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Folkert Muller-Hoissen

Publications and source records attributed to Folkert Muller-Hoissen.

At least 19 recordsLinked to original sources
arXiv2010

Solutions of matrix NLS systems and their discretisations: A unified treatment

Using a bidifferential graded algebra approach to integrable partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework, by specifying a representation of the bidifferential graded algebra and imposing a reduction. By application of a general result, corresponding families of exact solutions are obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The solutions are parametrised in terms of constant matrix data subject to a Sylvester equation (which previously appeared as a rank condition in the integrable systems literature). These data exhibit a certain redundancy, which we diminish to a large extent. More precisely, we first consider more general AKNS-type systems from which two different matrix NLS systems emerge via reductions. In the continuous case, the familiar Hermitian conjugation reduction leads to a continuous matrix (including vector) NLS equation, but it is well-known that this does not work as well in the discrete cases. On the other hand there is a complex conjugation reduction, which apparently has not been studied previously. It leads to square matrix NLS systems, but works in all three cases (continuous, semi- and fully-discrete). A large part of this work is devoted to an exploration of the corresponding solutions, in particular regularity and asymptotic behaviour of matrix soliton solutions.

nlin.SI↗
arXiv2009

Quasi-symmetric functions and the KP hierarchy

Quasi-symmetric functions show up in an approach to solve the Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product and the outer coproduct. In particular, supplied with this new product and the outer coproduct, the algebra of quasi-symmetric functions becomes an infinitesimal bialgebra. Using these results we derive a sequence of identities in the algebra of quasi-symmetric functions that are in formal correspondence with the equations of the KP hierarchy.

math-ph↗
arXiv2008

BKP and CKP revisited: The odd KP system

Restricting a linear system for the KP hierarchy to those independent variables t\_n with odd n, its compatibility (Zakharov-Shabat conditions) leads to the "odd KP hierarchy". The latter consists of pairs of equations for two dependent variables, taking values in a (typically noncommutative) associative algebra. If the algebra is commutative, the odd KP hierarchy is known to admit reductions to the BKP and the CKP hierarchy. We approach the odd KP hierarchy and its relation to BKP and CKP in different ways, and address the question whether noncommutative versions of the BKP and the CKP equation (and some of their reductions) exist. In particular, we derive a functional representation of a linear system for the odd KP hierarchy, which in the commutative case produces functional representations of the BKP and CKP hierarchies in terms of a tau function. Furthermore, we consider a functional representation of the KP hierarchy that involves a second (auxiliary) dependent variable and features the odd KP hierarchy directly as a subhierarchy. A method to generate large classes of exact solutions to the KP hierarchy from solutions to a linear matrix ODE system, via a hierarchy of matrix Riccati equations, then also applies to the odd KP hierarchy, and this in turn can be exploited, in particular, to obtain solutions to the BKP and CKP hierarchies.

nlin.SI↗
arXiv2008

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.

nlin.SI↗
arXiv2008

From the Kadomtsev-Petviashvili equation halfway to Ward's chiral model

The "pseudodual" of Ward's modified chiral model is a dispersionless limit of the matrix Kadomtsev-Petviashvili (KP) equation. This relation allows to carry solution techniques from KP over to the former model. In particular, lump solutions of the su(m) model with rather complex interaction patterns are reached in this way. We present a new example.

nlin.SI↗
arXiv2008

Weakly nonassociative algebras, Riccati and KP hierarchies

It has recently been observed that certain nonassociative algebras (called "weakly nonassociative", WNA) determine, via a universal hierarchy of ordinary differential equations, solutions of the KP hierarchy with dependent variable in an associative subalgebra (the middle nucleus). We recall central results and consider a class of WNA algebras for which the hierarchy of ODEs reduces to a matrix Riccati hierarchy, which can be easily solved. The resulting solutions of a matrix KP hierarchy then determine (under a rank 1 condition) solutions of the scalar KP hierarchy. We extend these results to the discrete KP hierarchy. Moreover, we build a bridge from the WNA framework to the Gelfand-Dickey formulation of the KP hierarchy.

