This question concerns the Korteweg-de Vries equation.
It is known that the transform $F=f^2+f_x$ transforms $$F_t-6FF_x+F_{xxx}=0$$ into $$f_t-6f^2f_x+f_{xxx}=0$$ where $F=F(x,t), f=f(x,t)$
However, I have read that the converse does not work. In other words, given $F$, it is not necessarily true that $f$ implicitly given by $F=f^2+f_x$ is a solution to the second differential equation. Is there an example to illustrate this?
Thank you.
Anyone?