The generalized Korteweg-de Vries equation is


$u\_t + u\_{xxx} + (u^p)\_x=0$


for integer $p$. (The original Korteweg-de Vries equation is the case $p=2$.) I need to understand solutions for $p=1$, but I haven't been able to find this case addressed in the literature despite extensive searching. Is there a reason why $p=1$ isn't interesting? Does it reduce to a simpler form? Is it intractable?


If not, what the best way to learn about this case?


Thanks!



What are the interesting cases of the generalized Korteweg-de Vries equation?
For $p=1$ your equation indeed reduces to a simpler form. Put $X=x-t, T=t$. Then $u\_t=u\_T-u\_X$, $u\_x=u\_X$, and hence in the new independent variables $X,T$ your equation (with $p=1$) becomes a well-studied linear third-order equation
$$
u\_{T}+u\_{XXX}=0,
$$
which is, *inter alia*, a linearized version of the "standard" KdV equation with $p=2$,
see e.g. the discussion in [this book](http://books.google.com/books?id=6fr_gQfHna0C) by Ablowitz and Segur (and cf. also Willie's answer).


Note that the case of $p=3$ (the *modified* KdV equation) is also quite interesting. It is integrable by the inverse scattering transform just as the "usual" KdV and, in fact, is related to it through the Miura transformation (see e.g. the above book for details).


