Non-integrability of non-linear diffusion-convection equations in two-spatial dimensions
- 11 May 1986
- journal article
- Published by IOP Publishing in Journal of Physics A: General Physics
- Vol. 19 (7) , 1245-1257
- https://doi.org/10.1088/0305-4470/19/7/025
Abstract
It is generally accepted that integrable partial differential time evolution equations possess Lie-Backlund symmetries of arbitrarily high finite order. It has already been established that in (1+1) dimensions the full class of nonlinear Fokker-Planck (convection-diffusion) equations having this property consists of the known integrable examples. These are closely related either to the Burgers equations or to the Fokas-Yortsos-Rosen equation, both of which have been applied to unsaturated flow in porous media. Here the author shows that higher-order Lie-Backlund symmetries do not exist for any examples of the nonlinear Fokker-Planck equation in (1+2) (one time and two space) dimensions.Keywords
This publication has 23 references indexed in Scilit:
- Analytically solvable dynamical systems which are not integrablePhysica D: Nonlinear Phenomena, 1984
- Two-dimensional Burgers equationLettere al Nuovo Cimento (1971-1985), 1983
- On the exactly solvable equation$S_t = [ ( \beta S + \gamma )^{ - 2} S_x ]_x + \alpha ( \beta S + \gamma )^{ - 2} S_x $ Occurring in Two-Phase Flow in Porous MediaSIAM Journal on Applied Mathematics, 1982
- BURGERS' EQUATIONSoil Science, 1981
- A symmetry approach to exactly solvable evolution equationsJournal of Mathematical Physics, 1980
- On the remarkable nonlinear diffusion equation (∂/∂x)[a (u+b)−2(∂u/∂x)]−(∂u/∂t)=0Journal of Mathematical Physics, 1980
- Lie-Bäcklund tangent transformationsJournal of Mathematical Analysis and Applications, 1977
- The Exact Pattern of a Concentration-Dependent Diffusion in a Semi-infinite Medium, Part ITextile Research Journal, 1952
- On a quasi-linear parabolic equation occurring in aerodynamicsQuarterly of Applied Mathematics, 1951
- The partial differential equation ut + uux = μxxCommunications on Pure and Applied Mathematics, 1950