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On the KdV Equation with Hysteresis
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World Journal of Mechanics·Volume 01 (2011)·Pages 1–5·Published 30 January 2011·DOI10.4236/wjm.2011.11001
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Abstract
This paper discusses the generalized play hysteresis operator in connection with the KdV equation. Results from the nonlinear semigroup theory are applied to assure the existence and uniqueness. The KdV equation with hysteresis is reduced to a system of differential inclusions and solved.
KeywordsHysteresis OperatorKdV Equations with Hysteresis
- J. A. Ewing, “Experimental Research in Magnetism,” Philosophical Transactions of the Royal Society of London, Vol. 176, No. 2, 1895, pp. 131-159.
- M. A. Kranoselskii and A. V. Pokrovskii, “Systems with Hysteresis,” Springer, Berlin, 1989.
- M. Brokate and J. Sprekels, “Hysteresis and Phase Transitions,” Springer, Berlin, 1996.
- P. Kre?í, “Convexity, Hysteresis and Dissipation in Hyperbolic Equations,” Gakkotosho, Tokyo, 1997.
- A. Visintin, “Differential Models of Hysteresis,” Springer-Verlag, Berlin 1995.
- A. Visintin, “Quasi-Linear Hyperbolic Equations with Hysteresis,” Annales de l’ Institute Henri Poincaré, Nonlinear Analysis, Vol. 19, No. 4, 2002, pp. 451-476. doi:10.1016/S0294-1449(01)00086-5
- Y. Kōmura, “Nonlinear Semi-Groups in Hilbert Space,” Journal of the Mathematical Society of Japan, Vol. 19, No. 4, 1967, pp. 493-507. doi:10.2969/jmsj/01940493
- M. C. Crandall and T. M. Liggett, “Generation of Semigroups of Nonlinear Transformations on General Banach spaces,” American Journal of Mathematics, Vol. 93, No. 2, April 1971, pp. 265-298. doi:10.2307/2373376
- V. Barbu, “Nonlinear Semigroups and Differential Equations in Banach Spaces,” Noordhoff, Leyden, 1976.
- J. Kopfová, “Nonlinear Semigroup Methods in Problems with Hysteresis,” Discrete and Continuous Dynamical Systems Supplement, 2007, pp. 580-589.
- A. Visintin, “Hysteresis and Semigroups,” In: A. Visintin, Ed., Models of Hysteresis, Longman, Harlow, 1993, pp. 192-206.
- D. Damjanovic, “Hysteresis in Piezoelectric and Ferroelectric Materials,” In: G. Bertotti, I. Mayergoyz, Eds., The Science of Hysteresis, Elsevier, 2006, pp. 338-46.
- I. D. Mayergoyz, “Mathematical Models of Hysteresis and Their Applications,” Elsevier, Amsterdam, 2003.
- G. Bertotti, “Hysteresis in Magnetism,” Academic Press, Boston, 1998.
- A. Visintin, “Homogenization of Some Models of Hysteresis,” Physica B: Condensed Matter, Vol. 403, No. 2-3, February 2008, pp. 245-249. doi:10.1016/j.physb.2007.08. 020
- V. Mosnegutu and V. Chiroiu, “On the Dynamics of Systems with Friction,” Proceedings of the Romanian Academy, Series A: Mathematics, Physics, Technical Sciences, Information Science, Vol. 11, No. 1, 2010, pp. 63-68.