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ISRN Mathematical Physics
Volume 2013 (2013), Article ID 109170, 7 pages
Approximate Symmetries of the Harry Dym Equation
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran
Received 27 October 2013; Accepted 17 November 2013
Academic Editors: B. Bagchi and Z. Qiao
Copyright © 2013 Mehdi Nadjafikhah and Parastoo Kabi-Nejad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- W. Hereman, P. P. Banerjee, and M. R. Chatterjee, “Derivation and implicit solution of the Harry Dym equation and its connections with the Korteweg-de Vries equation,” Journal of Physics A, vol. 22, no. 3, pp. 241–255, 1989.
- M. D. Kruskal, Lecture Notes in Physics, vol. 38, Springer, Berlin, Germany, 1975.
- P. C. Sabatier, “On some spectral problems and isospectral evolutions connected with the classical string problem—II: evolution equation,” Lettere Al Nuovo Cimento, vol. 26, no. 15, pp. 483–486, 1979.
- L. Yi-Shen, “Evolution equations associated with the eigenvalue problem based on the equation ,” Lettere Al Nuovo Cimento, vol. 70, no. 1, pp. 1–12, 1982.
- Z. J. Qiao, “A completely integrable system associated with the Harry Dym hierarchy,” Journal of Nonlinear Mathematical Physics, vol. 1, no. 1, pp. 65–74, 1994.
- Z. Qiao, “Commutator representations of nonlinear evolution equations: Harry Dym and Kaup-Newell cases,” Journal of Nonlinear Mathematical Physics, vol. 2, no. 2, pp. 151–157, 1995.
- F. Calogero and A. Degasperis, Spectral Transform and Solitons, vol. 1, North-Holland, Amsterdam, The Netherlands, 1982.
- M. Wadati, Y. H. Ichikawa, and T. Shimizu, “Cusp soliton of a new integrable nonlinear evolution equation,” Progress of Theoretical Physics, vol. 64, no. 6, pp. 1959–1967, 1980.
- M. Wadati, K. Konno, and Y. H. Ichikawa, “New integrable nonlinear evolution equations,” Journal of the Physical Society of Japan, vol. 47, no. 5, pp. 1698–1700, 1979.
- F. Magri, “A simple model of the integrable Hamiltonian equation,” Journal of Mathematical Physics, vol. 19, no. 5, pp. 1156–1162, 1978.
- P. J. Olver, Application of Lie Groups To Differential Equations, Springer, New York, NY, USA, 2nd edition, 1993.
- V. A. Baĭkov, R. K. Gazizov, and N. Kh. Ibragimov, “Approximate symmetries of equations with a small parameter,” Matematicheskiĭ Sbornik, vol. 136, no. 4, pp. 435–450, 1988, English Translation in Mathematics of the USSR, vol. 64, pp. 427–441, 1989.
- V. A. Baikov, R. K. Gazizov, and N. H. Ibragimov, “Approximate transformation groups and deformations of symmetry Lie algebras,” in CRC Handbook of Lie Group Analysis of Differential Equation, N. H. Ibragimov, Ed., vol. 3, chapter 2, CRC Press, Boca Raton, Fla, USA, 1996.
- W. I. Fushchich and W. M. Shtelen, “On approximate symmetry and approximate solutions of the nonlinear wave equation with a small parameter,” Journal of Physics A, vol. 22, no. 18, pp. L887–L890, 1989.
- N. Euler, M. W. Shulga, and W.-H. Steeb, “Approximate symmetries and approximate solutions for a multidimensional Landau-Ginzburg equation,” Journal of Physics A, vol. 25, no. 18, pp. L1095–L1103, 1992.
- M. Euler, N. Euler, and A. Kohler, “On the construction of approximate solutions for a multidimensional nonlinear heat equation,” Journal of Physics A, vol. 27, no. 6, pp. 2083–2092, 1994.
- M. Pakdemirli, M. Yürüsoy, and I. T. Dolapçi, “Comparison of approximate symmetry methods for differential equations,” Acta Applicandae Mathematicae, vol. 80, no. 3, pp. 243–271, 2004.
- R. Wiltshire, “Two approaches to the calculation of approximate symmetry exemplified using a system of advection-diffusion equations,” Journal of Computational and Applied Mathematics, vol. 197, no. 2, pp. 287–301, 2006.
- N. H. Ibragimov and V. F. Kovalev, Approximate and Renormgroup Symmetries, Nonlinear Physical Science, Higher Education Press, Beijing, China, 2009, A. C.J Luo and N.H Ibragimov, Eds.
- J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, “Invariants of real low dimension Lie algebras,” Journal of Mathematical Physics, vol. 17, no. 6, pp. 986–994, 1975.