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What does "no explicit time dependence" mean in this context? :

A symmetry of the KdV is given by $$\tilde x=x, \tilde t=t+\epsilon, \tilde u =u$$ as there is no explicit time dependence in the KdV.

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In Physics, the notion of explicit time dependence denotes equations where the time parameter t occurs "freely" and not only as a $\frac{d}{dt}$. So an equation of the form $v(x)=a_0t+v_0$ is explicitly time dependent, but $a(x)=a_0$ is not.

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Thank you, Dominik. Just to confirm my understanding of your explanation: the partial derivative wrt $t$ does not count as explicitly time-dependent, only an actual factor of $t$ in the equation does? –  Henry Nov 24 '12 at 19:59
    
That's right. Another commonly used word for this is that the equation is autonomous, a word which for unfathomable reasons doesn't appear in the wikipedia article on differential equations. –  Harald Hanche-Olsen Nov 24 '12 at 20:24
    
Thank you, @HaraldHanche-Olsen ! –  Henry Nov 24 '12 at 21:45

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