A new high-order spectral problem of the mKdV and its associated integrable decomposition

A new approach to formulizing a new high-order matrix spectral problem from a normal 2× 2 matrix modified Korteweg--de Vries (mKdV) spectral problem is presented. It is found that the isospectral evolution equation hierarchy of this new higher-order matrix spectral problem turns out to be the well-known mKdV equation hierarchy. By using the binary nonlinearization method, a new integrable decomposition of the mKdV equation is obtained in the sense of Liouville. The proof of the integrability shows that r-matrix structure is very interesting.     

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.     

Algebro-Geometric Solutions of the TD Hierarchy

Resorting to the Lenard recursion scheme, we derive the TD hierarchy associated with a 2?×?2 matrix spectral problem and establish Dubrovin-type equation in terms of the introduced elliptic variables. Based on the theory of algebraic curve, all the flows associated with the TD hierarchy are straightened under the Abel-Jacobi coordinates. An algebraic function ?, also called the meromorphic function, carrying the data of the divisor is introduced on the underlying hyperelliptic curve $\mathcal {K}_{n}$ . The known zeros and poles of ? allow to find theta function representations for ? by referring to Riemann’s vanishing theorem, from which we obtain algebro-geometric solutions for the entire TD hierarchy with the help of asymptotic expansion of ? and its theta function representation.     

New integrable lattice hierarchies

In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula.     

An extension of the modified Sawada—Kotera equation and conservation laws

Based on the modified Sawada-Kotera equation, we introduce a 3 × 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawada-Kotera equation, by which a Lax pair of the modified Sawada-Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawada-Kotera equation, the Kaup-Kupershmidt equation, and the modified Sawada-Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada-Kotera equation are constructed with the aid of their Lax pairs.     

A lattice hierarchy and its continuous limits

By introducing a discrete spectral problem, we derive a lattice hierarchy which is integrable in Liouville's sense and possesses a multi-Hamiltonian structure. It is show that the discrete spectral problem converges to the well-known AKNS spectral problem under a certain continuous limit. In particular, we construct a sequence of equations in the lattice hierarchy which approximates the AKNS hierarchy as a continuous limit.     

Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions

The modified Boussinesq hierarchy associated with the 3×3 matrix spectral problem is derived with the help of Lenard recursion equations. Based on the characteristic polynomial of Lax matrix for the modified Boussinesq hierarchy, we introduce an algebraic curve K_m−1 of arithmetic genus m−1, from which we establish the associated Baker-Akhiezer function, meromorphic function and Dubrovin-type equations. The straightening out of various flows is exactly given through the Abel map. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker-Akhiezer function, the meromorphic function, and in particular, that of solutions for the entire modified Boussinesq hierarchy.     

An integrable coupling system of lattice hierarchy and its continuous limits

In [E.G. Fan, Phys. Lett. A 372 (2008) 6368], Fan present a lattice hierarchy and its continuous limits. In this Letter, we extend this method, by introducing a complex discrete spectral problem, a coupling lattice hierarchy is derived. It is shown that a new sequence of combinations of complex lattice spectral problem converges to the integrable coupling couplings of soliton equation hierarchy, which has the integrable coupling system of AKNS hierarchy as a continuous limit.     

Continuous limits for an integrable coupling system of Toda equation hierarchy

In this Letter, we present an integrable coupling system of lattice hierarchy and its continuous limits by using of Lie algebra sl(4). By introducing a complex discrete spectral problem, the integrable coupling system of Toda lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable couplings of discrete soliton equation hierarchy, which has the integrable coupling system of C-KdV hierarchy as a new kind of continuous limit.     

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.     

Dynamics days : Twente,June 7–9, 1983 annual informal workshop on nonlinear dynamics

A hierarchy of isospectral deformation equations related to the Kaup-Newell spectral problem is obtained. The generalized gauge transformations of the eigenmatrix of the spectral problem are used to generate Bäcklund and self-Bäcklund transformations. In a special case, the Bäcklund transformation of the null solution for all the equations in the hierarchy has been explicitly solved.     

Two hierarchies of integrable lattice equations associated with a discrete matrix spectral problem

Two hierarchies of nonlinear integrable positive and negative lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws of the positive hierarchy, then, the integrable coupling systems of the positive hierarchy are derived from enlarging Lax pair.     

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 Differential-Difference Equations and Their Integrable Couplings

Starting from a discrete spectral problem, the corresponding hierarchy of nonlinear differential-difference equation is proposed. It is shown that the hierarchy possesses the bi-Hamiltionian structures. Further, two integrable coupling systems for the hierarchy are constructed through enlarged Lax pair method.     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems (FDHS) in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.     

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

Starting from a new discrete spectral problem, the corresponding hierarchy of nonlinear lattice equations is proposed. It is shown that the lattice soliton hierarchy possesses the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, infinite conservation laws of the hierarchy are presented.     

Exact Solutions of Ablowitz-Ladik Hierarchy with Self-Consistent Sources Revisited and Reductions

In the paper, Ablowitz-Ladik hierarchy with new self-consistent sources is investigated. First the source in the hierarchy is described as φ_nφ_n+1, where φ_n is related to the Ablowitz-Ladik spectral problem, instead of the corresponding adjoint spectral problem. Then by means of the inverse scattering transform, the multi-soliton solutions for the hierarchy are obtained. Two reductions to the discrete mKdV and nonlinear Schrödinger hierarchies with self-consistent sources are considered by using the uniqueness of the Jost functions, as well as their N-soliton solutions.     

A Discrete Spectral Problem and Related Hierarchy of Discrete Hamiltonian Lattice Equations

Staring from a discrete matrix spectral problem, a hierarchy of lattice soliton equations is presented though discrete zero curvature representation. The resulting lattice soliton equations possess non-local Lax pairs. The Hamiltonian structures are established for the resulting hierarchy by the discrete trace identity. Liouville integrability of resulting hierarchy is demonstrated.     

A Discrete Spectral Problem and Related Hierarchy of Discrete Hamiltonian Lattice Equations

Staring from a discrete matrix spectral problem, a hierarchy of lattice soliton equations is presented though discrete zero curvature representation. The resulting lattice soliton equations possess non-local Lax pairs. The Hamiltonian structures are established for the resulting hierarchy by the discrete trace identity. Liouville integrability of resulting hierarchy is demonstrated.     

A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations.