Infinitely many integrable/conserved quantities? Soliton?

[hanson]

Hi all.
I would like to know what's so special about those "integratable systems"? I heard that KdV and NLS models belong to these systems and so they have soliton solution? But why? What's the importance of this?

And what's the significance of many conserved quantities? I know, say, KdV has many conserved quantities like energy, momentum and mass. But it is natural to see these three quantities to be conserved, isn't it? Even after collision, they should still conserved, isn't it? Please kindly clear my doubts.

 

[HallsofIvy]

That's a rather puzzling question! "Integrable systems" are important because they are integrable! For example $(x+ y)dx+ (y^2+ x)dy$ is integrable because it is equal to $d((1/2)x^2+ xy+ (1/3)y^3)$, a much simpler expression. $(x+ y)dx+ (y^2- x)dy$ is NOT integrable: there is no function F(x,y) such that $dF= (x+ y)dx+ (y^2- x)dy$. Because the first is integrable the equation $(x+ y)dx+ (y^2+ x)dy= 0$ is equivalent to $d((1/2)x^2+ xy+ (1/3)y^3)= 0$ which has "solution" $(1/2)x^2+ xy+ (1/3)y^3= C$, a constant. Because $(x+ y)dx+ (y^2+ x)dy$ is integrable, $(1/2)x^2+ xy+ (1/3)y^3$ is a "conserved quantity", it is a constant. Of course, things that are constant are easier to work with than things that change!

 

[jdg812]

Hi all.
I would like to know what's so special about those "integratable systems"? I heard that KdV and NLS models belong to these systems and so they have soliton solution? But why? What's the importance of this?

And what's the significance of many conserved quantities? I know, say, KdV has many conserved quantities like energy, momentum and mass. But it is natural to see these three quantities to be conserved, isn't it? Even after collision, they should still conserved, isn't it? Please kindly clear my doubts.

Actually, KdV, NLS and other "integratable systems" have infinitely many conserved quantities, or moments. Even after collision of solitons they are the same.

Hovever, another initial condition gives another set of conserved quantities.