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.

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