A generalization of the Lax equation (Q5943186)

The original Lax scheme replaces the Korteweg-de Vries equation \(u_t= 6uu_x- u_{xxx}\) with a certain operator equation \(L_t=[N,L]\) and then the spectrum of the operator \(L=\partial^2_x -u(t,x)\) provides first integrals of the original KdV equation. An analogous scheme is proposed by employing Lie groups: Let \({\mathcal M}\) be a matrix differential manifold, \({\mathcal G}\subset Gl (n)\) a closed Lie matrix subgroup with Lie algebra \({\mathfrak g}\), \(\rho:{\mathfrak g} \times {\mathcal M}\to T{\mathcal M}((N,L) \mapsto\rho (N,L))\) an action of \({\mathfrak g}\) on \(M\); we will denote \(\rho_N(L)= \rho(N,L)\in T{\mathcal M}\). Then the initial value problem (*) \(\partial L/ \partial t=\rho_{N(t,L)} (L)\), \(L(0)=L_0\) (where \(L,L_0\in {\mathcal M}\) and \(N=N(t,L) \in{\mathfrak g})\) has a unique solution \(L(t)= \Phi_{G(t)} (L_0)\) (where \(\Phi\) is the action of \({\mathcal G}\) on \({\mathcal M}\) corresponding to the action \(\rho\) of \({\mathfrak g}\), and \(G(t)\in {\mathcal G}\) satisfies \(\partial G/ \partial t=N(t,L)G\), \(G(0)=1)\). On the other hand, the equation (*) may be expressed as a certain dynamical system \(dx^i/dt= F^i(t,x^1, \dots, x^n)\); \(i=1, \dots,n\), such that the \({\mathcal G}\)-invariant functions are first integrals of (*).NEWLINENEWLINENEWLINENumerous examples are stated and the Hamiltonian structures generated by certain Poisson brackets (the Sklyanin brackets) are determined.