Action-Angle Representation of Multisolitons by Potentials of Mastersymmetries

Abstract

395Using the algebra of symmetries/mastersymmetries a purely algebraic construction for the action/angle representatibn of multisolitons is given. By the 'same method an explicit construction of the potentials of the eigenstates of the recursion operator is perforined in terms of partial derivatives of a fundamental scalar field.

Multi-Hamiltonian Theory of Dynamical Systems

“…In the submanifold MN we are now going to give a new parametrization in the following way [23]' [145]. Define a map II which assigns to each UN the set of asymptotic data (ql, ... ,qN,Cl"",CN) or (reqi, ... ,imcN)' Of course, this map cannot be written down explicitly; however this is not necessary for our further considerations.…”

Unified approach to action-angle representation of real and complex multisolitons

“…So, in particular, these systems are translation invariant systems. Analogous to [5], [6], [10] we assume that the following conditions hold:…”

“…For real-analytic integrable equations in (1 + 1)-dimensions, which were considered as flows on infinite dimensional manifolds, we constructed in [5], [6], [10] the action/angle variables for multisoliton solutions. Using only the algebraic properties of symmetries and mastersymmetries for these equations, it was possible to give explicit formulas for the action/angle variables, and their corresponding hamiltonian vector fields in terms of the physical field variable.…”

“…In this letter we will examine the structure of those complex equations, which, like the Nonlinear Schrödinger Equation (NLS), involve complex conjugation in their tangent fields. Complex conjugation is not an analytic operation, hence these equations are not in the class of equations considered in [5], [6], [10]. Since these equations necessarily involve complex quantities we term them complex equations.…”

“…In most cases, these complex equations can be considered as reductions of complex-analytic systems. Therefore in chapter 1 we will briefly present results for integrable evolution equations on complex manifolds analogous to the ones given in [10]. In chapter 2 the notion of extended systems as a special case of complex-analytic equations on complex manifolds is introduced.…”