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

Gilson, Claire R.; Nimmo, J. J. C.; Sooman, C M

doi:10.1088/1751-8113/41/8/085202

Cited by 30 publications

(29 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Let us mention that in the early 1990 the Gelf'and school [51] already noticed the role quasi-determinants for some integrable systems, see also [94] for some recent work in this direction regarding non-Abelian Toda and Painlevé II equations. Jon Nimmo and his collaborators, the Glasgow school, have studied the relation of quasi-determinants and integrable systems, in particular we can mention the papers [55,56,67,54,68]; in this direction see also [58,124,59]. All this paved the route, using the connection with orthogonal polynomials a la Cholesky, to the appearance of quasi-determinants in the multivariate orthogonality context.…”

Section: 2mentioning

confidence: 99%

Multivariate orthogonal polynomials and integrable systems

Ariznabarreta

Mañas

2016

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, its second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants -as well as Schur complements-of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasideterminants, lead to expressions for the multivariate orthogonal polynomials and its second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm those that relate the Baker and adjoint Baker functions with ratios of Miwa shifted τ -functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R D , of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involves only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials are given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

show abstract

Section: 2mentioning

confidence: 99%

Multivariate orthogonal polynomials and integrable systems

Ariznabarreta

Mañas

2016

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2009

View full text Add to dashboard Cite

We present Bäcklund transformations for the non-commutative anti-self-dual YangMills equation where the gauge group is G = GL(2), and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants. We also explain the origins of all the ingredients of the Bäcklund transformations within the framework of non-commutative twistor theory. In particular, we show that the generated solutions belong to a non-commutative version of the Atiyah-Ward ansatz.

show abstract

“…Recently, there has been a lot of interest in non-commutative versions of some well-known soliton equations, such as the KP equation, the KdV equation, the Hirota-Miwa equation, the modified KP equation and the twodimensional Toda lattice [1,[5][6][7][8][9]12,14,16,18,19,[21][22][23]. There are a number of reasons for this lack of commutativity.…”

Section: Introductionmentioning

confidence: 99%