ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35, 22, 92 Cherniha, Roman M., Institute of Mathematics, Ukrainian Academy of Sciences, Tereshchenkivs’ka Str. 3, Kyiv 4, Ukraine Nonlinear reaction-diffusion-convection equations of the form

U_{t} = [A(U)U_{x}]_{x} + B(U)U_{x} + C(U),  U = U(t, x),  (1)

are considered. A complete describing Lie symmetries of equation (??) is presented [1]. A new constructive method [1] for obtaining new non-Lie ansätze and exact solutions of nonlinear PDEs with trivial Lie symmetries is suggested. The method is based on the consideration of a fixed nonlinear PDE together with an additional generating condition in the form of a linear high-order ODE. New exact solutions are obtained for equations of the form (1) and for some nonlinear diffusion systems.

[1] Serov M. I. and Cherniha R. Ukrainian Math. J. 49 (1997), 1262-70; Cherniha R. M. and Serov M. European J. Appl. Math. (to appear).

[2] Cherniha R. Rep. Math. Phys. 38.3 (1996), 301-312; Cherniha R. Ukrainian Math. J. 49 (1997), 814-827; Cherniha R. and Fehribach J. J. Phys. A: Math. Gen. 31 (1998), 3815-29.