Timeline for What is the structure of the space of solutions of a non linear ODE?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 16, 2021 at 22:16 | comment | added | Pietro Majer | Note the correct spelling: Riccati (btw, the word “ricatti” does exist in Italian, and the meaning is: “blackmails”) | |
| Apr 19, 2016 at 6:46 | comment | added | Tom Copeland | @L.Spice. tandfonline.com/doi/abs/10.1142/S1402925110000635 | |
| Oct 15, 2015 at 16:16 | comment | added | LSpice | In my quest for links: Bernoulli numbers and solitons—revisited. It's not on the arXiv, and I can't find a home page for Rzadkowski, so I'm not sure if there's a free version out there. | |
| Oct 10, 2015 at 18:44 | comment | added | Tom Copeland | Two others with interesting links: 4) "Bernoulli numbers and solitons - revisited" - Rzadkowski, and 5) "Lie algebras, representations, and analytic semigoups through dual vector fields" - Feinsilver (pp. 44-45), chanoir.math.siu.edu/MATH/Merida/PDF/Merida.pdf. | |
| Oct 10, 2015 at 18:16 | comment | added | Tom Copeland | For more on Lie and Ricatti: 1) "An introduction to Lie groups and symplectic geometry" - Bryant, math.duke.edu/~bryant/ParkCityLectures.pdf, 2) "Elie Cartan and geometric duality" - Bryant, math.duke.edu/~bryant/Cartan.pdf, 3) numerous papers by Carinena and associates, e.g., "Integrability of Lie systems through Ricatti equations" arxiv.org/abs/1002.0530. | |
| Dec 9, 2012 at 14:36 | comment | added | Robert Bryant | That's true. The point is that the 'evaluation map' from the space of solutions with its natural $\mathbb{RP}^1$-structure to $\mathbb{R}$ given by $u\mapsto u(t_0)$ is an affine chart on the $\mathbb{RP}^1$ for each $t_0$. The constancy of the cross-ratio is then a reflection of the fact that projective transformations of $\mathbb{RP}^1$ preserve the cross-ratio of $4$ points. Of course, this generalizes to all equations of Lie type, replacing $\mathbb{RP}^1$ with the appropriate homogeneous space $M$ and the cross-ratio with appropriate invariant function(s) on products of $M$ with itself. | |
| Dec 9, 2012 at 13:47 | comment | added | Denis Serre | The formula given in this answer is related to the fact any four solutions of the Ricatti equation are in constant Cross-ratio. | |
| Dec 8, 2012 at 23:29 | history | answered | Robert Bryant | CC BY-SA 3.0 |