In the original paper of Korteweg and de Vries, they derive the equation $$ \frac{\partial \eta}{\partial t} = \frac{3}{2}\sqrt{\frac{g}{l}} \partial_{x}(\eta^2/2 + 2\alpha \eta/3 + \sigma \partial_x^2 \eta/3) $$ where "$\alpha$ is a small but arbitrary constant, which is in close connexion with the exact velocity of the uniform motion given to the liquid".
However, in later works, it appears that $\alpha$ has either been set to zero, or set to one.
Is this constant not super meaningful? Why can it be disregarded, or set to one?
Well, after you non-dimensionalize away as many constants as you can, absorbing them into the normalizations of x and t, you note your remainder equation is mostly linear except in one term, which I isolate in the second term, together with the linear object of your puzzlement, $$ (\partial_t +\partial_x^3)\eta + \sigma \partial_x \Bigl (\eta^2/2 + 2\alpha \eta/3 \Bigr ) =0, $$ where different authors chose the freedom of this non-dimensionalization to tweak the respective constants in all kinds of ways... experiment with these options yourself.
Here, you are meant to observe that the $\alpha$ component in the second term is phony, since you may complete the square in the second term, $$ \partial_x \Bigl ( (\eta +2\alpha/3)^2 - 4\alpha^2/9\Bigr ) = \partial_x (\eta +2\alpha/3)^2 , $$ so redefining $u=\eta+2\alpha/3$ completely eliminates it from the problem: it is a phantom parameter,
$$ u_t+u_{xxx} + \sigma u u_x=0 . $$