Isospectral Hierarchy Research Articles

Some symmetries and similarity solutions of the long-water wave hierarchy

We introduce isospectral and non-isospectral Lax pairs, then apply the Tu scheme to generate the isospectral integrable hierarchy and the non-isospectral hierarchy, whose Hamiltonian structure, hereditary operator, symmetries are followed to obtain. In addition, a kind of Bäcklund transformation of the long-water wave hierarchy for the isospectral hierarchy is constructed. Through reductions of the isospectral hierarchy, we again get the long-water wave system whose similarity solutions, nonlinear self-adjointness and the non-invariant solutions are investigated, respectively, by the use of symmetry analysis. Finally, we make use of the weight method of the variables appearing in the long-water wave system to analyze the conservation laws of the system.

A τ-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter

A matrix spectral problem is researched with an arbitrary parameter. Through zero curvature equations, two hierarchies are constructed of isospectral and nonisospectral generalized derivative nonlinear schrödinger equations. The resulting hierarchies include the Kaup-Newell equation, the Chen-Lee-Liu equation, the Gerdjikov-Ivanov equation, the modified Korteweg-de Vries equation, the Sharma-Tasso-Olever equation and a new equation as special reductions. The integro-differential operator related to the isospectral and nonisospectral hierarchies is shown to be not only a hereditary but also a strong symmetry of the whole isospectral hierarchy. For the isospectral hierarchy, the corresponding τ -symmetries are generated from the nonisospectral hierarchy and form an infinite-dimensional symmetry algebra with the K-symmetries.

Conservation Laws and $$\tau $$τ-Symmetry Algebra of the Gerdjikov–Ivanov Soliton Hierarchy

Two hierarchies of isospectral and nonisospectral Gerdjikov–Ivanov equations are constructed. Conservation laws of the isospectral soliton hierarchy are explicitly derived from a specific Riccati equation that a ratio of two eigenfunctions satisfies. The corresponding K-symmetries and $$\tau $$-symmetries formulated from the isospectral and nonisospectral hierarchies constitute an infinite-dimensional $$\tau $$-symmetry algebra for the isospectral hierarchy.

Lie algebras and Hamiltonian structures of multi-component Ablowitz–Kaup–Newell–Segur hierarchy

Isospectral and non-isospectral hierarchies of multi-component Ablowitz–Kaup–Newell–Segur (AKNS) are obtained from a matrix spectral problem, then by means of the zero curvature representations of the isospectral flows {Km} and non-isospectral flows {σn}, we construct the symmetries and their algebraic structures for isospectral multi-component AKNS hierarchies, demonstrate the recursive operator L is a strong and hereditary symmetry for the isospectral hierarchy. We also derive that there are implectic operator θ and symplectic operator J such that L = θJ, and discuss the multi-Hamiltonian structures and the Liouville integrability of the isospectral hierarchies.

Hamiltonian structure of discrete soliton systems

We describe an approach for investigating the Hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem. By using the so-called implicit representations of the isospectral flows, we demonstrate the existence of the recursion operator L, which is a strong and hereditary symmetry of the flows. It is then proven that every equation in the isospectral hierarchy possesses the Hamiltonian structure if L has a skew-symmetric factorization and the first equation (ut = K(0)) in the hierarchy satisfies some simple condition. We obtain related properties, such as the implectic-symplectic factorization of L, the Liouville complete integrability and the multi-Hamiltonian structures of the isospectral hierarchy. Four examples are given.

Coupled Integrable Systems Associated with a Polynomial Spectral Problem and Their Virasoro Symmetry Algebras

An isospectral hierarchy of commutative integrable systems associated with a polynomial spectral problem is proposed. The resulting hierarchy possesses a recursion structure controlled by a hereditary operator. The nonisospectral flows generate the time first order dependent symmetries of the isospectral hierarchy, which constitute Virasoro symmetry algebras together with commutative symmetries.