Computation of Some Geometric Properties for New Nonlinear PDE Models

Nassar Hassan Abdel-All, Mohammed Abdullah Abdullah Hamad, Mohammed Abdullah Abdel-Razek, Amal Aboelwafa Khalil

.

DOI: 10.4236/am.2011.26088                

Abstract

The purpose of the present work is to construct new geometrical models for motion of plane curve by Darboux transformations. We get nonlinear partial differential equations (PDE). We have obtained the exact solutions of the resulting equations using symmetry groups method. Also, the Gaussian and mean curvatures of Monge form of the soliton surfaces have been calculated and discussed.

Keywords

Motion of Curve, Darboux Transformations, Gaussian and Mean Curvatures, Symmetry Groups

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	B. Kimia, A. Tannenbaum and S. Zucker, “On the Evolution of Curves via a Function of Curvature. I. The Classical Case,” Journal Mathematical Analysis and Application, Vol. 163, No. 2, 1992, pp. 438-458. doi:10.1016/0022-247X(92)90260-K
[2]	B. Kimia, A. Tannenbaum and S. Zucker, “Shapes, Shochs, and Deformations I: The Components of Two-Dimensional Shape and the Reaction-Diffusion Space,” International Journal of Computer Vision, Vol. 15, No. 3, 1995, pp. 189-224. doi:10.1007/BF01451741
[3]	C. Rogers and W. K. Schief, “Backlund and Darboux Transformations Geometry and Modern Application in Soliton Theory,” Cambridge University press, Cambridge, 2002. doi:10.1017/CBO9780511606359
[4]	M. do Carmo, “Differerntial Geometry of Curves and Surfaces,” Prentice-Hall, Upper Saddle River, 1976.
[5]	L. P. Eisenhart, “A Treatise on the Differential Geometry of Curves and Surfaces,” Ginn, Boston, 1909.
[6]	G. W. Bluman and S. Kumei, “Symmetries and Differential Equations,” Springer, New York, 1989.
[7]	P. J. Olver, “Applications of Lie Groups to Differential Equations,” Springer, New York, 1986.
[8]	P. Winternitz, “Lie Groups and Solutions of Nonlinear Partial Differential Equations, Integrable Systems, Quantum Groups, and Qunatum Field Theories,” Kluwer Academic, Dordrecht, 1992.
[9]	C. H. Gu, “Soliton Theory and Its Applications,” Sprin-ger-Verlag, Berlin, 1995.
[10]	C. H. Gu and H. S. Hu, “On the Determination of Nonlinear PDE Admitting Integrable System,” Scientia Sinica, Series A, 1986, pp. 704-719.