Negative Times of the Davey-Stewartson Integrable Hierarchy

We use example of the Davey-Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.

Kadomtsev–Petviashvili Hierarchy: Negative Times

The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator identities. This approach enables introduction of linear differential equations that admit lifts up to nonlinear integrable ones by means of the special dressing procedure. Thus, one can construct not only nonlinear equations, but corresponding Lax pairs as well. The Lax operator of this evolution coincides with the Lax operator of the “positive” hierarchy. We also derive (1 + 1)-dimensional reductions of equations of this hierarchy.

Symmetries of the multi-component Boussinesq hierarchy

In this paper, we construct the Lax operator of the multi-component Boussinesq hierarchy. Based on the Sato theory and the dressing structure of the multi-component Boussinesq hierarchy, the adjoint wave function and the Orlov–Schulman’s operator are introduced, which are useful for constructing the additional symmetry of the multi-component Boussinesq hierarchy. Besides, the additional flows can commute with the original flows, and these flows form an infinite dimensional [Formula: see text] algebra. Taking the above discussion into account, we mainly study the additional symmetry flows and the generating function for both strongly and weakly multi-component of the Boussinesq hierarchies. By the way, using the [Formula: see text] constraint of the multi-component Boussinesq hierarchy, the string equation can be derived.

The reduction of the q-Wronskian solutions of the q-deformed modified KP hierarchy to its constrained case

By using the eigenfunction symmetry constraints of the [Formula: see text]-deformed modified Kadomtsev–Petviashvili ([Formula: see text]-mKP) hierarchy, we show a necessary and sufficient condition to reduce [Formula: see text]-Wronskian solutions of the [Formula: see text]-mKP hierarchy to the [Formula: see text]-cmKP ([Formula: see text]-deformed constrained modified Kadomtsev–Petviashvili) hierarchy defined by setting the Lax operator as [Formula: see text]. Then an illustrative example is given.

Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds

This paper is dedicated to proving the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by $$\mathcal {U}_N$$ U N . We construct generalized action–angle coordinates which establish a real analytic symplectomorphism from $$\mathcal {U}_N$$ U N onto some open convex subset of $${\mathbb {R}}^{2N}$$ R 2 N and allow to solve the equation by quadrature for any such initial datum. As a consequence, $$\mathcal {U}_N$$ U N is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gérard–Kappeler (Commun Pure Appl Math, 2020. 10.1002/cpa.21896. arXiv:1905.01849). The global well-posedness of the BO equation on $$\mathcal {U}_N$$ U N is given by a polynomial characterization and a spectral characterization of the manifold $$\mathcal {U}_N$$ U N . Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup.

On quantum separation of variables beyond fundamental representations

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called “non-fundamental” models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin’s approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl_{2})Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a QQ-operator in terms of the fused transfer matrices. Finally, we show that the QQ-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.

Sato–Bäcklund transformations and string equations of the mKP hierarchy

Additional symmetry is an important kind of symmetries depending explicitly on the time and space variables, which can be expressed through Sato–Bäcklund transformations. In this paper, we construct Sato–Bäcklund transformations of the modified KP hierarchy and its constrained cases. Then the string equations of the [Formula: see text]-reduced modified KP hierarchy are established by requiring the system independent on some additional symmetry flows, which are expressed by the Lax operator [Formula: see text] and the Orlov–Shulman’s operator [Formula: see text]. At last, we obtain the negative Virasoro constraint on the two tau functions of the 2-reduced modified KP hierarchy satisfying the string equations.

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II

This article is a direct continuation of where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the KK-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary KK-matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary KK-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in . The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter’s T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.