 #1
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Homework Statement
Given that [itex] u(x,t) [/itex] satisfies [itex] u_t6uu_x+u_{xxx}=0 [/itex] (*) and [itex] u, u_x, u_{xx} \to 0 [/itex] as [itex] x \to \infty [/itex] show that
[itex] I_1 = \int_{\infty}^{\infty}u^2dx [/itex]
is independent of time ([itex] \frac{\partial I_1}{\partial t}=0 [/itex]).
Homework Equations

The Attempt at a Solution
I guess it would suffice to show that [itex] I_1 [/itex] is equal to some function of (only) [itex] x [/itex]. I can't think of a way to do this which involves the equation (*). My attempt so far has been to do some kind of substitution so that I integrate with respect to [itex] u [/itex] rather than [itex] x [/itex]. Something like
[itex] I_1 = \int_{\infty}^{\infty}u^2dx = \int_{u(\infty,t)}^{u(\infty, t)}u^2 \frac{dx}{du} du = \int_{0}^{0} \frac{u^2}{u_x} dx = 0 [/itex]
Am I on the right track?