Infinitely many integrable/conserved quantities? Soliton?

In summary: So, the solitons behave like particles, but particles which have infinitely many internal "degrees of freedom"!In summary, "integrable systems" are important because they are integrable, meaning they can be simplified to a much simpler expression. KdV, NLS and other such systems have soliton solutions, which have infinitely many conserved quantities that are the same even after a collision. These systems behave like particles, but with infinitely many internal "degrees of freedom".
  • #1
hanson
319
0
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.
 
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  • #2
That's a rather puzzling question! "Integrable systems" are important because they are integrable! For example [itex](x+ y)dx+ (y^2+ x)dy[/itex] is integrable because it is equal to [itex]d((1/2)x^2+ xy+ (1/3)y^3)[/itex], a much simpler expression. [itex](x+ y)dx+ (y^2- x)dy[/itex] is NOT integrable: there is no function F(x,y) such that [itex]dF= (x+ y)dx+ (y^2- x)dy[/itex]. Because the first is integrable the equation [itex](x+ y)dx+ (y^2+ x)dy= 0[/itex] is equivalent to [itex]d((1/2)x^2+ xy+ (1/3)y^3)= 0[/itex] which has "solution" [itex](1/2)x^2+ xy+ (1/3)y^3= C[/itex], a constant. Because [itex](x+ y)dx+ (y^2+ x)dy[/itex] is integrable, [itex](1/2)x^2+ xy+ (1/3)y^3[/itex] is a "conserved quantity", it is a constant. Of course, things that are constant are easier to work with than things that change!
 
  • #3
hanson said:
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.
 

1. What are integrable/conserved quantities in physics?

Integrable/conserved quantities are physical quantities that do not change over time, even as the system they are a part of evolves. They are important in physics because they provide a way to describe and understand the behavior of complex systems.

2. How are integrable/conserved quantities related to solitons?

Solitons are solitary waves that maintain their shape and speed while propagating through a medium. They are closely related to integrable/conserved quantities because they are formed from the interaction of multiple integrable/conserved quantities, which allows them to maintain their stability and coherence.

3. What are some examples of integrable/conserved quantities in physics?

Some examples of integrable/conserved quantities include energy, momentum, and angular momentum. Other examples include charge, baryon number, and lepton number in particle physics, as well as vorticity and circulation in fluid mechanics.

4. How are integrable/conserved quantities and symmetries related?

Integrable/conserved quantities and symmetries are closely related in physics. In fact, the existence of integrable/conserved quantities is often a result of symmetries in a system. For example, the conservation of energy is a result of time symmetry, while the conservation of momentum is a result of spatial symmetry.

5. How do integrable/conserved quantities and solitons impact our understanding of the universe?

The study of integrable/conserved quantities and solitons has led to significant advancements in our understanding of the universe. They have been used to describe and model a wide range of phenomena in various fields of physics, from quantum mechanics to cosmology. Additionally, the discovery of new integrable/conserved quantities and solitons continues to expand our understanding of the fundamental laws that govern the universe.

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