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## Banach Center Publications

1996 | 33 | 1 | 297-308
Tytuł artykułu

### Non-Euclidean geometry and differential equations

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper a geometrical link between partial differential equations (PDE) and special coordinate nets on two-dimensional smooth manifolds with the a priori given curvature is suggested. The notion of G-class (the Gauss class) of differential equations admitting such an interpretation is introduced. The perspective of this approach is the possibility of applying the instruments and methods of non-Euclidean geometry to the investigation of differential equations. The equations generated by the coordinate nets on the Lobachevsky plane $Λ^2$ (the hyperbolic plane) take a particular place in this study. These include sine-Gordon, Korteweg-de Vries, Burgers, Liouville and other equations. They form the so-called $Λ^2$-class (the Lobachevsky class). The theorems on the mutual transformation of solutions of $Λ^2$-class equations are formulated. On the base of the developed approach a transformation allowing one to construct global solutions of Liouville type equations from solutions of the Laplace equation is established. Natural generalizations of the well-known nonlinear PDE from the non-Euclidean geometry point of view are proposed. The possibility of the applications of the discussed formalism in the phase spaces theory is stressed.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
297-308
Opis fizyczny
Daty
wydano
1996
Twórcy
autor
• Chair of Mathematics, Department of Physics, Moscow State University, 119899 Moscow, Russia
Bibliografia
• [1] A. Artikbaev and D. D. Sokoloff, Geometry 'in the Large' in Plane Space-Time, Fan, Tashkent, 1991 (in Russian).
• [2] A. Barone and G. Paterno, The Josephson Effect, Mir, Moscow, 1984 (in Russian).
• [3] R. Beals, M. Rabelo and K. Tenenblat, Bäcklund transformation and inverse scattering for some pseudospherical surface equations, Stud. Appl. Math. 81 (1989), 125-151.
• [4] F. Calogero and X. Ji, C-integrable partial differential equations I, J. Math. Phys. 32 (4) (1991), 875-887.
• [5] F. Calogero and X. Ji, C-integrable PDEs II, ibid. 32 (10) (1991), 2703-2717.
• [6] S. S. Chern and K. Tenenblat, Pseudospherical surfaces and evolution equations, Stud. Appl. Math. 74 (1) (1986), 55-83.
• [7] M. Crampin and D. J. Saunders, The sine-Gordon equation, Tchebyshev nets and harmonic maps, Rep. Math. Phys. 23 (3) (1986), 327-340.
• [8] N. Kamran and K. Tenenblat, On differential equations describing pseudospherical surfaces, J. Differential Equations (to be published).
• [9] G. L. Lamb, Jr. An Introduction to Soliton Theory, Mir, Moscow, 1983 (in Russian).
• [10] V. S. Malakhovskiĭ, An Introduction to the Theory of Exterior Forms, Kaliningrad University Press, Kaliningrad, 1980 (in Russian).
• [11] A. G. Popov, Exact formulas for the construction of solutions of the Liouville equation $Δ_2 u = e^u$ from solutions of the Laplace equation $Δ_2 v = 0$, Dokl. Akad. Nauk 333 (4) (1993), 440-441 (in Russian).
• [12] A. G. Popov, The geometrical approach to the interpretation of solutions of the sine-Gordon equation, Dokl. Akad. Nauk SSSR 312 (5) (1990), 1109-1111 (in Russian).
• [13] A. G. Popov, On the transformation of local solutions of the equations connected with surface geometry, Izv. Vyss. Uchebn. Zaved. Mat. 9 (1993), 1-10 (in Russian).
• [14] A. G. Popov, The phase spaces of nonzero curvature and evolution of physical systems, Vestnik Moskov. Univ. 34 (6) (1993), 7-13.
• [15] E. G. Poznyak and A. G. Popov, The geometry of the sine-Gordon equation, Itogi Nauki i Tekh., Problems in Geometry 23 (1991), 99-130 (in Russian).
• [16] E. G. Poznyak and A. G. Popov, The Lobachevsky geometry and equations of mathematical physics, Dokl. Akad. Nauk 332 (4) (1993), 418-421 (in Russian).
• [17] E. G. Poznyak and E. V. Shikin, Differential Geometry, Moscow University Press, Moscow, 1990 (in Russian).
• [18] M. Rabelo, On evolution equations which describe pseudospherical surface, Stud. Appl. Math. 81 (1989), 221-248.
• [19] A. Sanchez and L. Vazquez, Nonlinear wave propagation in disorded media, Internet. J. Modern Phys. B 5 (18) (1991), 2825-2882.
• [20] R. Sasaki, Geometrization of soliton equations, Phys. Lett. A 71 (1979), 390-392.
• [21] R. Sasaki, Soliton equations and pseudospherical surfaces, Nucl. Phys. B 154 (1979), 343-357.
• [22] A. Sym, Soliton surfaces, Lett. Nuovo Cimento 33 (12) (1982), 394-400.
• [23] P. Tchebychef [P. Chebyshev], Sur la coupe des vêtements, Assoc. franç. pour l'avancement des sciences, 1878; see also Œ uvres II.
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Bibliografia
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