation in which the constrained singular manifold plays the role of a similarity variable. The method is capable of yielding new solutions corresponding to either classical or nonclassical (conditional) Lie symmetries.
MSC:
| 35Q80 | Applications of PDE in areas other than physics (MSC2000) |
| 37-XX | Dynamical systems and ergodic theory |
Keywords:
Ginzburg-Landau-equation; singular manifold expansions; exact solutions; nonintegrable evolution equations; rescaling ansatz; Lie symmetriesReferences:
| [1] | Tabor, M., Chaos and Integrability in Nonlinear Dynamics (1989), Wiley Interscience: Wiley Interscience New York · Zbl 0682.58003 |
| [2] | Weiss, J.; Tabor, M.; Carnevale, G., J. Math. Phys., 24, 256-522 (1983) · Zbl 0514.35083 |
| [3] | Cariello, F.; Tabor, M., Physica D, 39, 77-94 (1989) · Zbl 0687.35093 |
| [4] | Weiss, J., J. Math. Phys., 25, 2226-2235 (1984) · Zbl 0557.35107 |
| [5] | Fournier, J.-D.; Spiegel, E. A.; Thual, O., (Turchetti, G., Conference on Nonlinear Dynamics. Conference on Nonlinear Dynamics, Bologna, 1988 (1989), World Scientific: World Scientific Singapore) |
| [6] | Conte, R.; Musette, M., J. Phys. A: Math. Gen., 22, 169-177 (1989) · Zbl 0687.35087 |
| [7] | Newell, A.; Tabor, M.; Zeng, Y. B., Physica D, 29, 1-68 (1987) · Zbl 0643.35096 |
| [8] | Fournier, J. D.; Levine, G.; Tabor, M., J. Phys. A: Math. Gen., 21, 33-54 (1988) · Zbl 0641.34034 |
| [9] | Levine, G.; Tabor, M., Physica D, 33, 189-210 (1988) · Zbl 0663.58042 |
| [10] | Bluman, G. W.; Cole, J. D., Similarity Methods for Differential Equations, (Appl. Math. Sci., 13 (1974), Springer: Springer New York) · Zbl 0292.35001 |
| [11] | Abramowitz, M.; Stegun, I. A., (Handbook of Mathematical Functions (1965), Dover: Dover New York) |
| [12] | Keefe, L. R., Phys. Fluids, 29, 3135-3141 (1986) · Zbl 0602.76061 |
| [13] | Skierski, M.; Grunland, A. M.; Tuszynski, J. A., J. Phys. A: Math. Gen., 22, 199-219 (1989) |
| [14] | Ablowitz, M. J.; Zeppetella, A., Bull. Math. Biol., 41, 835-840 (1979) · Zbl 0423.35079 |
| [15] | Clarkson, P.; Kruskal, M. D., J. Math. Phys., 30, 2201-2213 (1989) · Zbl 0698.35137 |
| [16] | Levi, D.; Winternitz, J. Phys. A, 22, 2915-2924 (1989) · Zbl 0694.35159 |
| [17] | Clarkson, P., J. Phys. A: Math. Gen., 22, 3821-3848 (1989) · Zbl 0711.35113 |
| [18] | Gibbon, J. D.; Newell, A. C.; Tabor, M.; Zeng, Y. B., Nonlinearity, 1, 481-490 (1988) · Zbl 0682.34011 |
| [19] | Weiss, J., Phys. Lett. A, 105, 387-389 (1984) |
| [20] | T. Sen, private communication.; T. Sen, private communication. |
| [21] | J.A. Powell, A.C. Newell and C.K.R.T. Jones, Competition between generic and nongeneric fronts in envelope equations, preprint.; J.A. Powell, A.C. Newell and C.K.R.T. Jones, Competition between generic and nongeneric fronts in envelope equations, preprint. |
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