Hierarchy Of Soliton Equations Research Articles

A novel kind of a multicomponent hierarchy of discrete soliton equations and its application

A novel kind of a multicomponent hierarchy of discrete soliton equations and its application

A novel kind of a multicomponent hierarchy of discrete soliton equations and its application

Представлен метод построения многокомпонентной интегрируемой иерархии дискретных солитонных уравнений на основе алгебры Ли $\hat g$. В качестве приложения метода рассматривается модифицированная спектральная задача Тоды. Получена новая многокомпонентная интегрируемая иерархия решеточных уравнений с двумя произвольными константами. Эта новая иерархия может быть сведена к двум многокомпонентным интегрируемым системам, одной из которых является знаменитая система Тоды.

Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

Matrix Integral Solutions to the q-Difference Two-Dimensional Toda Lattice Equation and Its Pfaffianized System

Matrix integrals used in random matrix theory have been revealed to provide τ-functions for several hierarchies of soliton equations. In this article, we extend this relation by showing that such i...

A note on “Lattice soliton equation hierarchy and associated properties”**Project supported by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2014AM001).

We demonstrate that the new hierarchy of integrable lattice equations in Chin. Phys. B 21 090202 (2012) can be changed into the integrable lattice hierarchy in Chin. Phys. B 13 1009 (2004) by using a very simple transformation.

Linear integral equations, infinite matrices, and soliton hierarchies

A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all (2 + 1)- and (1 + 1)-dimensional soliton hierarchies associated with scalar differential spectral problems. The integrability characteristics for the obtained soliton hierarchies, including Miura-type transforms, τ-functions, Lax pairs, and soliton solutions, are also derived within this framework.

Reduction in soliton hierarchies and special points of classicalr-matrices

Abstract We propose the most general approach to construction of the U – V pairs of hierarchies of soliton equations in two dimensions based on the theory of classical non-skew-symmetric r -matrices with spectral parameters and infinite-dimensional Lie algebras. We show that reduction in integrable hierarchies is connected with “special points” of classical r -matrices in which they become singular or degenerated. We prove that “Mikhailov’s reduction” or reduction with the help of automorphism is a partial case of our construction. We consider two types of integrable hierarchies and the corresponding soliton equations: the so-called “positive” and “negative flow” equations. We show that the “negative flow” equations can be written in the most general case without a specification of the concrete form of classical r -matrix. They coincide with a generalization of chiral field equation and its different reductions, with a generalization of abelian and non-abelian Toda field equations and new class of integrable equations which we call “double-shift” equations. For the case of “positive” hierarchies and “positive” flows we explicitly write general U – V pairs of “nominative” equations of hierarchy. We consider examples of new equations of such the types and present new soliton equations coinciding with elliptic deformation of dNS equation and its “negative flow” equation and ultimate generalization of the abelian and non-abelian modified Toda field equations.

Unitarily-invariant integrable systems and geometric curve flows in SU (n + 1)/U(n) and SO(2n)/U(n)

Bi-Hamiltonian hierarchies of soliton equations are derived from geometric non-stretching (inelastic) curve flows in the Hermitian symmetric spaces and . The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, new integrable multi-component versions of the Sine–Gordon (SG) equation and the modified Korteveg–de Vries (mKdV) equation, as well as a novel nonlocal multi-component version of the nonlinear Schrödinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These integrable systems are unitarily invariant and correspond to geometric curve flows given by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map in the case of the SG and mKdV systems, and a generalization of the vortex filament bi-normal equation in the case of the NLS systems.

Finite-dimensional Liouville integrable Hamiltonian systems generated from Lax pairs of a bi-Hamiltonian soliton hierarchy by symmetry constraints

We construct finite-dimensional Hamiltonian systems by means of symmetry constraints from the Lax pairs and adjoint Lax pairs of a bi-Hamiltonian hierarchy of soliton equations associated with the 3-dimensional special linear Lie algebra, and discuss the Liouville integrability of these systems based on the existence of sufficiently many integrals of motion.

