Non-Euclidean geometry and differential equations
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.
-  A. Artikbaev and D. D. Sokoloff, Geometry 'in the Large' in Plane Space-Time, Fan, Tashkent, 1991 (in Russian).
-  A. Barone and G. Paterno, The Josephson Effect, Mir, Moscow, 1984 (in Russian).
-  R. Beals, M. Rabelo and K. Tenenblat, Bäcklund transformation and inverse scattering for some pseudospherical surface equations, Stud. Appl. Math. 81 (1989), 125-151.
-  F. Calogero and X. Ji, C-integrable partial differential equations I, J. Math. Phys. 32 (4) (1991), 875-887.
-  F. Calogero and X. Ji, C-integrable PDEs II, ibid. 32 (10) (1991), 2703-2717.
-  S. S. Chern and K. Tenenblat, Pseudospherical surfaces and evolution equations, Stud. Appl. Math. 74 (1) (1986), 55-83.
-  M. Crampin and D. J. Saunders, The sine-Gordon equation, Tchebyshev nets and harmonic maps, Rep. Math. Phys. 23 (3) (1986), 327-340.
-  N. Kamran and K. Tenenblat, On differential equations describing pseudospherical surfaces, J. Differential Equations (to be published).
-  G. L. Lamb, Jr. An Introduction to Soliton Theory, Mir, Moscow, 1983 (in Russian).
-  V. S. Malakhovskiĭ, An Introduction to the Theory of Exterior Forms, Kaliningrad University Press, Kaliningrad, 1980 (in Russian).
-  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).
-  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).
-  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).
-  A. G. Popov, The phase spaces of nonzero curvature and evolution of physical systems, Vestnik Moskov. Univ. 34 (6) (1993), 7-13.
-  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).
-  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).
-  E. G. Poznyak and E. V. Shikin, Differential Geometry, Moscow University Press, Moscow, 1990 (in Russian).
-  M. Rabelo, On evolution equations which describe pseudospherical surface, Stud. Appl. Math. 81 (1989), 221-248.
-  A. Sanchez and L. Vazquez, Nonlinear wave propagation in disorded media, Internet. J. Modern Phys. B 5 (18) (1991), 2825-2882.
-  R. Sasaki, Geometrization of soliton equations, Phys. Lett. A 71 (1979), 390-392.
-  R. Sasaki, Soliton equations and pseudospherical surfaces, Nucl. Phys. B 154 (1979), 343-357.
-  A. Sym, Soliton surfaces, Lett. Nuovo Cimento 33 (12) (1982), 394-400.
-  P. Tchebychef [P. Chebyshev], Sur la coupe des vêtements, Assoc. franç. pour l'avancement des sciences, 1878; see also Œ uvres II.