Zero-curvature Representation Research Articles

Quasi-periodic solutions of a discrete integrable equation with a finite-dimensional integrable symplectic structure

Through the paper, we research the quasi-periodic solutions to a semi-discrete hierarchy which has Hamiltonian structure and integrable symplectic map. Firstly, a semi-discrete hierarchy is derived by use of the discrete zero-curvature representation and then its integrability is proved under the Liouville condition. Using the binary nonlinearization approach, the integrable symplectic map and finite-dimensional Hamiltonian system of the hierarchy are obtained. Moreover, the trigonal curve Kq−1 is denoted through the characteristic polynomials for the Lax pair, as well as the related Baker–Akhiezer function and meromorphic function are introduced, from which the asymptotic properties and divisors of the two functions mentioned above are analyzed. Finally, we introduce the three kinds of Abel differentials and straighten out of corresponding continuous and discrete flows, the quasi-periodic solutions of the equations are received via the Riemann theta function.

Darboux transformaiton and exact solutions of a four-component Volterra lattice system

In this letter, by means of the discrete zero curvature representation, four-component Volterra lattice (FCVL) hierarchy and its first flow, i.e., FCVL system are presented from a 4 × 4 matrix spectral problem with four potential functions. Moreover, a Darboux transformation for the FCVL system is established with the aid of the gauge transformation between the corresponding 4 × 4 matrix spectral problems, by which some exact solutions of the FCVL system are obtained in different cases.

Davydov–Kyslukha model as the starting point in the development of integrable multi-component nonlinear dynamical systems on quasi-one-dimensional lattices

The Davydov–Kyslukha nonlinear exciton-phonon model on a regular one-dimensional lattice is asserted to be the driving force for the development of integrable multi-component nonlinear dynamical systems encompassing excitonic, vibrational and orientational degrees of freedom. The two most representative quasi-one-dimensional integrable multi-component nonlinear systems inspired by the Davydov–Kyslukha model are presented explicitly in their concise Hamiltonian forms. The new six-subsystem integrable nonlinear model on a regular quasi-one-dimensional lattice is proposed and its derivation based upon the appropriate zero-curvature representation is presented. The constructive aspect of the famous Davydov motto is illustrated by the examples of semi-discrete integrable nonlinear dynamical systems canonicalizeable via the proper point transformations to the physically motivated field variables.

A Zero-Curvature Representation of Electromagnetism and the Conservation of Electric Charge

We show that the laws of electromagnetism in $(D+1)$-dimensional Minkowski space-time $\mathcal{M}$, explicitly for $D=1$, $2$ and $3$, can be obtained from an integral representation of the zero-curvature equation in the corresponding loop space $\mathcal{L}^{(D-1)}(\mathcal{M})$. The conservation of the electric charge can be seen as the result of a hidden symmetry in this representation of the dynamical equations.

The Integrability of a New Fractional Soliton Hierarchy and Its Application

Two fractional soliton equations are presented generated from the same spectral problem involved in a fractional potential by the zero-curvature representations. They are a kind of special reductions of the famous AKNS system. The two equations are integrable for they both possess explicit soliton solutions constructed by the N − fold Darboux transformation. As an application of the obtained solutions, new soliton solutions of the classic 2 + 1 -dimensional Kadometsev-Petviashvili (KP) equation are soughed out by a cubic polynomial relation. Dynamic properties are analyzed in detail.

Nonlinear dynamics of an integrable gauge-coupled exciton-phonon system on a regular one-dimensional lattice

A one-dimensional nonlinear dynamical system of intra-site excitations and lattice vibrations coupled via gauge-like mechanism is studied. The system admits the semi-discrete zero-curvature representation and therefore it proves to be integrable in the Lax sense. Relaying upon an appropriately developed Darboux–Bäcklund dressing technique the explicit four-component analytical solution to the system is found and analyzed in details. Due to mutual influence between the interacting subsystems the physically meaningful solution arises as the essentially nonlinear superposition of two principally distinct types of traveling waves. The interplay between the two typical spatial scales relevant to these traveling waves causes the criticality of system’s dynamics manifested as the dipole-monopole transition in the spatial distribution of intra-site excitations.

