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Register a new userThe $n$-component KP hierarchy and representation theory
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Starting from free charged fermions we give equivalent definitions of the $n/$-component KP hierarchy, in terms of $tau/$-functions $tau_alpha/$ (where $alpha in M =/$ root lattice of $sl_n/$), in terms of $n times n/$ matrix valued wave functions $W_alpha(alphain M)/$, and in terms of pseudodifferential wave operators $P_alpha(alphain M)/$. These imply the deformation and the zero curvature equations. We show that the 2-component KP hierarchy contains the Davey-Stewartson system and the $ngeq3/$ component KP hierarchy continues the $n/$-wave interaction equations. This allows us to construct theis solutions.
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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.
The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plucker relations from geometry. We give a solution to the Plucker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the KP hierarchy. These factorizations appear in geometry as encodings of branched covers, and thus by specializing our transitive factorization result, we are able to prove that the generating series for two classes of branched covers satisfies the KP hierarchy. For the first of these, the double Hurwitz series, this result has been previously given by Okounkov. The second of these, that we call the m-hypermap series, contains the double Hurwitz series polynomially, as the leading coefficient in m. The m-hypermap series also specializes further, first to the series for hypermaps and then to the series for maps, both in an orientable surface. For the latter series, we apply one of the KP equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces. This recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces, in recent work by Bender, Gao and Richmond.
A N=4 supersymmetric matrix KP hierarchy is proposed and a wide class of its reductions which are characterized by a finite number of fields are described. This class includes the one-dimensional reduction of the two-dimensional N=(2|2) superconformal Toda lattice hierarchy possessing the N=4 supersymmetry -- the N=4 Toda chain hierarchy -- which may be relevant in the construction of supersymmetric matrix models. The Lax pair representations of the bosonic and fermionic flows, corresponding local and nonlocal Hamiltonians, finite and infinite discrete symmetries, the first two Hamiltonian structures and the recursion operator connecting all evolution equations and the Hamiltonian structures of the N=4 Toda chain hierarchy are constructed in explicit form. Its secondary reduction to the N=2 supersymmetric alpha=-2 KdV hierarchy is discussed.
A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It provides a general way to construct soliton equations with sources and their Lax representations.
We consider solutions of the matrix KP hierarchy that are trigonometric functions of the first hierarchical time $t_1=x$ and establish the correspondence with the spin generalization of the trigonometric Calogero-Moser system on the level of hierarchies. Namely, the evolution of poles $x_i$ and matrix residues at the poles $a_i^{alpha}b_i^{beta}$ of the solutions with respect to the $k$-th hierarchical time of the matrix KP hierarchy is shown to be given by the Hamiltonian flow with the Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the spin trigonometric Calogero-Moser system with coordinates $x_i$ and with spin degrees of freedom $a_i^{alpha}, , b_i^{beta}$. By considering evolution of poles according to the discrete time matrix KP hierarchy we also introduce the integrable discrete time version of the trigonometric spin Calogero-Moser system.
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