Kortweg-de Vries: Parabolic PDE Homework

[iamkratos]

Homework Statement

The equation is u_t + uu_x + u_xxx = 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 = u_x, etc.

The Attempt at a Solution

So i took p = u_x

doesn't that just give me:

u_t + pu + p_xx = 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!

 

[hunt_mat]

So introduce another variable $$v=\partial_{x}p$$ and you have a system.

 

[iamkratos]

How would that help? I don't get it.
Can you please elaborate?

 

[hunt_mat]

Your system of equations is:
$$\begin{array}{ccc} v & = & \partial_{p} \\ p & = & \partial_{x}u \\ \partial_{t}u+pu+\partial_{x}v & = & 0 \end{array}$$

This system can be written in the form:

$$\mathbf{A}\partial_{t}\mathbf{U}+\mathbf{B}\partial_{x}\mathbf{U}=\mathbf{c}$$

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