Timeline for Coupled semilinear PDEs
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 4, 2010 at 19:26 | vote | accept | user9728 | ||
| Oct 3, 2010 at 7:55 | answer | added | Denis Serre | timeline score: 3 | |
| Oct 2, 2010 at 3:43 | comment | added | Deane Yang | There is a systematic theory known as Cartan-Kahler theory that is useful for analyzing a system like this. But for a system like this you can probably do it by hand. Basically, you should differentiate all of the equations and try to find all possible linear combinations of these second order equations that cause all of the second derivatives to disappear. By doing this, you will derive additional first order equations. You will find that either there are no solutions at all, or you can find a solution as I describe above. | |
| Oct 2, 2010 at 2:20 | comment | added | Deane Yang | Without the integrability condition this is just an ode along the vector field $\partial_x + a\partial_y$ | |
| Oct 2, 2010 at 1:56 | comment | added | Deane Yang | Given the integrability condition, this is an overdetermined system. If you substitute for $q_x$ in PDE2 using the integrability condition, you get an ODE in $y$ only. So you can specify initial data along the curve $x = 0$ that satisfies the ODE and then use PDE1 and PDE2 to propagate the solution in $x$. However, you need to verify that the integratibility condition is preserved by PDE1 and PDE2. | |
| Oct 2, 2010 at 1:42 | comment | added | Willie Wong | Using the integrability constraint, your equation reduces to $\partial_x^2 z + a^2 \partial_y^2 z = b (\partial_y z + a \partial_x z)$, which is elliptic type. So I assume you are prescribing some sort of boundary data? | |
| Oct 2, 2010 at 1:35 | comment | added | Willie Wong | How is this semilinear? Ignoring the integrability constraint it looks rather linear to me. | |
| Oct 2, 2010 at 0:59 | history | asked | user9728 | CC BY-SA 2.5 |