Bäcklund transformations: a tool to study Abelian and non-Abelian nonlinear evolution equations

Abstract

The KdV eigenfunction equation is considered: some explicit solutions are constructed. These, to the best of the authors' knowledge, new solutions represent an example of the powerfulness of the method devised. Specifically, Bäcklund transformation are applied to reveal algebraic properties enjoyed by nonlinear evolution equations they connect. Indeed, Bäcklund transformations, well known to represent a key tool in the study of nonlinear evolution equations, are shown to allow the construction of a net of nonlinear links, termed Bäcklund chart, connecting Abelian as well as non Abelian equations. The present study concerns third order nonlinear evolution equations which are all connected to the KdV equation. In particular, the Abelian wide Bäcklund chart connecting these nonlinear evolution equations is recalled. Then, the links, originally established in the case of Abelian equations, are shown to conserve their validity when non Abelian counterparts are considered. In addition, the non-commutative case reveals a richer structure related to the multiplicity of non-Abelian equations which correspond to the same Abelian one. Reduction from the nc to the commutative case allow to show the connection of the KdV equation with KdV eigenfunction equation, in the scalar case. Finally, recently obtained matrix solutions of the mKdV equations are recalled.

The Algebraic Structure of the Non-Commutative Nonlinear Schrodinger and Modified Korteweg-De Vries Hierarchy

“…Note that the polynomial partial differential field includes even degree terms. Such alternative forms can be obtained from non-commutative modified Korteweg-De Vries equation via a suitable gauge transformation as, for example, outlined in detail in Carillo and Schiebold [23]. The combinatorial algebraic structure we have developed would seem a natural context to investigate such alternative hierarchy forms further.…”

“…If we follow the sub-diagonal with the view of retaining non-zero terms along it, we observe we meet a obstruction in the first 4 × 4 block with matrix A 2 characterised by the composition component 32. The problem is that while the sub-diagonal of A 2 has non-zero entries, the next term along the diagonal that lies in the last column of that A 2 block, but beneath the entire block, and in fact the row corresponding to the basis element [23] × (1, 0, 0, 0) is zero. However there is a quick fix to this obstruction.…”

“…We can see from Tables 3 and 4 that this column-swap procedure would guarantee that the next entry in the sub-diagonal would be non-zero. We can then continue to consider the sub-diagonal in the A 2 block corresponding to the basis elements [23] × β i for i = 1, 2, 3, 4. However we observe a similar obstruction necessitating an anologous swap of the columns of A corresponding to the final two columns of this second A 4 block.…”