Timeline for Characteristic Variety of the Principal Symbol solves PDE system?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 15, 2022 at 21:29 | comment | added | Bran | @RobertBryant yes, clear. Thank you! | |
| May 15, 2022 at 14:02 | comment | added | Robert Bryant | @Bran: For the case in hand, a solution of the first system is a $2$-dimensional integral manifold of the $1$-form $\theta_1 = \mathrm{d}\phi+\phi\,\mathrm{d}z-\phi\,\mathrm{d}z'$ while a solution of the second system is a $2$-dimensional integral manifold of the $1$-form $\theta_2 = \mathrm{d}\phi+z\phi\,\mathrm{d}z-z\phi\,\mathrm{d}z'$. Both $1$-forms have integral curves, but $\theta_1=0$ is `involutive' while $\theta_2=0$ is not 'involutive' in the sense of $2$-plane fields. Does that help? | |
| May 14, 2022 at 15:52 | comment | added | Bran | @RobertBryant can you please elaborate on what involutivity means in this sense? | |
| Oct 14, 2015 at 4:40 | comment | added | Robert Bryant | First, involutivity has nothing to do with linearity; they are independent notions. Second, I don't know what it means to have a solution 'in the micro-local sense', so I can't help you there. What's certainly true is that there are solution curves in both cases (i.e., 1-dimensional integral manifolds in the space of $1$-jets), but the only $2$-dimensional integral manifold in the second case is the one given by the (unique) solution $\psi = 0$. | |
| Oct 14, 2015 at 3:27 | comment | added | alphanzo | Thank you for the comment. That is indeed the correct way to go about it in the Frobenius picture when the equations are linear. However, in the second link the general case is considered micro locally. It seems that the second system does have a solution if only in the micro local sense of the term. I'm wondering if it implies there exists a hypersurface M⊂C2 on which a solution exists for restricted values (z,z′)∈M? | |
| Oct 13, 2015 at 20:16 | history | answered | Robert Bryant | CC BY-SA 3.0 |