Timeline for characteristic surface
Current License: CC BY-SA 3.0
20 events
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| Aug 4, 2011 at 21:57 | comment | added | Will Jagy | as you seem to want a tutorial, try posting at math.stackexchange.com/questions?sort=active | |
| Aug 4, 2011 at 21:52 | history | edited | Uri | CC BY-SA 3.0 |
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| Aug 4, 2011 at 21:41 | history | edited | Uri | CC BY-SA 3.0 |
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| Aug 4, 2011 at 21:36 | comment | added | Uri | Dear Robert, Thanks for the helpful info. I was wondering if it is trivial to conclude that $z=0$ is a charateristic surface to all the equations besides the fifth one. I did some reading and found a procedure for that which I appended to the question above. Would you care to comment? Is that the way to go or did I overcomplicate things? Thanks again! | |
| Aug 4, 2011 at 21:30 | history | edited | Uri | CC BY-SA 3.0 |
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| Aug 4, 2011 at 21:18 | history | edited | Uri | CC BY-SA 3.0 |
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| Aug 4, 2011 at 11:28 | comment | added | Robert Bryant | I'm not sure what more to say. Yes, adding $G_{xx}-G_{zz}=0$ to the other $4$ equations yields a system with empty characteristic variety, so the solutions of the $5$-equation system depend only on constants. (Note that adding $G_{xx}=0$ instead would not eliminate the characteristics.) I guess that you would like to know how to compute such things for more general systems, i.e., how to determine the generality of solutions and how Cauchy data are properly posed for overdetermined systems of PDE. There is a theory; for example, see Exterior Differential Systems (our book). | |
| Aug 4, 2011 at 6:19 | history | edited | Uri | CC BY-SA 3.0 |
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| Aug 4, 2011 at 5:52 | history | edited | Gerry Myerson | CC BY-SA 3.0 |
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| Aug 4, 2011 at 4:58 | history | rollback | Jorge Vitório Pereira |
Rollback to Revision 8
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| Aug 4, 2011 at 4:56 | history | edited | Jorge Vitório Pereira | CC BY-SA 3.0 |
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| Aug 4, 2011 at 4:51 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Aug 4, 2011 at 4:51 | history | rollback | Uri |
Rollback to Revision 5
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| Aug 4, 2011 at 4:51 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Aug 4, 2011 at 4:47 | history | edited | Uri | CC BY-SA 3.0 |
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| Aug 4, 2011 at 4:47 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Aug 4, 2011 at 4:41 | history | edited | Uri | CC BY-SA 3.0 |
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| Aug 4, 2011 at 4:35 | history | edited | Uri | CC BY-SA 3.0 |
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| Aug 3, 2011 at 22:05 | comment | added | Robert Bryant |
Yes, the problem is exactly that the plane $z=0$ is a characteristic surface for this (overdetermined) system of PDE. In fact, the surfaces on which $dz$ vanishes are precisely the domain surfaces such that no amount of information about $G$ and its derivatives along the surface will be sufficient to determine a solution $G$ in a neighborhood of the surface. Overdetermined systems don't generally break up into hyperbolic', elliptic', or `parabolic' in any natural way without knowing more about the system. This system does have real, distinct characteristics, though.
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| Aug 3, 2011 at 20:56 | history | asked | Uri | CC BY-SA 3.0 |