Some Classification Results for Hyperbolic Equations F(x, y, u, ux, uy, uxx, uxy, uyy)=0
✍ Scribed by Martin Juráš
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 173 KB
- Volume
- 164
- Category
- Article
- ISSN
- 0022-0396
No coin nor oath required. For personal study only.
✦ Synopsis
We provide a contact invariant characterization for equations of the form u xy +a(x, y, u) u x +b(x, y, u) u y +c(x, y, u)=0, u xy +a(x, y) u x +b(x, y) u y +c(x, y, u)=0,
We classify all equations of the form u xy + f (x, y, u, u x , u y )=0 for which the two Ovsiannikov's invariants are constants. These results include characterization of the Klein Gordon equation u xy =u, the Liouville equation u xy =e u , and the class of Euler Poisson Darboux equations. It is shown that the wave equation u xy =0, Liouville equation, and the linear equation u xy =2uÂ(x+ y) 2 are the only variational equations Darboux integrable at level one. We also show that a hyperbolic Monge Ampe re equation Darboux integrable at level one is equivalent to an equation of type u xy + f (x, y, u, u x , u y )=0. We prove that the hyperbolic Fermi Ulam Pasta (FPU) equation u yy =}(u x ) 2 u xx is contact equivalent to a linear equation of type u xy =c(x+ y) u and we classify all FPU equations Darboux integrable at level one. We also apply our results to equations of type u xy =F(u, u x ) that describe pseudospherical surfaces.