Boussinesq Hierarchies Research Articles

Reciprocal transformations of generalized negative flows in integrable hierarchies

We construct some reciprocal transformations for the first negative flows of the Korteweg–de Vries, the modified Korteweg–de Vries, the Ablowitz–Kaup–Newell–Segur, the Fordy–Gibbons, the Boussinesq and the modified Boussinesq hierarchies or their reductions. With the help of these reciprocal transformations, the first negative flows of the above hierarchies are converted to the Camassa–Holm type and their short-wave limit, the generalized short pulse type as well as some new equations.

Lagrangian Multiforms for Kadomtsev–Petviashvili (KP) and the Gelfand–Dickey Hierarchy

Abstract We present, for the first time, a Lagrangian multiform for the complete Kadomtsev–Petviashvili hierarchy—a single variational object that generates the whole hierarchy and encapsulates its integrability. By performing a reduction on this Lagrangian multiform, we also obtain Lagrangian multiforms for the Gelfand–Dickey hierarchy of hierarchies, comprising, among others, the Korteweg–de Vries and Boussinesq hierarchies.

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.

Commutator representations and roots of pseudo differential operators

Based on the fundamental commutator representation proposed by Cao [4] we established two explicit expressions for roots of a third order differential operator. By using those expressions we succeeded in clarifying the relationship between two major approaches in theory of integrable systems: the zero curvature and the Lax representations for the KdV and the Boussinesq hierarchies. The proposed procedure could be extended to the general case of higher order of differential operators that leads to the Gel’fand-Dickey hierarchy.

Dispersionless noncommutative Kadomtsev–Petviashvili hierarchy: Multi-Hamiltonian structure and large N-like V-algebras

We propose a dispersionless limit of the NC Kadomtsev–Petviashvili hierarchy. Multi-Hamiltonian formulation, l-reductions and relevant W-algebras are investigated. As examples, Gelfand–Dickey (GD) Poisson brackets for dispersionless NCKdV and NC Boussinesq hierarchies are constructed explicitly. The associated GD Poisson algebras are shown large N- like analogs of classical V-algebras.

Extended N = 2 supersymmetric matrix (1, s)-KdV hierarchies

We propose the Lax operators for N = 2 supersymmetric matrix generalization of the bosonic (1, s)-KdV hierarchies. The simplest examples — the N = 2 supersymmetric a = 4 KdV and a = 5 2 Boussinesq hierarchies — are discussed in detail.

On the integrable hierarchies associated with the N = 2 super Wn algebra

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with the N = 2 super W n algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulting hierarchy contains the N = 2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of the N = 2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of the N = 2 supersymmetric Boussinesq hierarchies.

New integrable extensions of N = 2 KdV and Boussinesq hierarchies

We construct a new variety of N = 2 supersymmetric integrable systems by the junction of pseudo-differential superspace Lax operators for a = 4, N = 2 KdV and multi-component N = 2 NLS hierarchies. As an important special case, we obtain the Lax operator for the N = 4 super KdV system. A similar extension of one of the N = 2 super Boussinesq hierarchies is given. We also present a minimal N = 4 supersymmetric extension of the second flow of the N = 4 KdV hierarchy, and comment on its possible integrability.

Exact correlators of two-matrix models

We compute exact solutions of two-matrix models, i.e. detailed genus by genus expressions for the correlation functions of these theories, calculated without any approximation. We distinguish between two types of models, the unconstrained and the constrained ones. Unconstrained two-matrix models represent perturbations of c = 1 string theory, while the constrained ones correspond to topological field theories coupled to topological gravity. Among the latter we treat, in particular detail, the ones based on the KdV and on the Boussinesq hierarchies.

Double Wronskian Solutions of the AKNS and the Classical Boussinesq Hierarchies

The elementary and simple method of Freeman and Nimmo for verification of solutions of the bilinear soliton equations is developed to the Ablowitz-Kaup-Newell-Segur (AKNS) and the classical Boussinesq hierarchies in the bilinear form. Solutions in terms of double Wronskian determinants are obtained. Several determinantal identities which can be derived from the Laplace expansion theorem are utilized in the proof.

The constraints of potentials and the finite-dimensional integrable systems

Restricting potential to the space spanned by the eigenvectors of the recursion operator leads to a natural constraint of potential and a finite-dimensional integrable Hamiltonian system. The general method for proving the consistency of the two systems stemming from the Lax pair and obtaining the constants of the motion for the Hamiltonian system is illustrated by the classical Boussinesq and AKNS hierarchies. By using gauge transformation, similar results for the Jaulent–Miodek and Kaup–Newell hierarchies are presented.