nlin.SI↗
arXiv2007

A new approach to deformation equations of noncommutative KP hierarchies

Partly inspired by Sato's theory of the Kadomtsev-Petviashvili (KP) hierarchy, we start with a quite general hierarchy of linear ordinary differential equations in a space of matrices and derive from it a matrix Riccati hierarchy. The latter is then shown to exhibit an underlying 'weakly nonassociative' (WNA) algebra structure, from which we can conclude, refering to previous work, that any solution of the Riccati system also solves the potential KP hierarchy (in the corresponding matrix algebra). We then turn to the case where the components of the matrices are multiplied using a (generalized) star product. Associated with the deformation parameters, there are additional symmetries (flow equations) which enlarge the respective KP hierarchy. They have a compact formulation in terms of the WNA structure. We also present a formulation of the KP hierarchy equations themselves as deformation flow equations.

math-ph↗
arXiv2007

With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices

In case of the KP hierarchy where the dependent variable takes values in an (arbitrary) associative algebra A, it is known that there are solutions which can be expressed in terms of quasideterminants of a Wronski matrix which solves the linear heat hierarchy. We obtain these solutions without the help of quasideterminants in a simple way via solutions of matrix KP hierarchies (over A) and by use of a Cole-Hopf transformation. For this class of exact solutions we work out a correspondence with 'weakly nonassociative' algebras.

nlin.SI↗
arXiv2006

Burgers and KP hierarchies: A functional representation approach

From a 'discrete' functional zero curvature equation, functional representations of (matrix) Burgers and potential KP (pKP) hierarchies (and others), as well as corresponding Backlund transformations, can be obtained in a surprisingly simple way. With their help we show that any solution of a Burgers hierarchy is also a solution of the pKP hierarchy. Moreover, the pKP hierarchy can be expressed in the form of an inhomogeneous Burgers hierarchy. In particular, this leads to an extension of the Cole-Hopf transformation to the pKP hierarchy. Furthermore, these hierarchies are solved by the solutions of certain functional Riccati equations.

nlin.SI↗
arXiv2006

From AKNS to derivative NLS hierarchies via deformations of associative products

Using deformations of associative products, derivative nonlinear Schrodinger (DNLS) hierarchies are recovered as AKNS-type hierarchies. Since the latter can also be formulated as Gelfand-Dickey-type Lax hierarchies, a recently developed method to obtain 'functional representations' can be applied. We actually consider hierarchies with dependent variables in any (possibly noncommutative) associative algebra, e.g., an algebra of matrices of functions. This also covers the case of hierarchies of coupled DNLS equations.

nlin.SI↗
arXiv2006

From nonassociativity to solutions of the KP hierarchy

A recently observed relation between 'weakly nonassociative' algebras A (for which the associator (A,A^2,A) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus A' of A) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with A' a matrix algebra, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.

nlin.PS↗
arXiv2006

Nonassociativity and Integrable Hierarchies

Let A be a nonassociative algebra such that the associator (A,A^2,A) vanishes. If A is freely generated by an element f, there are commuting derivations delta_n, n=1,2,..., such that delta_n(f) is a nonlinear homogeneous polynomial in f of degree n+1. We prove that the expressions delta_{n_1} ... delta_{n_k}(f) satisfy identities which are in correspondence with the equations of the Kadomtsev-Petviashvili (KP) hierarchy. As a consequence, solutions of the `nonassociative hierarchy' partial_{t_n}(f) = delta_n(f), n=1,2,..., of ordinary differential equations lead to solutions of the KP hierarchy. The framework is extended by introducing the notion of an A-module and constructing, with the help of the derivations delta_n, zero curvature connections and linear systems.

nlin.SI↗
arXiv2006

Functional representations of integrable hierarchies

We consider a general framework for integrable hierarchies in Lax form and derive certain universal equations from which `functional representations' of particular hierarchies (like KP, discrete KP, mKP, AKNS), i.e. formulations in terms of functional equations, are systematically and quite easily obtained. The formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative (typically matrix) algebra. The obtained functional representations can be understood as `noncommutative' analogs of `Fay identities' for the KP hierarchy.

nlin.SI↗
arXiv2005

An algebraic scheme associated with the noncommutative 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.

nlin.SI↗
arXiv2004

Extension of Moyal-deformed hierarchies of soliton equations

Moyal-deformed hierarchies of soliton equations can be extended to larger hierarchies by including additional evolution equations with respect to the deformation parameters. A general framework is presented in which the extension is universally determined and which applies to several deformed hierarchies, including the noncommutative KP, modified KP, and Toda lattice hierarchy. We prove a Birkhoff factorization relation for the extended ncKP and ncmKP hierarchies. Also reductions of the latter hierarchies are briefly discussed. Furthermore, some results concerning the extended ncKP hierarchy are recalled from previous work.

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