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$$\partial_t u + \partial^{3}_x u = 0$$ show that this PDE has a particular solution in the form of a plane wave, $$u_k(x,t) = \exp(-i\,w\,t - i\,k\,x)$$ if the frequency $w$ obeys the appropriate dispersion relation, $w = w(k)$.

In the past I have solved up to 2nd order PDE's using separation of variables, Sturm-Liouville method. But since the all coefficients in the PDE has different powers I am confused how to solve this problem. Seems like I can not solve this problem using the methods I used to solve 2nd order PDE's. Any help?

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  • $\begingroup$ What happens when you plug this into the PDE. Can you show that it works, provided there is a certain relation (which you must find) between $w$ and $k$? $\endgroup$
    – GEdgar
    Dec 12, 2016 at 18:09

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Since you have a linear PDE you can use the principle of superposition to form solutions with $$ u \sim \exp(−iwt−ikx) $$ then inserting the plane wave solution you obtain $$ -i\omega u + (-ik)^3u = \left(-i\omega-i^3k^3\right)u = -i(\omega -k^3)u = 0 $$ or the dispersion relation is $$ w(k) = k^3 $$ if this relation is true then you can use the plane wave above.

If you want to solve you are missing quite a few pieces of information such as initial conditions etc..

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    $\begingroup$ I get $w=k^3$. Here $i=\sqrt{-1}$ $\endgroup$ Jun 4, 2018 at 6:39
  • $\begingroup$ @AleksasDomarkas I agree. I am not sure why I did it that way - I must have had a mini brain lapse. Thanks! $\endgroup$
    – Chinny84
    Jun 4, 2018 at 14:30

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