I would like to show a short but beautiful Pohazav argument to exclude “moving-to-left” solitons in the form of for the focusing generalized Korteweg de Vries equation

This is to show that .

Firstly we assume that is a solution for the equation above and also assume that decays to in the infinity (to justify the arguments). Then

Then an integration yields .

We multiply (1) by and integrate,

We multiply (1) by and integrate,

which is simiplified to

Now we see that yields that

,

This forces .

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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