Constrained KP Research Articles

Generalized bigraded Toda hierarchy

Bigraded Toda hierarchy L1M(n)=L2N(n) is generalized to L1M(n)=L2N(n)+∑j∈Z∑i=1mqn(i)Λjrn+1(i), which is the analogue of the famous constrained KP hierarchy Lk=(Lk)≥0+∑i=1mqi∂−1ri. It is known that different bosonizations of fermionic KP hierarchy will give rise to different kinds of integrable hierarchies. Starting from the fermionic form of constrained KP hierarchy, bilinear equation of this generalized bigraded Toda hierarchy (GBTH) are derived by using 2–component boson–fermion correspondence. Next based upon this, the Lax structure of GBTH is obtained. Conversely, we also derive bilinear equation of GBTH from the corresponding Lax structure.

Virasoro symmetries of the dispersionless coupled constrained KP hierarchy

In this paper, we construct the dispersionless coupled constrained KP hierarchy. We also introduce vital formal Laurent series [Formula: see text] to construct new flows, which lead out the additional symmetries of the dispersionless coupled constrained KP hierarchy. The additional flows construct a sub-algebra of the Virasoro algebra. In addition, we give the additional flows acting on [Formula: see text] and [Formula: see text] of the dispersionless coupled constrained KP hierarchy.

Constrained discrete KP hierarchy: The constraint on the tau functions and gauge transformations

Constrained discrete KP hierarchy: The constraint on the tau functions and gauge transformations

Darboux transformations of the supersymmetric constrained B and C type KP hierarchies

Based on Darboux transformations for the supersymmetric constrained KP(ScKP) hierarchy, we construct the supersymmetric constrained B type KP(ScBKP) and supersymmetric constrained C type KP(ScCKP) hierarchies of Manin–Radul and Jacobian types, and derive Darboux transformations on them. On the basis of the seed solutions, the new solutions can be generated by these Darboux transformations.

Modified constrained KP hierarchy and bi-Hamiltonian structures

We consider new modifications for the constrained KP hierarchy associated with a sequence of energy-dependent Lax operators. When $$n=1, 2,$$ the hierarchies are shown to be the KN hierarchy and a modified Yajima–Oikawa hierarchy respectively. When $$n=3$$ , it is the hierarchy of a new modified Melnikov type with a bi-Hamiltonian structure.

Recursion relations for the constrained multi-component KP hierarchy

In this paper, we define a new constrained multi-component KP(cMKP) hierarchy which contains the constrained KP(cKP) hierarchy as a special case. We derive the recursion operator of the constrained multi-component KP hierarchy. As a special example, we also give the recursion operator of the constrained two-component KP hierarchy.

Virasoro symmetry of the constrained multicomponent Kadomtsev–Petviashvili hierarchy and its integrable discretization

In this paper, we construct the Virasoro type additional symmetries of a kind of constrained multi-component KP hierarchy and give the Virasoro flow equation on eigenfunctions and adjoint eigenfunctions. It can also be seen that the algebraic structure of the Virasoro symmetry is kept after discretization from the constrained multi-component KP hierarchy to the discrete constrained multi-component KP hierarchy.

Classical Affine $${\mathcal{W}}$$-Algebras for $${\mathfrak{gl}_N}$$ and Associated Integrable Hamiltonian Hierarchies

We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper, to show that all W-algebras W(gl_N,f) carry such a hierarchy. As an application, we show that all vector constrained KP hierarchies and their matrix generalizations are obtained from these hierarchies by Dirac reduction, which provides the former with a bi-Poisson structure.

On the modified constrained Kadomtsev–Petviashvili equations

Three reduction sub-hierarchies in constrained KP hierarchies are studied. The corresponding matrix integrable systems are presented. Furthermore, Bäcklund transformations, bi-Hamiltonian structures, and τ-functions are formulated.

Gauge transformations for the constrained CKP and BKP hierarchies

A special chain of the gauge transformations of the two-component constrained KP is introduced, and then the transformed components and the τ-function of this hierarchy are presented. Based on this, by the different reductions, the gauge transformations of the constrained BKP and the constrained CKP hierarchies are obtained. Furthermore, the transformed component functions and τ-functions of them are expressed by the determinants with the help of the determinant representation of the gauge transformation operators.

Q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

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 x -> 0 and q -> 1, simultaneously.

Symmetries of supersymmetric integrable hierarchies of KP type

This article is devoted to the systematic study of additional (non-isospectral) symmetries of constrained (reduced) supersymmetric integrable hierarchies of KP type—the so-called SKP(R;MB,MF) models. The latter are supersymmetric extensions of ordinary constrained KP hierarchies which contain as special cases basic integrable systems such as (m)KdV, AKNS, Fordy–Kulish, Yajima–Oikawa, etc. As a first main result it is shown that any SKP(R;MB,MF) hierarchy possesses two different mutually (anti-)commuting types of superloop superalgebra additional symmetries corresponding to the positive- and negative-grade parts of certain superloop superalgebras. The second main result is the systematic construction of the full algebra of additional Virasoro symmetries of SKP(R;MB,MF) hierarchies, which requires nontrivial modifications of the Virasoro flows known from the general case of unconstrained Manin–Radul super-KP hierarchies (the latter flows do not define symmetries for constrained SKP(R;MB,MF) hierarchies). As a third main result we provide systematic construction of the supersymmetric analogs of multi-component (matrix) KP hierarchies and show that the latter contain, among others, the supersymmetric version of the Davey–Stewartson system. Finally, we present an explicit derivation of the general Darboux–Bäcklund solutions for the SKP(R;MB,MF) super-tau functions (supersymmetric “soliton”-like solutions) which preserve the additional (non-isospectral) symmetries.

