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Kortweg-de Vries PDE

  • Thread starter iamkratos
  • Start date
  • #1
7
0

Homework Statement



The equation is ut + uux + uxxx = 0

I need to show that this is a parabolic pde.

Homework Equations



Hint : convert to an equivalent system of 1st order equations by introducing an auxiliary variable p = ux, etc.

The Attempt at a Solution



So i took p = ux

doesn't that just give me:

ut + pu + pxx = 0

This i think is a parabolic pde by inspection.

But the hint says i need to get a system of 1st order equations. What am i missing? I am pretty sure I've made a giant error. Help please!
 

Answers and Replies

  • #2
hunt_mat
Homework Helper
1,739
18
So introduce another variable [tex]v=\partial_{x}p[/tex] and you have a system.
 
  • #3
7
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How would that help? I don't get it.
Can you please elaborate?
 
  • #4
hunt_mat
Homework Helper
1,739
18
Your system of equations is:
[tex]
\begin{array}{ccc}
v & = & \partial_{p} \\
p & = & \partial_{x}u \\
\partial_{t}u+pu+\partial_{x}v & = & 0
\end{array}
[/tex]

This system can be written in the form:

[tex]
\mathbf{A}\partial_{t}\mathbf{U}+\mathbf{B}\partial_{x}\mathbf{U}=\mathbf{c}
[/tex]

Now the condition for parabolic equation comes in with the determinants of A and B (look this up, this should be in your notes)
 

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