One of the most interesting freatures of the KdV is the existence of
infinitely many conservation laws. Lets begin with some basics of
conservation laws. If we can write our equation of the form
Then we can integrate this equation from [math]\displaystyle{ -\infty }[/math] to [math]\displaystyle{ \infty }[/math] to obtain
[math]\displaystyle{
\int_{-\infty }^{\infty }\partial _{t}T\left( u\right) \mathrm{d} x = -\int_{-\infty
}^{\infty }\partial _{x}X\left( u\right) \mathrm{d} x
}[/math]
The second integral will be zero if [math]\displaystyle{ X(u)\rightarrow 0 }[/math] as [math]\displaystyle{ x\rightarrow \pm
\infty . }[/math] Therefore
[math]\displaystyle{
\int_{-\infty }^{\infty }u^{2}\mathrm{d} x
}[/math]
must be conserved. This corresponds to conservation of momentum. It turns out
that there is an infinite number of conserved quantities and we give here
the proof of this.
Since we can rewrite the equation into the conservation form [math]\displaystyle{ \partial _{t}w+\partial_{x}\left(3w^2-2\varepsilon ^{2}w^{3}+\partial _{x}^{2}w\right)=0 }[/math], then
[math]\displaystyle{
w_{2} = \partial_{x}^2 u + u^{2}
}[/math]
[math]\displaystyle{
w_{3} = \partial _{x}^3 u + 4 u \partial_{x} u = \partial _{x}(\partial _{x}^2 u + 2 u^2)
}[/math]
[math]\displaystyle{
w_{4} = \partial _{x}^4 u + 6 u \partial_{x}^2 u + 5 (\partial_{x} u)^2 + 2 u^3 = \partial _{x}(\partial _{x}^3 u + 6 u \partial_{x} u) + 2 u^3 - (\partial_{x} u)^2
}[/math]
Note that each of the odd conservation laws ([math]\displaystyle{ w_1, w_3 }[/math] etc.) are just an [math]\displaystyle{ \partial _{x} }[/math] of some [math]\displaystyle{ X(u) }[/math] and therefore does not actually provide a conservation law.
As [math]\displaystyle{ \int_{-\infty }^{\infty }\partial_{x} X(u) \mathrm{d} x = 0 }[/math], [math]\displaystyle{ w_0 }[/math], [math]\displaystyle{ w_2 }[/math], and [math]\displaystyle{ w_4 }[/math] correspond to conservation of [math]\displaystyle{ u }[/math] (mass), [math]\displaystyle{ u^2 }[/math] (momentum), and [math]\displaystyle{ 2 u^3 - (\partial_{x} u)^2 }[/math] (energy).
