A new extended q-deformed KP hierarchy

Abstract

A method is proposed in this paper to construct a new extended q-deformed KP (q-KP) hiearchy and its Lax representation. This new extended q-KP hierarchy contains two types of q-deformed KP equation with self-consistent sources, and its two kinds of reductions give the q-deformed Gelfand-Dickey hierarchy with self-consistent sources and the constrained q-deformed KP hierarchy, which include two types of q-deformed KdV equation with sources and two types of q-deformed Boussinesq equation with sources. All of these results reduce to the classical ones when q goes to 1. This provides a general way to construct (2+1)- and (1+1)-dimensional q-deformed soliton equations with sources and their Lax representations.     

The gauge transformation of the q-deformed modified KP hierarchy

In this paper, we mainly study three types of gauge transformation operators for the q-mKP hierarchy. The successive applications of these gauge transformation operators are derived. And the corresponding communities between them are also investigated.     

Additional symmetries and string equations of the noncommutative B and C type KP hierarchies

In this paper, we construct the noncommutative B and C type KP hierarchies using pseudo-differential operators and reducing conditions. Further a series of additional flows of the noncommutative B and C type KP hierarchies will be defined and the additional symmetries constitute the B and C type infinite dimensional Lie algebra W_1+∞. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions. Further, the string equations of the noncommutative B and C type KP hierarchies are derived.     

Perturbative aspects of q-deformed dynamics

Within the framework of the q-deformed Heisenberg algebra a dynamical equation of q-deformed quantum mechanics is discussed. The perturbative aspects of the q-deformed Schr?dinger equation are analyzed. General representations of the additional momentum-dependent interaction originating from the q-deformed effects are presented in two approaches. As examples, such additional interactions related to the harmonic-oscillator potential and the Morse potential are demonstrated. Received: 26 February 2001 / Published online: 11 May 2001     

q-deformed probability and binomial distribution

We introduce the concept ofq-deformed probability and discuss theq-deformed binomial distribution.     

Remarks on Hamiltonian structures of the KP hierarchy and W KP( q ) algebra

It is known that second Hamiltonian structures of the KP hierarchy are parameterized by a continuous complex parameter q and correspond to the W-infinite algebra of W _infKP ^sup(q) . In this Letter, by constructing a Miura map, we first show a generalized decomposition theorem to the second Hamiltonian structures and then establish a relation between those structures which corresponds to values (q+1) and q of the parameter, respectively. This discussion also gives a better understanding to the structures of W _infKP ^sup(q) , its reduced algebras, and their free fields realizations.     

Spectral inverse problem for q-deformed harmonic oscillator

The supersymmetric quantization condition is used to study the wave functions of SWKB equivalent q-deformed harmonic oscillator which are obtained by using only the knowledge of bound-state spectra of q-deformed harmonic oscillator. We have also studied the nonuniqueness of the obtained interactions by this spectral inverse method.     

Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in 2 + 1 dimensions

We present two hierarchies of partial differential equations in 2 + 1 dimensions. Since there exist reciprocal transformations that connect these hierarchies to the Calogero-Bogoyavlenski-Schiff equation and its modified version, we can prove that one of the hierarchies can be considered as a modified version of the other. The connection between them can be achieved by means of a combination of reciprocal and Miura transformations.     

Solving the constrained modified KP hierarchy by gauge transformations

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of modified KP hierarchy T_D(Φ) = (Φ^?1)^?_x¹? Φ^?1 and T_I (Ψ) = Ψ^?1? ^?1Ψ_x, become the ones in the constrained case. Finally, the corresponding successive applications of T_D and T_I on the eigenfunction Φ and the adjoint eigenfunction Ψ are discussed.     

Hirota polynomials for the KP and BKP hierarchies

Using symmetric function techniques, we derive closed-form expressions for the Hirota polynomials for thepth modified KP and BKP hierarchies in terms of Schur and SchurQ-polynomials, respectively. The Hirota polynomials for the BKP hierarchy can also be expressed as Pfaffians while those for thepth modified KP hierarchies can, under certain conditions, be expressed as determinants.     

q-deformed Lorentz-algebra in Minkowski phase space

In the present paper we show that the Lorentz algebra as defined in [5] is isomorphic to an algebra closely related to a q-deformed algebra. On this algebra we define a Hopf algebra structure and show its action on q-spinor modules. This algebra is related to the q-deformed Minkowski space algebra by a non invertible factorisation. Received: 12 June 1998 / Published online: 5 October 1998     

Bilinear identities for the constrained modified KP hierarchy

In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ, the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ.     

Some comments on q-deformed oscillators and q-deformed su(2) algebras

The various relations between q-deformed oscillators algebras and the q-deformed su(2) algebras are discussed. In particular, we exhibit the similarity of the q-deformed su(2) algebra obtained from q- oscillators via Schwinger construction and those obtained from q-Holstein-Primakoff transformation and show how the relation between $su_{\sqrt q } (2)$ and Hong Yan q-oscillator can be regarded as an special case of Inöuë- Wigner contraction. This latter observation and the imposition of positive norm requirement suggest that Hong Yan q-oscillator algebra is different from the usual $su_{\sqrt q } (2)$ algebra, contrary to current belief in the literature.     

On centralisers in q-deformed Heisenberg algebras

Algebraic structure of q-deformed Heisenberg algebras is investigated with emphasis on the properties of centralisers of elements of the algebra.     

Self-Consistent Sources Extensions of Modified Differential-Difference KP Equation

In this paper, we investigate a modified differential-difference KP equation which is shown to have a continuum limit into the mKP equation. It is also shown that the solution of the modified differential-difference KP equation is related to the solution of the differential-difference KP equation through a Miura transformation. We first present the Grammian solution to the modified differential-difference KP equation, and then produce a coupled modified differential-difference KP system by applying the source generation procedure. The explicit N-soliton solution of the resulting coupled modified differential-difference system is expressed in compact forms by using the Grammian determinant and Casorati determinant. We also construct and solve another form of the self-consistent sources extension of the modified differential-difference KP equation, which constitutes a Bäcklund transformation for the differential-difference KP equation with self-consistent sources.     

q-plane wave solutions of q-deformed equations with quantum conformal symmetry

We construct explicit solutions of a hierarchy of q-deformed equations which are quantum conformal invariant. The solutions are given in terms of two different q-deformations of the plane wave written in conjugated bases.     

RECURSION OPERATORS FOR KP,mKP AND HARRY DYM HIERARCHIES

In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev–Petviashvili (KP), modified Kadomtsev–Petviashvili (mKP) and Harry Dym under n-reduction. This shows a new inherent relationship between them. To illustrate our construction, the recursion operator are calculated explicitly for 2-reduction and 3-reduction.     

Many boson realizations of universal nonlinearW ∞-algebras,modified KP hierarchies,and graded lie algebras

An infinite number of free field realizations of the universal nonlinear ^(N) ( ₁₊ ^(N) ) algebras, which are identical to the KP Hamiltonian structures, are obtained in terms ofp plusq scalars of different signatures withp –q =N. They are generalizations of the Miura transformation, and naturally give rise to the modified KP hierarchies via corresponding realizations of the latter. Their characteristic Liealgebraic origin is shown to be the graded SL(p, q).     

Hopf-type deformed oscillators,their quantum double and a q-deformed Calogero-Vasiliev algebra

A q-deformed oscillator Hopf algebra is presented and the quantum double construction is carried out to obtain an R-matrix. Investigation of the algebra's structure and Fock-type representation leads to a new q-deformed Calogero-Vasiliev algebra.