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Register a new userThe Coxeter relations and KP map for non-commuting symbols
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We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Malcev-Newmann division ring. The action is constructed from the non-Abelian Hirota-Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.
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Using the determinant representation of gauge transformation operator, we have shown that the general form of $tau$ function of the $q$-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP ($q$-cKP) hierarchy, i.e. $l$-constraints of $q$-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of $q$-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear $q$-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of $q$-deformation ($q$-effects) in single $q$-soliton from the simplest $tau$ function of the $q$-KP hierarchy and in multi-$q$-soliton from one-component $q$-cKP hierarchy, and their dependence of $x$ and $q$, were also presented. Finally, we observe that $q$-soliton tends to the usual soliton of the KP equation when $xto 0$ and $qto 1$, simultaneously.
We consider solutions of the KP hierarchy which are elliptic functions of $x=t_1$. It is known that their poles as functions of $t_2$ move as particles of the elliptic Calogero-Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian $H_k$ of the elliptic Calogero-Moser model which governs the dynamics of poles with respect to the $k$-th hierarchical time. The Hamiltonians $H_k$ are obtained as coefficients of the expansion of the spectral curve near the marked point in which the Baker-Akhiezer function has essential singularity.
In the recent paper (R. Willox and M. Hattori, arXiv:1406.5828), an integrable discretization of the nonlinear Schrodinger (NLS) equation is studied, which, they think, was discovered by Date, Jimbo and Miwa in 1983 and has been completely forgotten over the years. In fact, this discrete NLS hierarchy can be directly obtained from an elementary auto-Backlund transformation for the continuous NLS hierarchy and has been known since 1982. Nevertheless, it has been rediscovered again and again in the literature without attribution, so we consider it meaningful to mention overlooked original references on this discrete NLS hierarchy.
We consider solutions of the matrix KP hierarchy that are elliptic functions of the first hierarchical time $t_1=x$. It is known that poles $x_i$ and matrix residues at the poles $rho_i^{alpha beta}=a_i^{alpha}b_i^{beta}$ of such solutions as functions of the time $t_2$ move as particles of spin generalization of the elliptic Calogero-Moser model (elliptic Gibbons-Hermsen model). In this paper we establish the correspondence with the spin elliptic Calogero-Moser model for the whole matrix KP hierarchy. Namely, we show that the dynamics of poles and matrix residues of the solutions with respect to the $k$-th hierarchical time of the matrix KP hierarchy is Hamiltonian with the Hamiltonian $H_k$ obtained via an expansion of the spectral curve near the marked points. The Hamiltonians are identified with the Hamiltonians of the elliptic spin Calogero-Moser system with coordinates $x_i$ and spin degrees of freedom $a_i^{alpha}, , b_i^{beta}$.
The action of a Backlund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary Backlund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schrodinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because Backlund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a new Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices.
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