Bidifferential calculus approach to AKNS hierarchies and their solutions

@article{Dimakis2010BidifferentialCA,
  title={Bidifferential calculus approach to AKNS hierarchies and their solutions},
  author={Aristophanes Dimakis and Folkert Mueller-Hoissen},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2010},
  volume={6},
  pages={055}
}
We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the… Expand

Figures from this paper

Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
We present a general solution-generating result within the bidifferential calcu- lus approach to integrable partial differential and difference equations, based on a binary Darboux-typeExpand
“Riemann equations” in bidifferential calculus
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, weExpand
The non-autonomous chiral model and the Ernst equation of General Relativity in the bidifferential calculus framework
The non-autonomous chiral model equation for an m m matrix function on a two-dimensional space appears in particular in general relativity, where for m = 2 a certain reduction of it determinesExpand
Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy
New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of (2 + 1)dimensional integrable equations, including the DS-IIIExpand
Comment on "Discretisations of constrained KP hierarchies"
In the recent paper (R. Willox and M. Hattori, arXiv:1406.5828), an integrable discretization of the nonlinear Schr\"odinger (NLS) equation is studied, which, they think, was discovered by Date,Expand
Bidifferential calculus, matrix SIT and sine-Gordon equations
We express a matrix version of the self-induced transparency (SIT) equations in the bidifferential calculus framework. An infinite family of exact solutions is then obtained by application of aExpand
The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation
The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r sT we introduce a scalarExpand
Soliton Fay identities: I. Dark soliton case
We derive a set of bilinear identities for the determinants of the matrices that have been used to construct dark soliton solutions for various models. To give examples of the application of theExpand
Skew-selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations
In this paper we study direct and inverse problems for discrete and continuous time skew-selfadjoint Dirac systems with rectangular (possibly non-square) pseudo-exponential potentials. For such aExpand
A novel multi-component generalization of the short pulse equation and its multisoliton solutions
We propose a novel multi-component system of nonlinear equations that generalizes the short pulse (SP) equation describing the propagation of ultra-short pulses in optical fibers. By means of theExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 52 REFERENCES
Solutions of matrix NLS systems and their discretizations: 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 discreteExpand
Systems of PDEs Obtained from Factorization in Loop Groups
AbstractWe propose a generalization of a Drinfeld–Sokolov scheme of attaching integrable systems of PDEs to affine Kac–Moody algebras. With every affine Kac–Moody algebra $$\mathfrak{g} $$ and aExpand
Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies
An analytic-bilinear approach for the construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allowsExpand
Linear integral transformations and hierarchies of integrable nonlinear evolution equations
Abstract Integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations. These are (Riemann-Hilbert type of) integral transformations which leaveExpand
On generating functions in the AKNS hierarchy
It is shown that the self-induced transparency equations can be interpreted as a generating function for as positive so negative flows in the AKNS hierarchy. Mutual commutativity of these flows leadsExpand
Linear integral equations and multicomponent nonlinear integrable systems I
In a previous paper (I) an extension of the direct linearization method was developed for obtaining solutions of multicomponent generalizations of integrable nonlinear partial differential equationsExpand
On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type
A deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parametrize different extensions of the AKNS hierarchy to include negative flows. This construction establishes aExpand
Generalised KdV and MKdV equations associated with symmetric spaces
The authors extend previous results on the linear spectral problem introduced by Fordy and Kulish (1983). The odd-order isospectral flows admit both a KdV and MKdV type reduction. The non-linearExpand
Negative powers of Olver recursion operators
Olver’s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator. Some negative order KdV equations are derived in this way. InExpand
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 (such as KP, discreteExpand
...
1
2
3
4
5
...