×

Integrable equations and their evolutions based on intrinsic geometry of Riemann spaces. (English) Zbl 1177.53052

Summary: The intrinsic geometry of surfaces and Riemannian spaces will be investigated. It is shown that many nonlinear partial differential equations with physical applications and soliton solutions can be determined from the components of the relevant metric for the space. The manifolds of interest are surfaces and higher-dimensional Riemannian spaces. Methods for specifying integrable evolutions of surfaces by means of these equations will also be presented.

MSC:

53C40 Global submanifolds
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] D. G. Gross, C. N. Pope, and S. Weinberg, Two Dimensional Quantum Gravity and Random Surfaces, World Scientific, Singapore, 1992.
[2] D. Nelson, T. Piran, and S. Weinberg, Statistical Mechanics of Membranes and Surfaces, World Scientific, Singapore, 1992. · Zbl 1059.82002
[3] P. Bracken and A. M. Grundland, “Symmetry properties and explicit solutions of the generalized Weierstrass system,” Journal of Mathematical Physics, vol. 42, no. 3, pp. 1250-1282, 2001. · Zbl 1016.53008
[4] P. Bracken, “Partial differential equations which admit integrable systems,” International Journal of Pure and Applied Mathematics, vol. 43, no. 3, pp. 409-421, 2008. · Zbl 1142.53011
[5] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0762.35001
[6] P. Bracken, A. M. Grundland, and L. Martina, “The Weierstrass-Enneper system for constant mean curvature surfaces and the completely integrable sigma model,” Journal of Mathematical Physics, vol. 40, no. 7, pp. 3379-3403, 1999. · Zbl 0967.53004
[7] S. S. Chern and K. Tenenblat, “Pseudospherical surfaces and evolution equations,” Studies in Applied Mathematics, vol. 74, no. 1, pp. 55-83, 1986. · Zbl 0605.35080
[8] R. Sasaki, “Soliton equations and pseudospherical surfaces,” Nuclear Physics B, vol. 154, no. 2, pp. 343-357, 1979.
[9] F. Calogero and A. Degasperis, Spectral Transform and Solitons. Vol. I, vol. 13 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1982. · Zbl 0501.35072
[10] B. G. Konopelchenko, “Soliton curvatures of surfaces and spaces,” Journal of Mathematical Physics, vol. 38, no. 1, pp. 434-457, 1997. · Zbl 0867.53003
[11] B. G. Konopelchenko, Introduction to Multidimensional Integrable Equations, Plenum, New York, NY, USA, 1992. · Zbl 0877.35118
[12] S. S. Chern, W. H. Chen, and K. S. Lam, Lectures on Differential Geometry, vol. 1 of Series on University Mathematics, World Scientific, Singapore, 1999. · Zbl 0940.53001
[13] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press, Oxford, UK, 1997. · Zbl 0081.22207
[14] R. Beutler and B. G. Konopelchenko, “Surfaces of revolution via the Schrödinger equation: construction, integrable dynamics and visualization,” Applied Mathematics and Computation, vol. 101, no. 1, pp. 13-43, 1999. · Zbl 0979.53011
[15] P. Bracken, “Dynamics of induced surfaces in four-dimensional Euclidean space,” Pacific Journal of Applied Mathematics, vol. 1, no. 2, pp. 207-220, 2008. · Zbl 1192.37098
[16] I. A. Taimanov, “Surfaces of revolution in terms of solitons,” Annals of Global Analysis and Geometry, vol. 15, no. 5, pp. 419-435, 1997. · Zbl 0896.53007
[17] B. G. Konopelchenko, “Induced surfaces and their integrable dynamics,” Studies in Applied Mathematics, vol. 96, no. 1, pp. 9-51, 1996. · Zbl 0869.58027
[18] B. W. Char, K. O. Geddes, B. L. Leong, M. Monagen, and S. Watt, Maple V Language Reference Manual, Springer, New York, NY, USA, 1991. · Zbl 0758.68038
[19] V. E. Zakharov, “Integrable systems in multidimensional spaces,” in Mathematical Problems in Theoretical Physics, vol. 153 of Lecture Notes in Physics, pp. 190-216, Springer, Berlin, Germany, 1982.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.