Verifying Galilean Invariance of the KdV Equation

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Homework Help Overview

The discussion revolves around demonstrating the Galilean invariance of the Korteweg-de Vries (KdV) equation, specifically the equation ut + 6uux + uxxx = 0, under a specified transformation involving changes in variables.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of the chain rule in the context of variable transformation to verify the invariance of the KdV equation. There is a focus on how to express derivatives after the transformation.

Discussion Status

Some participants have provided guidance on the approach to take, specifically suggesting the use of the chain rule and variable changes. The discussion appears to be progressing with attempts to clarify the mathematical steps involved.

Contextual Notes

Participants are working within the constraints of a homework assignment that requires them to show invariance without providing a complete solution or final verification.

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Homework Statement


Show that the KdV has Galilean invariance.
That is ut + 6uux + uxxx = 0 is invariant under the transformation xi = x - ct, tau = t, psi = phi - c/6


Homework Equations





The Attempt at a Solution


Do we use the chain rule on these and plug into the KdV?
 
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That would be what I would do. Essentially you just want to make that "change" of variables and show that you get exactly the same equation again. And changing variables in a differential equation involves the chain rule.
 
Thanks.
So d/dx = xix*d/dxi + taux*d/dtau = d/dxi
and d/dt = xit*d/dxi + taut*d/dtau = -cd/dxi + d/dtau
 
Got it.
Thanks.
 

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