Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2+1 dimensions
arXiv:0706.1373 · doi:10.1088/1751-8113/41/26/265205
Abstract
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.
37 pages, 10 figures, 2nd version: some extensions (Fig 3, Appendix A, additional references), 3rd version: some minor changes, additional references
References in corpus (40)
- Integrable Hierarchies and Dispersionless Limit
- Conformal maps and dispersionless integrable hierarchies
- Currents, Charges, and Canonical Structure of Pseudodual Chiral Models
- Einstein-Weyl geometry, the dKP equation and twistor theory
- Noncommutative Multi-Solitons in 2+1 Dimensions
- Scattering of Noncommutative Solitons in 2+1 Dimensions
- Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability
- A Geometry for Multidimensional Integrable Systems
- Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system
- Dispersionless scalar integrable hierarchies, Whitham hierarchy and the quasi-classical dbar-dressing method
- Bi-differential calculi and integrable models
- Soliton Solutions and Nontrivial Scattering in an Integrable Chiral Model in (2+1) Dimensions
- Soliton-Antisoliton Scattering Configurations in a Noncommutative Sigma Model in 2+1 Dimensions
- Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and II
- On the Miura map between the dispersionless KP and dispersionless modified KP hierarchies
- Reduced dynamics of Ward solitons
- The Moyal bracket and the dispersionless limit of the KP hierarchy
- Noncommutative Extended Waves and Soliton-like Configurations in N=2 String Theory
- Topological conformal field theory with a rational W potential and the dispersionless KP hierarchy
- Functional representations of integrable hierarchies
- The Energy of Scattering Solitons in the Ward Model
- Topology and Energy of Time Dependent Unitons
- Lump dynamics in the CP^1 model on the torus
- Solutions of the Modified Chiral Model in (2+1) Dimensions
- Interaction of Noncommutative Plane Waves in 2+1 Dimensions
- Burgers and KP hierarchies: A functional representation approach
- Symmetries and tau function of higher dimensional dispersionless integrable hierarchies
- Time-Space Noncommutative Abelian Solitons
- Ward's solitions
- Bicomplexes and Backlund transformations
- Moduli-Space Dynamics of Noncommutative Abelian Sigma-Model Solitons
- Remarks on gauge vortex scattering
- Soliton scattering in the O(3) model on a torus
- Conserved quantities for integrable chiral equations in 2+1 dimensions
- Existence Theorems for Vortex Scattering
- Lagrangian Formulation of the General Modified Chiral Model
- A Note on Ward's Chiral Model
- Integrable Yang-Mills-Higgs Equations in 3-Dimensional De Sitter Space-Time
- Emergence of Time from Dimensional Reduction in Noncommutative Geometry
- Periodic and homoclinic solutions of the modified 2+1 Chiral model