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

Abstract

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 twosoliton solutions.

A new bidirectional generalization of (2+1)-dimensional matrix k-constrained Kadomtsev-Petviashvili hierarchy

“…Elaborated dressing technique method provides us also with a possibility to investigate exact solutions of matrix equations that (2+1)-BDk-cKP hierarchy (30) contains. We shall note that the interest to noncommutative equations has also arisen recently [52][53][54][55] . Thus one of problems for further investigation is the generalization of (2+1)-BDk-cKP hierarchy to the case of noncommutative algebras 5,52-55 (namely, the elements q, r, u j and v i belong to some noncommutative ring).…”

Bäcklund transformations and the Atiyah–Ward ansatz for non-commutative anti-self-dual Yang–Mills equations

“…This approach gives both finite action solutions (instantons) and infinite action solutions (such as nonlinear plane waves). The solutions obtained are written in terms of quasideterminants (Gelfand and Retakh (1991); Gelfand and Retakh (1992)) which appear also in the construction of exact soliton solutions in lower-dimensional noncommutative integrable equations such as the Toda equation (Etingof, Gelfand and Retakh (1997); Etingof, Gelfand and Retakh (1998); Li and Nimmo (2008); ), the KP and KdV equations (Dimakis and Müller-Hoissen (2007); Etingof, Gelfand and Retakh (1997); ; Hamanaka (2007)), the Hirota-Miwa equation (Gilson, Nimmo and Ohta (2007); Li, Nimmo and Tamizhmani (2009);Nimmo (2006)), the mKP equation (Gilson, Nimmo and Sooman (2008a); Gilson, Nimmo and Sooman (2008b)), the Schrödinger equation (Goncharenko and Veselov (1998); Samsonov and Pecheritsin (2004)), the Davey-Stewartson equation (Gilson and Macfarlane (2009)), the dispersionless equation (Hassan (2009)), and the chiral model (Haider and Hassan (2008)), where they play the role that determinants do in the corresponding commutative integrable systems. We also clarify the origin of the results from the viewpoint of noncommutative twistor theory by using noncommutative Penrose-Ward correspondence or by solving a noncommutative Riemann-Hilbert problem.…”

A new (1+1)-dimensional matrix k-constrained KP hierarchy

“…From Corollary 4 we see that the functions Φ = ϕ∆ −1 and Ψ= ψ∆ ⊤,−1 where ∆ = C + D −1 {ψ ⊤ ϕ} (see formulae (43)) satisfy equations (63). After the change q := Φ, r := Ψ, 63) and ( 69) we obtain that N × K-matrix functions q, r, an N × N-matrix function v 0 , a K × K-matrix function S = 2α(∆ −1 ) t and a K × Kmatrix M 0 satisfy equations (27).…”

“…Assume that functions ϕ and ψ satisfy problems (39) and (52). Then the functions Φ = W {ϕ}C −1 and Ψ = W −1,τ {ψ}C ⊤,−1 (see formulae (43)) satisfy the equations:…”

“…where S 1 = µ −1 v 0 + c 1 S. This is the well-known stationary Davey-Stewartson system (DS-I) and ( 28) is therefore a matrix (noncommutative) generalization. The interest in noncommutative versions of DS systems and some other noncommutative nonlinear equations (in particular, solution generating technique) has also arisen recently in [41][42][43][44] .…”