On the KdV Equation with Hysteresis

Marius-Florinel Ionescu, Ligia Munteanu, Veturia Chiroiu

.

DOI: 10.4236/wjm.2011.11001                

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.

Keywords

Hysteresis Operator, KdV Equations with Hysteresis

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. A. Ewing, “Experimental Research in Magnetism,” Philosophical Transactions of the Royal Society of London, Vol. 176, No. 2, 1895, pp. 131-159.
[2]	M. A. Kranoselskii and A. V. Pokrovskii, “Systems with Hysteresis,” Springer, Berlin, 1989.
[3]	M. Brokate and J. Sprekels, “Hysteresis and Phase Transitions,” Springer, Berlin, 1996.
[4]	P. Kre?í, “Convexity, Hysteresis and Dissipation in Hyperbolic Equations,” Gakkotosho, Tokyo, 1997.
[5]	A. Visintin, “Differential Models of Hysteresis,” Springer-Verlag, Berlin 1995.
[6]	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
[7]	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
[8]	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
[9]	V. Barbu, “Nonlinear Semigroups and Differential Equations in Banach Spaces,” Noordhoff, Leyden, 1976.
[10]	J. Kopfová, “Nonlinear Semigroup Methods in Problems with Hysteresis,” Discrete and Continuous Dynamical Systems Supplement, 2007, pp. 580-589.
[11]	A. Visintin, “Hysteresis and Semigroups,” In: A. Visintin, Ed., Models of Hysteresis, Longman, Harlow, 1993, pp. 192-206.
[12]	D. Damjanovic, “Hysteresis in Piezoelectric and Ferroelectric Materials,” In: G. Bertotti, I. Mayergoyz, Eds., The Science of Hysteresis, Elsevier, 2006, pp. 338-46.
[13]	I. D. Mayergoyz, “Mathematical Models of Hysteresis and Their Applications,” Elsevier, Amsterdam, 2003.
[14]	G. Bertotti, “Hysteresis in Magnetism,” Academic Press, Boston, 1998.
[15]	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
[16]	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.
[17]	P. P.Teodorescu, V. Chiroiu, L. Munteanu and D. Dumitriu, “On the Resonance Wave Interaction Phenomenon,” Revue Roumaine des Sciences Techniques, série de Mécanique Appliquée, Vol. 55, No. 1, 2010, pp. 31-38.
[18]	V. Preda, M. F. Ionescu, V. Chiroiu and T. Sireteanu, “A Preisach Model for the Analysis of the Hysteretic Phenomena,” Revue Roumaine des Sciences Techniques, série de Mécanique Appliquée, Vol. 55, No. 3, 2010, pp. 75-86.
[19]	A. S. Gliozzi, L. Munteanu, T. Sireteanu and V. Chiroiu, “An Identification Problem from Input-Output Data,” Revue Roumaine des Sciences Techniques, série de Mécanique Appliquée, Vol. 55, No. 3, 2010, pp. 54-67.
[20]	A. Visintin, “Mathematical models of hysteresis,” In: G. Bertotti, I. Mayergoyz, Eds., The Science of Hysteresis, Elsevier, 2006, pp. 1-123.
[21]	J. A. Biello and A. J. Majda, “Boundary Layer Dissipation and the Nonlinear Interaction of Equatorial Baroclinic and Barotropic Rossby Waves,” Geophysical & Astrophysical Fluid Dynamics, Vol. 98, No. 2, 2004, pp. 85-127. doi:10.1080/03091920410001686712
[22]	J. A. Biello and A. J. Majda, “The Effect of Meridional and Vertical Shear on the Interaction of Equatorial Baroclinic and Barotropic Rossby Waves,” Studies in Applied Mathematics, Vol. 112, No. 4, 2004, pp. 341-390. doi:10. 1111/j.0022-2526.2004.01518.x
[23]	A. J. Majda and J. A. Biello, “The Nonlinear Interaction of Barotropic and Equatorial Baroclinic Rossby Waves,” Journal of the Atmospheric Sciences, Vol. 60, No.15, 2003, pp. 1809-1821. doi:10.1175/1520-0469(2003)060< 1809:TNIOBA>2.0.CO;2
[24]	L. Munteanu and S. Donescu, “Introduction to Soliton Theory: Applications to Mechanics,” Book Series Fundamental Theories of Physics, Vol. 143, Kluwer Academic Publishers, Dordrecht, 2004.
[25]	D. Levi and O. Ragnisco, “Dressing Method and B?cklund and Darboux Transfomations,” CRM Proceedings and Lecture Notes, Centre de Recherches Mathématiques, Vol. 29, 2001, pp. 29-51.
[26]	R. V. Lapshin, “Analytical Model for the Approximation of Hysteresis Loop and Its Application to the Scanning Tunneling Microscope,” Review of Scientific Instruments, Vol. 66, No. 9, 1995, pp. 4718-4730. doi:10.1063/1.1145 314