Posted by: Shuanglin Shao | February 18, 2009

## Pohazav argument: no moving-to-left “solitions” for the critical gKdV

I would like to show a short but beautiful Pohazav argument to exclude “moving-to-left” solitons in the form of $u(t,x):=u(x-ct)$ for the focusing generalized Korteweg de Vries equation
$u_t+u_{xxx}+ (u^5)_x=0.$
This is to show that $c>0$.

Firstly we assume that $u(x,t)=u(x-ct)$ is a solution for the equation above and also assume that $u$ decays to $0$ in the infinity (to justify the arguments). Then
$-cu_x+u_{xxx}+(u^5)_x=0.$
Then an integration yields $-cu+u_{xx}+u^5=0. \qquad (1)$.

We multiply (1) by $u$ and integrate,
$-c\int u^2-\int u_x^2 + \int u^6=0. \qquad (2)$

We multiply (1) by $xu$ and integrate,
$-c\int xuu_x +\int xu u_{xxx}+\lambda \int xu (u^5)_x=0,$
which is simiplified to
$\frac c2 \int u^2 +\frac 32 \int u_x^2-\int u^6=0. \qquad (3)$

Now we see that $(2)\times \frac 32+(3)$ yields that
$-c\int u^2 +\frac {4}3 \int \lambda^6=0$,
This forces $c>0$.

In general, for the defocusing critical gKdV, to exclude solutions in the “solitions” sense is really a hard topic.
A decay estimate on the right is desired.