Kortweg-de Vries: Parabolic PDE Homework

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In summary, the conversation discusses converting an equation to a parabolic PDE by introducing an auxiliary variable and creating a system of first-order equations. The method of introducing another variable is suggested, which would result in a system that meets the condition for a parabolic equation.
  • #1
iamkratos
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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!
 
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  • #2
So introduce another variable [tex]v=\partial_{x}p[/tex] and you have a system.
 
  • #3
How would that help? I don't get it.
Can you please elaborate?
 
  • #4
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)
 

1. What is Kortweg-de Vries equation?

Kortweg-de Vries (KdV) equation is a nonlinear partial differential equation (PDE) that describes the evolution of weakly nonlinear and dispersive waves in one dimension. It was first proposed by Dutch physicists Diederik Kortweg and Gustav de Vries in 1895 to model shallow water waves.

2. What are the applications of KdV equation?

KdV equation has many applications in various fields of physics and engineering, such as fluid dynamics, optics, and plasma physics. It is used to study the behavior of waves in shallow water, to understand the formation of rogue waves, and in the design of optical fibers for long-distance communication.

3. How is KdV equation solved?

KdV equation can be solved using various methods such as the inverse scattering transform, perturbation theory, and numerical methods. The most widely used method is the inverse scattering transform, which allows for the construction of exact solutions to the equation.

4. What are the key properties of the KdV equation?

The KdV equation has several important properties, including soliton solutions, conservation laws, and integrability. Soliton solutions are stable, localized waves that can maintain their shape and speed while traveling. Conservation laws ensure that physical quantities such as mass and energy are conserved over time. Integrability means that the equation can be solved exactly using the inverse scattering transform.

5. What are the limitations of the KdV equation?

The KdV equation is a simplified model of wave behavior and has some limitations. It does not account for wave breaking, dissipation, or dispersion in more than one dimension. It also assumes small amplitude and weakly nonlinear waves, which may not accurately describe all physical systems. Therefore, it should be used with caution and in conjunction with other models for a more comprehensive understanding of wave dynamics.

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