Linear Aspects in the Theory of Solitons and Nonlinear Integrable Equations

Abstract

In this survey we show how to obtain from the analytic structure of one-soliton solutions, the complete action angle variable representation of arbitrary multi-solitons. Special attention is paid to the interacting solitons and their relation to singularity analysis.

Structural properties of the noncommutative KdV recursion operator

“…Here we can closely follow earlier results in Ref. 7. In the short appendix, we finally verify some claims from the discussion in Sec.…”

On the Recursion Operator for the Noncommutative Burgers Hierarchy

“…The proof of hereditariness is given in the next section. The section is concluded with a verification that the hierarchy generated by the recursion operator (7) is the same as the one obtained in another manner by Kuperschmidt [11].…”

“…In the present context, the most involved and interesting part is the proof of hereditariness, a concept introduced by Fuchssteiner in [6]. Once an operator is known to be hereditary, the proof of several further important properties of the whole hierarchy generated by the operator (strong symmetry, pairwise involutivity of the flows of the hierarchy) reduce to the verification of the appropriate property for the first member in the hierarchy or of identities involving the operator (see [6,7]). On the other hand, verification that an operator is hereditary can require computations of considerable complexity even in the commutative setting (see [1,8]).…”

A novel noncommutative KdV-type equation, its recursion operator, and solitons

“…If Φ is a strong symmetry, so also the powers Φ n , n ∈ N, and we get a hierarchy of vector fields K n = Φ n K which all commute with K. For Φ hereditary, it was proved in [13] that these vector fields all commute pairwise. In all cases considered in the present article, the hierarchy can be extended taking as a base member K 0 (U ) = U x , the symmetry expressing translation invariance.…”