Integrable system

Finite degree of freedom

It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional symplectic manifold on which the Hamiltonian vector field is defined, it is possible to find $n$ integrals of motion in involution. Moreover, they are, locally, superintegrable systems, in the sense that they have $(2n-1)$ first integral of motions.

Then, the question of integrability and superintegrability it is interesting only if we:

• make it a global question (Liouville integrability), or
• still in a local set up, restrict the class of "allowed functions", for example, to polynomial functions.

An special case consists of that systems which can be expressed with a Lax pair.

Liouville integrability

The classical Hamiltonian system $(M,\omega ,H)$ is said to be Liouville integrable (or completely integrable) if there exists a set of $n$ smooth functions $f_1,f_2,\dots ,f_n:M\to \mathbb{R}$ satisfying the following conditions:

1. Functional Independence:
   The functions $f_1,f_2,\dots ,f_n$ are functionally independent, meaning their differentials $d{f}_1,d{f}_2,\dots ,d{f}_n$ are linearly independent almost everywhere on $M$. This implies that the map $F=(f_1,f_2,\dots ,f_n):M\to {\mathbb{R}}^n$ is a submersion on an open dense subset of $M$.
2. Involution:
   The functions $f_1,f_2,\dots ,f_n$ are in involution with respect to the Poisson bracket induced by $\omega$. That is, for all $i,j=1,2,\dots ,n$,$$\{f_i,f_j\}=0,$$where the Poisson bracket $\{\cdot ,\cdot \}$ is defined by:$$\{f,g\}=\omega (X_f,X_g),$$and $X_f,X_g$ are the Hamiltonian vector fields associated with $f$ and $g$, respectively.
3. Commutation with the Hamiltonian:
   Each function $f_i$ Poisson-commutes with the Hamiltonian $H$, i.e.,$$\{f_i,H\}=0\text{for all }i=1,2,\dots ,n.$$This implies that the functions $f_i$ are constants of motion (or first integrals) of the system.

Liouville Foliation and Invariant Manifolds

The set of functions $F=(f_1,f_2,\dots ,f_n)$ defines a Liouville foliation of the phase space $M$. Specifically:

• The level sets of $F$, given by:

  $$M_c=\{x\in M\mid f_i(x)=c_i\text{ for all }i=1,2,\dots ,n\},$$

  where $c=(c_1,c_2,\dots ,c_n)\in {\mathbb{R}}^n$, are invariant submanifolds of $M$. This means that the Hamiltonian flow preserves $M_c$, i.e., if a trajectory starts on $M_c$, it remains on $M_c$ for all time.

• The tangent space to $M_c$ at any point is spanned by the Hamiltonian vector fields $X_{f_1},X_{f_2},\dots ,X_{f_n}$. These vector fields are tangent to $M_c$ because the $f_i$ are constants of motion.

Infinite degree of freedom (fields)

(According to this)
Many nonlinear PDEs possess families of traveling wave, periodic and solitary wave solutions. However, the addition of integrability endows an equation with a much deeper mathematical structure. In turn, integrability provides one with corresponding mathematical methods to probe more deeply the structure of the solution space, which allows one to do something quite rare in the study of PDEs: explicitly write down a large family of physically meaningful solutions that can be observed in nature and study in detail their dynamical behavior.

In the context of PDEs, integrable systems could generally be viewed as those equations that can be obtained from an overdetermined system of linear differential equations, with the original PDE playing the role of the compatibility condition. The overdetermined linear system is the Lax pair associated to the equation. If such a Lax pair representation is known for a PDE, then many methods of analysis can be applied to study the PDE. In particular, methods such as the Inverse Scattering Method (IST), which is a generalization of the Fourier transform to non-linear PDE and is an effective solution method for initial value problems.

Related: evolution equation.