R-matrix Hierarchies

Abstract

Almost all (perhaps, all) known integrable systems possess Lax representations. In the situation of systems described by ordinary differential equations, a Lax representation for a given system means that there exist two maps L: P→g and P→g from the system’s phase space P into some Lie algebrag such that the equations of motion are equivalent to

$$ \dot{L} = \left[ {L,B} \right]. $$

(2.1.1)

(In fact, as a rule, there exist several different Lax representations, with different algebras g; numerous examples of this circumstance will be given in this book.) The matrix L, or, better, the map g is called the Lax matrix, while the matrix B is called the auxiliary matrix of the Lax representation. The pair (L,B) is called the Lax pair (and sometimes one uses, somewhat loosely, this term for the equation (2.1.1) itself). Finding a Lax representation for a given system usually implies its integrability, due to the fact that Ad-invariant functions on the Lie algebra g are integrals of motion of the systems of the type (2.1.1), and therefore the values of such functions composed with the map L deliver functions on P serving as integrals of motion of the original system. of course, in the Hamiltonian context, there remains something to be done in order to establish the complete integrability, namely, to show that the number of functionally independent integrals thus found is large enough, and that they are in involution. There exists an approach which incorporates an involutivity property in the very construction of Lax equations, namely the r-matrix approach.