Nonlinear system of PT-symmetric excitations and Toda vibrations integrable by the Darboux–Bäcklund dressing method

The nonlinear dynamics of coupledPT-symmetric excitations and Toda-like vibrations on a one-dimensional lattice are studied analytically and elucidated graphically. The nonlinear exciton-phonon system as the whole is shown to be integrable in the Lax sense inasmuch as it admits the zero-curvature representation supported by the auxiliary linear problem of third order. Inspired by this fact, we have developed in detail the Darboux–Bäcklund integration technique appropriate to generate a higher-rank crop solution by dressing a lower-rank (supposedly known) seed solution. In the framework of this approach, we have found a rather non-trivial four-component analytical solution exhibiting the crossover between the monopole and dipole regimes in the spatial distribution of intra-site excitations. This effect is inseparable from the pronounced mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two essentially distinct types of travelling waves. We have established the criterion of monopole-dipole transition based upon the interplay between the localization parameter of Toda mode and the inter-subsystem coupling parameter.

A super KdV equation of Kupershmidt: Bäcklund transformation, Lax pair and related discrete system

A super extension of the Korteweg-de Vries (KdV) equation proposed by Kupershmidt is studied. Its Bäcklund transformation is explicitly presented and a Lax pair is derived, so the integrability of this system is confirmed. The corresponding nonlinear superposition formula is constructed from the Bäcklund transformation and is interpreted as a fully discrete system, whose integrability is demonstrated by verifying its consistency around the cube (CAC) property and supplying a zero-curvature representation.

A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

Coupled Nonlinear Dynamics in the Three-Mode Integrable System on a Regular Chain

The article suggests the nonlinear lattice system of three dynamical subsystems coupled both in their potential and kinetic parts. Due to its essentially multicomponent structure the system is capable to model nonlinear dynamical excitations on regular quasi-one-dimensional lattices of various physical origins. The system admits a clear Hamiltonian formulation with the standard Poisson structure. The alternative Lagrangian formulation of system’s dynamics is also presented. The set of dynamical equations is integrable in the Lax sense, inasmuch as it possesses a zero-curvature representation. Though the relevant auxiliary linear problem involves a spectral third-order operator, we have managed to develop an appropriate two-fold Darboux–Backlund dressing technique allowing one to generate the nontrivial crop solution embracing all three coupled subsystems in a rather unusual way.

Explicit Solutions of the Coupled Bogoyavlensky Lattice 1(2) Hierarchy

By means of the discrete zero curvature representation, the coupled Bogoyavlensky lattice 1(2) hierarchy related to a $$3\times 3$$ matrix problem is derived. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a trigonal curve $$\mathcal {K}_{m-1}$$ of arithmetic genus $$m-1$$ and construct the related Baker–Akhiezer function and meromorphic function. By analyzing the asymptotic expansion of the meromorphic function, the algebro-geometric solutions to the stationary coupled Bogoyavlensky lattice 1(2) are obtained. Moreover, the explicit theta function representations for the coupled Bogoyavlensky lattice hierarchy are given with the help of the Abelian differential.

Non-Abelian Evolution Systems with Conservation Laws

We find noncommutative analogs for well-known polynomial evolution systems with higher conservation laws and symmetries. The integrability of obtained non-Abelian systems is justified by explicit zero curvature representations with spectral parameter.

The Idea of a Lax Pair—Part I

In Part I [1], we introduced the idea of a Lax pair and explained how it could be used to obtain conserved quantities for systems of particles. Here, we extend these ideas to continuum mechanical systems of fields such as the linear wave equation for vibrations of a stretched string and the Kortewegde Vries (KdV) equation for water waves. Unlike the Lax matrices for systems of particles, here Lax pairs are differential operators. A key idea is to view the Lax equation as a compatibility condition between a pair of linear equations. This is used to obtain a geometric reformulation of the Lax equation as the condition for a certain curvature to vanish. This ‘zero curvature representation’ then leads to a recipe for finding (typically an infinite sequence of) conserved quantities.

