Constrained KP Hierarchy 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.

The constrained KP hierarchy and the bigraded Toda hierarchy of (M, 1)-type

The constrained KP hierarchy and the bigraded Toda hierarchy of (M, 1)-type

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.

Dubrovin–Frobenius manifolds associated with Bn and the constrained KP hierarchy

In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group Bn constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the 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

Several Integrable Properties of the Discrete KP Hierarchy Obtained by Means of Gauge Transformations

The gauge transformation is a powerful method to solve integrable hierarchies. In the discrete KP hierarchy, there are two basic gauge transformations, T D ( q ) = Λ( q ) ∘ Δ ∘ q −1 and T I ( r ) = Λ −1 ( r −1 ) ∘ Δ −1 ∘ r. This paper is concerned with several integrable properties of the discrete KP hierarchy found by means of gauge transformations. The successive applications of T I to the constrained discrete KP hierarchy have been presented. Meanwhile, we derive two novel discrete equations as the orbits of gauge transformations T D and T I for the constrained KP hierarchy. Moreover, with the help of the spectral representations and the gauge transformations, Fay-similar equations of the tau functions for the constrained discrete KP have been investigated. In addition, we also discuss the relation between the gauge transformation and an additional symmetry for the discrete KP hierarchy.

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.

The integral type gauge transformation and the additional symmetry for the constrained KP hierarchy

In this paper, the compatibility between the integral type gauge transformation and the additional symmetry of the constrained KP hierarchy is given. And the string-equation constraint in matrix models is also derived.

Commutativity of the extended KP flows

The commutativity problem of the extended KP hierarchy is analyzed. The compatibility equation of two extended KP flows is constructed, together with its Lax representations involving two extended Lax operators. The resulting theory shows that the extended KP hierarchy is a natural generalization of the KP flows, but does not commute unlike the constrained KP hierarchy. A few particular examples are computed, along with their Lax pairs.

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.

Two choices of the gauge transformation for the AKNS hierarchy through the constrained KP hierarchy

On the basis of the equivalence between the AKNS hierarchy and the cKP hierarchy with the constraint k=1, we point out that there exist two choices to keep the form of the Lax operator when we perform the gauge transformation for the AKNS hierarchy, which results in two classes of functions to trigger the gauge transformation. For the second choice, two theorems for two types of gauge transformation are established. Several new and more general forms of tau-functions for the AKNS hierarchy are obtained by means of gauge transformations of both types. The union of the two choices leads to new forms of τ-functions. We generate the AKNS hierarchy from the “free” Lax operator L(0)=∂ via a chain of gauge transformations.

The complex sine-Gordon equation as a symmetry flow of the AKNS hierarchy

It is shown how the complex sine-Gordon equation arises as a symmetry flow of the AKNS hierarchy. The AKNS hierarchy is extended by the ``negative'' symmetry flows forming the Borel loop algebra. The complex sine-Gordon and the vector Nonlinear Schrodinger equations appear as lowest negative and second positive flows within the extended hierarchy. This is fully analogous to the well-known connection between the sine-Gordon and mKdV equations within the extended mKdV hierarchy. A general formalism for a Toda-like symmetry occupying the ``negative'' sector of sl(N) constrained KP hierarchy and giving rise to the negative Borel sl(N) loop algebra is indicated.

On Grassmannian description of the constrained KP hierarchy

This note develops an explicit construction of the constrained KP hierarchy within the Sato Grassmannian framework. Useful relations are established between the kernel elements of the underlying ordinary differential operator and the eigenfunctions of the associated KP hierarchy as well as between the related bilinear concomitant and the squared eigenfunction potential.

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.

Matrix formulation of Hamiltonian structures of constrained KP hierarchy

We give a matrix formulation of the Hamiltonian structures of constrained KP hierarchy. First, we derive from the matrix formulation the Hamiltonian structure of the one-constraint KP hierarchy, which was originally obtained by Oevel and Strampp. We then generalize the derivation to the multiconstraint case and show that the resulting bracket is actually the second Gelfand–Dickey bracket associated with the corresponding Lax operator. The matrix formulation of the Hamiltonian structure of the one-constraint KP hierarchy in the form introduced in the study of matrix model is also discussed.

Solving the constrained KP hierarchy by gauge transformations

We solve the constrained Kadomtsev–Petviashvili (cKP) hierarchy by using the gauge transformation technique. We show that there are two kinds of gauge transformations which preserve the form of the Lax operator of the cKP hierarchy. One of them is differential type and the other is integral type. Through two such gauge transformations we obtain not only the Wronskian-type τ-functions for the cKP hierarchy, but also the binary-type τ-functions which have not been obtained before.

An Analytic Description of the Vector Constrained KP Hierarchy

In this paper we give a geometric description in terms of the Grassmann manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the vector $k$-constrained KP hierarchy. We also show in a geometric way that these hierarchies are equivalent to Krichever's general rational reductions of the KP hierarchy.