Noncommutative KP Equation Research Articles

Noncommutative versions of the KP and modified KP equations with self-consistent sources

The noncommutative KP equation with self-consistent sources (nc KPESCS) and the noncommutative modified KP equation with self-consistent sources (nc mKPESCS) are proposed and solved. Two types of quasideterminant solutions are presented and proved by general derivative formulae for each of them, respectively. Furthermore, Miura transformation is established between the nc KPESCS and the nc mKPESCS. As a byproduct, a useful quasideterminant identity is developed.

On integrability of a noncommutative q-difference two-dimensional Toda lattice equation

In our previous work (C.X. Li and J.J.C. Nimmo, 2009 [18]), we presented a generalized type of Darboux transformations in terms of a twisted derivation in a unified form. The twisted derivation includes ordinary derivatives, forward difference operators, super derivatives and q-difference operators as its special cases. This result not only enables one to recover the known Darboux transformations and quasideterminant solutions to the noncommutative KP equation, the non-Abelian two-dimensional Toda lattice equation, the non-Abelian Hirota–Miwa equation and the super KdV equation, but also inspires us to investigate quasideterminant solutions to q-difference soliton equations. In this paper, we first construct the bilinear Bäcklund transformations for the known bilinear q-difference two-dimensional Toda lattice equation (q-2DTL) and then derive a Lax pair whose compatibility gives a formally different nonlinear q-2DTL equation and finally obtain its quasideterminant solutions by iterating its Darboux transformations.

Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation

We consider matrix solutions of a noncommutative KP and a noncommutative mKP equation; these solutions can be expressed as quasideterminants. In particular, we investigate the interaction of two-soliton solutions.

On solutions to the non-Abelian Hirota–Miwa equation and its continuum limits

In this paper, we construct Grammian-like quasideterminant solutions of a non-Abelian Hirota–Miwa equation. Through continuum limits of this non-Abelian Hirota–Miwa equation and its quasideterminant solutions, we construct a cascade of non-commutative differential-difference equations ending with the non-commutative KP equation. For each of these systems, the quasideterminant solutions are constructed as well.

On a direct approach to quasideterminant solutions of a noncommutative modified KP equation

A noncommutative version of the modified KP equation and a family of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux transformations and the solutions are verified directly. We also verify directly an explicit connection between quasideterminant solutions of the noncommutative mKP equation and the noncommutative KP equation arising from the Miura transformation.

On a direct approach to quasideterminant solutions of a noncommutative KP equation

A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it is shown that these solutions may also be verified directly. This approach is reminiscent of the wronskian technique used for the Hirota bilinear form of the regular, commutative KP equation but, in the noncommutative case, no bilinearizing transformation is available.