True and Fake Lax Pairs: How to Distinguish Them

The gauge-invariant description of zero-curvature representations of evolution equations is applied to the problem of how to distinguish the fake Lax pairs from the true Lax pairs. The main difference between the true Lax pairs and the fake ones is found in the structure of their cyclic bases.

Integrable nonlinear triplet lattice system with the combined inter-mode couplings

In this article we introduce the nonlinear model of three dynamical subsystems coupled both in their kinetic and potential parts. Due to the quasi-one-dimensional spatial structure of its underlying lattice, the model grasps several degrees of freedom capable to imitate the dynamical behavior of long macromolecules both natural and synthesized origins. The model admits a clear Hamiltonian formulation with the standard form of fundamental Poisson brackets and it demonstrates the complex-conjugate symmetry between two subsystems of a Toda type. The integrability of system equations is supported by their zero-curvature representation based upon the auxiliary linear problem with the relevant spectral operator of third order. In view of these facts, we have developed an appropriate twofold Darboux–Backlund dressing technique capable to generate the nontrivial crop solution embracing all three coupled subsystems. The explicit crop solution to the nonlinear model is found to be of a pronounced pulson character.

Exact multisolitons of noncommutative and commutative Lakshmanan–Porsezian–Daniel equation

We propose and study a noncommutative generalization of Lakshmanan–Porsezian–Daniel equation. For this generalization, we demonstrate its integrability by constructing its zero curvature representation, Darboux transformation and multisoliton solutions. The soliton solutions up to fifth order are presented explicitly within framework of quasideterminants.

Flag Manifold Sigma Models and Nilpotent Orbits

In the present paper we study flag manifold sigma-models that admit a zero-curvature representation. It is shown that these models may be naturally considered as interacting (holomorphic and anti-holomorphic) $\beta\gamma$-systems. Besides, using the theory of nilpotent orbits of complex Lie groups, we establish a relation to the principal chiral model.

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In [13], for any (1+1)-dimensional scalar evolution equation E, we defined a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. In [12] we showed that, using these algebras, one obtains necessary conditions for existence of a Bäcklund transformation between two given equations.In this paper we show that, using the algebras F(E), one obtains some necessary conditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order 5. Also, we prove a result announced in [13] on the structure of the algebras F(E) for certain classes of equations of orders 3, 5, 7, which include KdV, mKdV, Kaup–Kupershmidt, Sawada–Kotera type equations. Among the obtained algebras for equations considered in this paper and in [12], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus 1 and 0.

An Integrable Matrix Camassa-Holm Equation**Supported by National Natural Science Foundation of China under Grant Nos. 11771186 and 11671177

We present an integrable sl(2)-matrix Camassa-Holm (CH) equation. The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws. This equation includes two undetermined functions, which satisfy a system of constraint conditions and may be reduced to a lot of known multi-component peakon equations. We find a method to construct constraint condition and thus obtain many novel matrix CH equations. For the trivial reduction matrix CH equation we construct its N-peakon solutions.

A Hermitian symmetric space Fokas–Lenells equation: Solitons, breathers, rogue waves

Based on the zero-curvature representation, we propose a Hermitian symmetric space Fokas–Lenells equation associated with a 4 × 4 matrix spectral problem. Resorting to the gauge transformation between spectral problems, we find a restrictive condition of the spectral functions that can ensure the symmetry of the new potential matrix, from which n-fold classical Darboux transformations and N-fold generalized Darboux transformations of this equation are constructed with the help of the limit technique and an iterative procedure. Utilizing the Darboux transformations and Mathematica software, we obtain various solutions for the Hermitian symmetric space Fokas–Lenells equation, including the multi-soliton solution, kink–breather solution, eye-shaped rogue wave solution, one-peaky shaped rogue wave solution, and dark two-peaky shaped rogue wave solution. By varying the values of the free parameters, we obtain different types of spatial–temporal distribution structures for the solutions, including the fundamental, line and triangular patterns. Furthermore, the dynamical behaviour of these solutions, including the fusion and fission processes of rogue waves, is analysed in detail.