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KdV equation and classical linear wave equation

Like we know, the standard form of KdV equation is $$u_{t}-6uu_{x}+u_{xxx}=0,\tag{1}$$ where this equation describes a solitary wave propagation and $u=u(x,t)$. On the other hand, we know the ...
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0answers
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Relation between solutions to Yang-Baxter equations, integrability and exact solvability?

Wikipedia mentions that there is an implication: Yang-Baxter solutions yield integrable models, what 1D systems concerns. In arbitrary dimensions, what is the relation, if any, between solutions to ...
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2answers
123 views

Learning about super-symmetric quantum mechanics and integrable systems

I'm an undergraduate interested in theory. I recently asked one of my professors, a physicist in the particle theory group at my school, if he'd be willing to take on an undergraduate for a senior ...
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1answer
59 views

Idea of integrable systems

I do not quite understand the idea an integrable dynamical system. Does it mean that the EOMs are analytically and exactly solvable? What are the necessary and sufficient conditions such that a system ...
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0answers
65 views

Liouville's theorem on integrable Hamiltonian systems is a let down

I read a proof of Liouville’s theorem on integrable Hamiltonian systems. According to the theorem, an autonomous Hamiltonian system can be integrated in quadratures, given $n$ involutive first ...
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2answers
125 views

Reference request for exactly solved models

Can someone recommend a textbook or review article that covers exactly solved models in statistical mechanics, such as the six- or eight-vertex models? If there is literature at the undergraduate ...
3
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1answer
119 views

Relation between (super)integrability and closed orbits

Inspired by this recent question, I would like to understand from a more general and mathematical perspective why closed orbits are only found for the Kepler ($V(r) \sim 1/r$) or harmonic ($V(r) \sim ...
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2answers
483 views

What is precisely a Yangian symmetry?

The terms Yangian and Yangian symmetry appear in a list of physical problems (spin chains, Hubbard model, ABJM theory, $\mathcal{N}= 4$ super Yang-Mills in $d=4$, $\mathcal{N}= 8$ SUGRA in $d=4$), ...
3
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1answer
183 views

Undefined amplitudes in the Coordinate Bethe Ansatz for the XXX model?

Rather specific question for someone familiar with the Coordinate Bethe Ansatz... I am considering the Heisenberg XXX-model, consisting of a one-dimensional chain of L sites with a spin-1/2 particle ...
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1answer
271 views

Is integrability necessary for the Amplituhedron?

It is well known that there exist mappings between operators in N = 4 Super Yang–Mills and spin chain states making the theory Bethe Ansatz integrable. Is integrability a necessity for the ...
3
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2answers
237 views

What is the definition of a quantum integrable model?

What is the definition of a quantum integrable model? To be specific: given a quantum Hamiltonian, what makes it integrable?
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0answers
107 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
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473 views

Linear sigma models and integrable systems

I'm a mathematician who recently became very interested in questions related to mathematical physics but somehow I have already difficulties in penetrating the literature... I'd highly appreciate any ...
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1answer
172 views

Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
6
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1answer
867 views

Constants of motion vs. integrals of motion

Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of ...
6
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4answers
486 views

Non-Integrable systems

Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities (n being the number of degrees of freedom), or n whose Poisson brackets with each other ...
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3answers
323 views

A nonintegrable quantum system whose classical limit is integrable?

In this discussion: http://chat.stackexchange.com/rooms/4243/discussion-between-arnold-neumaier-and-ron-maimon Arnold Neumaier suggested that there might be a close link between classical and quantum ...
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442 views

What is a good introduction to integrable models in physics?

I would be interested in a good mathematician-friendly introduction to integrable models in physics, either a book or expository article. Related MathOverflow question: what-is-an-integrable-system.