Mastersymmetries, angle variables, and recursion operator of the relativistic Toda lattice

Abstract
Conserved quantities, bi-Hamiltonian formulation, and recursive structure of the relativistic Toda lattice (RT) are obtained in an algorithmic way without making use of the Lax representation. Furthermore, for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries. A new hierarchy of completely integrable systems is discovered, which turns out to correspond to the ‘‘negative’’ of the hierarchy of RT. Thus it is shown that the full algebra of time-dependent symmetry group generators for each member of the RT hierarchy is isomorphic to the algebra of first order differential operators with Laurent polynomials as coefficients. The surprising phenomenon is revealed that the members of the RT hierarchy are connected to their negative counterparts by explicit Bäcklund transformation.