Darboux transformations for super-Schrödinger equation, super-Dirac equation and their exact solutions

The Darboux transformation (DT) for the super-integrable hierarchy has an essential difference from the general system. As we know, the super-integrable soliton equation hierarchies with four potentials are discussed. Starting from the spectral problems of super-AKNS hierarchy and super-Dirac hierarchy, a DT method for two super-integrable hierarchies is constructed, which is more complex than the general integrable system. Soliton solutions of super-Schrodinger equation and super-Dirac equation are presented by using DT, which contain some bright, dark and breather wave soliton solutions. Then, the properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients.

A New Soliton Hierarchy Associated withso⁡3,Rand Its Conservation Laws

Based on the three-dimensional real special orthogonal Lie algebraso(3,R), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy.

Algebro-Geometric Solutions to a New Hierarchy of Soliton Equations

Algebro-Geometric Solutions to a New Hierarchy of Soliton Equations

An integrable extension of a soliton hierarchy associated with so(3,ℝ) and its bi-Hamiltonian formulation

A hierarchy of soliton equations is constructed from an extended spectral problem associated with the three-dimensional special orthogonal real Lie algebra so(3,ℝ) by means of zero curvature equations. The bi-Hamiltonian structure of this hierarchy is also presented via the trace identity.

Tau Function Approach to Theta Functions

We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We describe the initial term of the expansion by the Schur function corresponding to the partition determined by the gap sequence of a certain flat line bundle. The other is the expansion of the theta function and its certain derivatives in one of the variables on the Abel-Jacobi images of k points on a Riemann surface with k less than or equal to g. We determine the initial term of the expansion as certain derivatives of the theta function successively. As byproducts, firstly we obtain a refinement of Riemann's singularity theorem. Secondly we determine normalization constants of higher genus sigma functions of a Riemann surface, defined by Korotkin and Shramchenko, such that they become modular invariant.

Soliton equations related to the affine Kac-Moody algebra D 4 (1)

We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D (1)4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.

A soliton hierarchy associated with a new spectral problem and its Hamiltonian structure

A hierarchy of soliton equations together with its Hamiltonian structure is constructed from a new spectral problem associated with the three-dimensional special orthogonal real Lie algebra, so(3,ℝ). The Liouville integrability of the presented soliton hierarchy is proved, based on the Hamiltonian structure.

A Generalization of the Wadati-Konno-Ichikawa Soliton Hierarchy and its Liouville Integrability

Abstract A new matrix spectral problem associated with sl(2,), which generalizes the Wadati-Konno-Ichikawa spectral problem, is introduced, and the corresponding hierarchy of soliton equations is generated from the associated zero curvature equations. A bi-Hamiltonian structure of the resulting generalized soliton hierarchy is furnished by using the trace identity, and thus, every system in the generalized hierarchy is Liouville integrable.

Integrable couplings, bi-integrable couplings and their Hamiltonian structures of the Giachetti-Johnson soliton hierarchy

On the basis of zero curvature equations from semi-direct sums of Lie algebras, we construct integrable couplings of the Giachetti–Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi-integrable couplings of the Giachetti–Johnson hierarchy and their Hamiltonian structures by applying a class of non-semisimple matrix loop algebras consisting of triangular block matrices. Copyright © 2014 John Wiley & Sons, Ltd.

Two soliton hierarchies associated with SO(4) and the applications of SU(2) ⊗ SU(2) ≅ SO(4)

Based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct two different isospectral problems. And the two hierarchies of soliton equations from zero curvature equations can be generated. At last, we use the property of the isomorphism SU(2) ⊗ SU(2) ≅ SO(4), and obtain the relations between the hierarchy of SU(2) ⊗ SU(2) and the hierarchy of SO(4).

A tri-Hamiltonian formulation of a new soliton hierarchy associated with so(3,R)

A third Hamiltonian operator is presented for a new hierarchy of bi-Hamiltonian soliton equations, thereby showing that this hierarchy is tri-Hamiltonian. Additionally, an inverse hierarchy of common commuting symmetries is also presented.