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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.

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# Infinitely many integrable/conserved quantities? Soliton?

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