Loop algebra and visaoro symmetries of integrable hierarchies of KP type

We propose a systematic treatment of symmetries of KP integrable systems, including constrained (reduced) KP models cKPR,M (generalized AKNS hierarchies), and their multi component (matrix) generalizations. Any cKPR,M integrable hierarchy is shown to possess loop algebra (additional) symmetry. Also we provide a systematic construction of the full algebra of Virasoro additional symmetries in the case of constrained KP models which requires a non trivial modification of the known Orlov Schulman construction for the general unconstrained KP hierarchy. Multi component KP hierarchies are identified as ordinary (scalar) one component KP hierarchies supplemented with the Cartan subalgebra of the additional symmetry algebra, which provides the basis of a new method for construction of soliton like solutions. Davey Stewartson and N wave resonant systems arise as symmetry flows of ordinary CKPR,M hierarchies.

Solitons from dressing in an algebraic approach to the constrained KP heirachy

The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras leads to a variety of 1+1 soliton equations. By varying the rank of the underlying $sl (n)$ algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy. The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine $sl (n)$ algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of the constrained KP hierarchy.

Constrained KP models as integrable matrix hierarchies

We formulate the constrained KP hierarchy (denoted by cKPK+1,M) as an affine sl̂(M+K+1) matrix integrable hierarchy generalizing the Drinfeld–Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld–Sokolov hierarchy, we are able to find several new universal results valid for the cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac–Moody current algebra. An explicit example is given for the case sl̂(M+K+1), for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple non-regular element E of sl(M+K+1) and the content of the center of the kernel of E.

ON MATRIX KP AND SUPER-KP HIERARCHIES IN THE HOMOGENEOUS GRADING

Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie-algebraic AKS matrix framework associated with the homogeneous grading. The role played by different regular elements in defining the corresponding hierarchies is analyzed, as well as the symmetry properties under the Weyl group transformations. The coset structure of higher order Hamiltonian densities is proven. For a generic Lie algebra the hierarchies considered here are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in Refs. 1 and 2 are obtained as special limit restrictions on Hermitian symmetric spaces. In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series. The bosonic hierarchies obtained from [Formula: see text] and the supersymmetric ones derived from the N=1 affinization of sl (2), sl (3) and osp (1|2) are explicitly constructed. An unexpected result is found: only a restricted subclass of the sl (3) bosonic hierarchies can be supersymmetrically extended while preserving integrability.

On the constrained KP hierarchy

An explanation for the so-called constrained hierarhies is presented by linking them with the symmetries of the KP hierarchy. While the existence of ordinary symmetries (belonging to the hierarchy) allows one to reduce the KP hierarchy to the KdV hierarchies, the existence of additional symmetries allows to reduce KP to the constrained KP.

Affine Lie algebraic origin of constrained KP hierarchies

An affine sl(n+1) algebraic construction of the basic constrained KP hierarchy is presented. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and it is shown that these approaches are equivalent. The model is recognized to be the generalized non-linear Schrödinger (GNLS) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-Bäcklund transformations and interpolate between GNLS and multi-boson KP-Toda hierarchies. Our construction uncovers the origin of the Toda lattice structure behind the latter hierarchy.

Classical extended conformal algebras associated with constrained KP hierarchy

The conformal property of the second Hamiltonian structure of constrained KP hierarchy derived by Oevel and Strampp is examined. It is found that it naturally gives a family of nonlocal extended conformal algebras. Two examples of such algebras are given and it is found that they are similar to Bilal’s V algebra. By taking a gauge transformation one can map the constrained KP hierarchy to Kupershmidt’s nonstandard Lax hierarchy. The second Hamiltonian structure in this representation is considered. It is shown that after mapping the Lax operator to a pure differential operator the second structure becomes the sum of the second and the third Gelfand–Dickey brackets defined by this differential operator. The corresponding conformally covariant form of this differential operator is also given.

Two-matrix string model as constrained (2+1)-dimensional integrable system

We show that the 2-matrix string model corresponds to a coupled system of 2+1-dimensional KP and modified KP ((m)KP 2+1) integrable equations subject to a specific “symmetry” constraint. The latter together with the Miura-Konopelchenko map for (m)KP 2+1 are the continuum incarnation of the matrix string equation. The (m)KP 2+1 Miura and Bäcklund transformations are natural consequences of the underlying lattice structure. The constrained (m)KP 2+1 system is equivalent to a 1+1-dimensional generalized KP-KdV hierarchy related to graded SL(3,1). We provide an explicit representation of this hierarchy, including the associated W(2,1)-algebra of the second Hamiltonian structure, in terms of free currents.