On the BKP Hierarchy: Additional Symmetries,Fay Identity and Adler–Shiota–van Moerbeke Formula

We give an alternative proof of the Adler–Shiota–van Moerbeke formula for the BKP hierarchy. The proof is based on a simple expression for the generator of additional symmetries and the Fay identity of the BKP hierarchy.      

Additional Symmetries of the Zakharov—Shabat Hierarchy,String Equaion and Isomonodromy

Isomonodromic deformations are nothing but symmetries of the Zakharov–Shabat (isospectral) hierarchy, both the basic ones (belonging to the hierarchy) and additionally, restricted to the submanifold of solutions to the string equation.     

Factorization of the scattering matrix and the location of the eigenvalues of the Manakov-Zakharov-Shabat system

We show the scattering matrix associated to the Manakov-Zakharov-Shabat (MZS) system can be factorized as the product of two scattering matrices associated to the Zakharov-Shabat (ZS) system of the Nonlinear Schrödinger (NLS) equation, whenever the initial conditions of the Manakov system have disjoint support. Moreover, if these initial conditions are assumed to be single-lobe, the eigenvalues of the MZS system are purely imaginary.     

New block matrix spectral problem and Hamiltonian structure of the discrete integrable coupling system

In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity.     

Hierarchies of multi-component mKP equations and theirs integrable couplings

First, a new multi-component modified Kadomtsev-Petviashvill (mKP) spectral problem is constructed by k-constraint imposed on a general pseudo-differential operator. Then, two hierarchies of multi-component mKP equations are derived, including positive non-isospectral mKP hierarchy and negative non-isospectral mKP hierarchy. Moreover, new integrable couplings of the resulting mKP soliton hierarchies are constructed by enlarging the associated matrix spectral problem.     

Classical and quantum integrability for a class of potentials in two dimensions

A method for the construction of the second constant of motion in fourth order is carried out. Correspondingly the fourth order potential equation is obtained whose solutions directly provide the classical integrable systems. Second constant of motion is obtained for a large class of potentials. Quantum invariants are also obtained with second order quantum corrections of the order O(?²) to the corresponding classical invariants. The phase space diagrams for these cases are drawn using a mathematical computer software package in two dimensions.     

Initial value problem of the Whitham equations for the Camassa-Holm equation

We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham solution matches the Burgers solution, which exists outside the cusp.     

Two hierarchies of multi-component Kaup-Newell equations and theirs integrable couplings

Two hierarchies of multi-component Kaup-Newell equations are derived from an arbitrary order matrix spectral problem, including positive non-isospectral Kaup-Newell hierarchy and negative non-isospectral Kaup-Newell hierarchy. Moreover, new integrable couplings of the resulting Kaup-Newell soliton hierarchies are constructed by enlarging the associated matrix spectral problem.     

A hierarchy of non-isospectral multi-component AKNS equations and its integrable couplings

A hierarchy of non-isospectral multi-component AKNS equations is derived from an arbitrary order matrix spectral problem. As a reduction, non-isospectral multi-component Schrödinger equations are obtained. Moreover, new non-isospectral integrable couplings of the resulting AKNS soliton hierarchy are constructed by enlarging the associated matrix spectral problem.     

Darboux transformation for the non-isospectral AKNS hierarchy and its asymptotic property

In this Letter, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-Bäcklund transformation, because the integral constants of the hierarchy will be changed after the transformation. The transform rule of the integral constants will be also derived. By this means, the soliton solutions of the nonlinear equations derived by the non-isospectral AKNS hierarchy can be found.     

An integrable coupling family of Merola-Ragnisco-Tu lattice systems, its Hamiltonian structure and related nonisospectral integrable lattice family

An integrable coupling family of Merola-Ragnisco-Tu lattice systems is derived from a four-by-four matrix spectral problem. The Hamiltonian structure of the resulting integrable coupling family is established by the discrete variational identity. Each lattice system in the resulting integrable coupling family is proved to be integrable discrete Hamiltonian system in Liouville sense. Ultimately, a nonisospectral integrable lattice family associated with the resulting integrable lattice family is constructed through discrete zero curvature representation.     

1-soliton solution of the generalized Radhakrishnan, Kundu, Lakshmanan equation

This Letter carries out the integration of the generalized Radhakrishnan, Kundu, Lakshmanan equation to obtain the 1-soliton solution. The solitary wave ansatz is used to carry out the integration, to obtain an exact solution of this equation.     

Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics

Taking into account many developments in fiber optics communications, we propose a higher nonlinear Schrödinger equation (HNLS) with variable coefficients, more general than that in [R. Essiambre, G.P. Agrawal, Opt. Commun. 131 (1996) 274], which governs the propagation of ultrashort pulses in a fiber optics with generic variable dispersion. The study of this equation is performed using the Painlevé test and the zero-curvature method. Also, we prove the equivalence between this equation and its anomalous integrable counterpart (the so-called Sasa-Satsuma equation). Finally, in view of its physical relevance, we present a soliton solution which represents the propagation of ultrashort pulses in a dispersion decreasing fiber.     

A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product

It is shown that the Kronecker product can be applied to construct a new integrable coupling system of discrete soliton equation hierarchy in this Letter. A direct application to the generalized Toda lattice spectral problem leads to a novel integrable coupling system. It is also indicated that the study of integrable couplings by using of the Kronecker product is an efficient and straightforward method.     

The double Wronskian solutions of a non-isospectral Kadomtsev-Petviashvili equation

The double Wronskian solutions of a non-isospectral Kadomtsev-Petviashvili equation (n-KPE) are derived. One-soliton solution and two-soliton solution are presented, the characteristics of one-soliton and two-soliton scattering are discussed also.     

A new extended KP hierarchy

A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It provides a general way to construct soliton equations with sources and their Lax representations.     

On generalized Lotka-Volterra lattices

Generalized matrix Lotka-Volterra lattice equations are obtained in a systematic way from a “master equation” possessing a bicomplex formulation. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.