Noncommutative KP Hierarchy Research Articles

Notes on the Boussinesq Integrable Hierarchy

This work is dedicated to some notes on the Moyal momentum algebras applied to the sl_3 Boussinesq integrable hierarchy. Starting from a brief review of the Moyal momentum algebra structures, we establish in detail the Non-commutative Boussinesq hierarchy by using the Lax pair Generating Technique. Then we shows that these equations can be obtained as 3-reduction of Non-commutative KP hierarchy in a similarly form via some conformal realizations.

Discretization of bilinear identities for the noncommutative KP and CKP hierarchies

The noncommutative KP hierarchy is known to be expressed by a bilinear identity for wavefunctions. Although the hierarchy does not seem to have any τ function, it is shown that direct discretization of the bilinear identity gives the noncommutative discrete KP equation. The bilinear identity for the noncommutative CKP hierarchy is also discretized and the noncommutative discrete CKP equation is obtained.

An extended noncommutative KP hierarchy

Introducing squared eigenfunctions in the Moyal-deformed Lax equations generates an extended noncommutative KP (ncKP) hierarchy. The compatibility equations between the ncKP flows and the extended ncKP flows and the compatibility equations among the extended ncKP flows themselves are constructed. The n-reductions and the k-constraints are analyzed, and thus, the ncKP hierarchy with self-consistent sources, the constrained ncKP hierarchy, and the constrained ncKP hierarchy with self-consistent sources are presented. Various illustrative examples are worked out, together with their Lax representations involving pseudodifferential Lax operators.

Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2 + 1 dimensions

The usual dispersionless limit of the KP hierarchy does not work in the case where the dependent variable has values in a noncommutative (e.g. matrix) algebra. Passing over to the potential KP hierarchy, there is a corresponding scaling limit in the noncommutative case, which turns out to be the hierarchy of a ‘pseudodual chiral model’ in 2 + 1 dimensions (‘pseudodual’ to a hierarchy extending Ward's (modified) integrable chiral model). Applying the scaling procedure to a method generating exact solutions of a matrix (potential) KP hierarchy from solutions of a matrix linear heat hierarchy, leads to a corresponding method that generates exact solutions of the matrix dispersionless potential KP hierarchy, i.e. the pseudodual chiral model hierarchy. We use this result to construct classes of exact solutions of the su(m) pseudodual chiral model in 2 + 1 dimensions, including various multiple lump configurations.

An algebraic scheme associated with the non-commutative KP hierarchy and some of its extensions

A well-known ansatz (`trace method') for soliton solutions turns the equations of the (noncommutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. We develop an algebraic formalism, in particular involving a (mixable) shuffle product, to explore their structure. More precisely, we show that the equations of the noncommutative KP hierarchy and its extension (xncKP) in the case of a Moyal-deformed product, as derived in previous work, correspond to identities in this algebra. Furthermore, the Moyal product is replaced by a more general associative product. This leads to a new even more general extension of the noncommutative KP hierarchy. Relations with Rota-Baxter algebras are established.

WDVV equation and triple-product relation

We study the relation between the WDVV equations and the τ-function of the noncommutative KP (NCKP) hierarchy. WDVV-like equations (Hirota triple-product relation) in the noncommutative context appear as a consequence of the nontrivial equation for τ-function of the NC KP hierarchy, while the prepotential in the Seiberg–Witten (SW) theory has been identified to the τ-function of the Whitham hierarchy. We show that the spectral curve for the SW theory is the same as the Toda-chain hierarchy. We also show explicitly that Whitham hierarchy includes commutative Toda/KP hierarchy. Further, we comment on the origin of the Hirota triple-product relation in the context of the SW theory.

Factorization methods for noncommutative KP and Toda hierarchy

We show that the solution space of the noncommutative KP hierarchy is the same as that of the commutative KP hierarchy owing to the Birkhoff decomposition of groups over the noncommutative algebra. The noncommutative Toda hierarchy is introduced. We derive the bilinear identities for the Baker--Akhiezer functions and calculate the $N$-soliton solutions of the noncommutative Toda hierarchy.