Hierarchy Of Nonlinear Evolution Equations Research Articles

New integrable peakon equation and its dynamic system

A two component generalization Camassa–Holm equation with peakon is proposed, which is associated with a 2 × 2 matrix spectral problem with two potentials. As its reduction, a new integrable peakon equation is derived. Then positive flows associated with the 2 × 2 matrix spectral problem are discussed, a new hierarchy of nonlinear evolution equations is obtained. Subsequently, N-peakon of the peakon equation are constructed and the dynamical system with which the N-peakon evolve is deduced. As an example, exact single peakon and 2-peakon are given and their figures are plotted. Moreover, infinitely many conserved quantities of the peakon equation are obtained.

An extended KdV6 hierarchy of nonlinear evolution equations: Painlevé integrability and variety of branches of resonances

PurposeThe purpose of this paper is to study an extended hierarchy of nonlinear evolution equations including the sixth-order dispersion Korteweg–de Vries (KdV6), eighth-order dispersion KdV (KdV8) and many other related equations.Design/methodology/approachThe newly developed models have been handled using the simplified Hirota’s method, whereas multiple soliton solutions are furnished using Hirota’s criteria.FindingsThe authors show that every member of this hierarchy is characterized by distinct dispersion relation and distinct resonance branches, whereas the phase shift retains the KdV type of shifts for any member.Research limitations/implicationsThis paper presents an efficient algorithm for handling a hierarchy of integrable equations of diverse orders.Practical implicationsMultisoliton solutions are derived for each member of the hierarchy, and then generalized for any higher-order model.Social implicationsThis work presents useful algorithms for finding and studying integrable equations of a hierarchy of nonlinear equations. The developed models exhibit complete integrability, by investigating the compatibility conditions for each model.Originality/valueThis paper presents an original work with a variety of useful findings.

Propagation of Correlations in a Hard-Sphere System

The paper develops an approach to the description of the evolution of correlations for many hard spheres based on a hierarchy of evolution equations for the cumulants of the probability distribution function governed by the Liouville equation. It is established that the constructed dynamics of correlations underlies the description of the evolution of the states of many hard spheres described by the BBGKY hierarchy for reduced distribution functions or the hierarchy of nonlinear evolution equations for reduced correlation functions. As an application of the developed approach to describing the evolution of the state of many hard spheres within the framework of dynamics of correlations, the challenges of the derivation of kinetic equations are discussed.

A Degasperis–Procesi equation II with multi-peakon solutions

Based on a 3 × 3 matrix spectral problem, a new hierarchy of nonlinear evolution equations is obtained by means of the zero-curvature equation and the Lenard recursion equation. With the help of the trace identity, the Hamiltonian structures of this hierarchy are established. Then a new nonlinear wave equation, called the Degasperis–Procesi equation II, is derived from a negative flow, which admits exact solutions with N-peakons. Finite-dimensional dynamical systems related to N-peakon solutions and an infinite sequence of conserved quantities of the Degasperis–Procesi equation II are obtained.

From generalized Fourier transforms to spectral curves for the Manakov hierarchy. I. Generalized Fourier transforms

The generalized Fourier transforms (GFTs) for the hierarchies of multi-component models of Manakov type are revisited. Our aim is to adapt GFT so that one could treat the whole hierarchy of nonlinear evolution equations simultaneously. To this end we consider the potential Q of the Lax operator L as local coordinate on some symmetric space depending on an infinite number of variables t, and $$z_k$$ , $$k=1, 2, \ldots $$ . The dependence on $$z_k$$ is determined by the k-th higher flow (conserved density) of the hierarchy. Thus we have an infinite set of commuting operators with common fundamental analytic solution. We analyze the properties of the resolvent and thus determine the spectral properties of L. Then we derive the generalized Fourier transforms that linearize this hierarchy of NLEE and establish their fundamental properties as well as dynamical compatibility of each two pairs of such flows. Using the classical R-matrix approach we derive the Poisson brackets between all conserved quantities first assuming that Q is a quasi-periodic function of t. Next taking the limit when the period tends to $$\infty $$ , we derive the Poisson brackets between the scattering data of L. In addition we analyze a possible relation of our approach to the one based on the $$\tau $$ -function that could lead to multi-dimensional integrable equations.

A computational procedure for exact solutions of Burgers’ hierarchy of nonlinear partial differential equations

This paper considers the exact solution of Burgers’ hierarchy of nonlinear evolution equations. We construct the general nth conservation law of the hierarchy and prove that these expressions may be transformed into ordinary differential equations. In particular, a coordinate transformation leads to the systematic reduction of the conservation law properties of the Burgers’ hierarchy. Such an approach yields a nonlinear equation, where a second transformation is derived to linearize the expression. Consequently, this approach describes a procedure for finding the exact solutions of the hierarchy. A formula of the nth solution is provided, and to demonstrate its application, we discuss the solution to several members of the nonlinear hierarchy.

Two integrable generalizations of WKI and FL equations: Positive and negative flows, and conservation laws**Project supported by the National Natural Science Foundation of China (Grant Nos. 11971441, 11871440, and 11931017) and Key Scientific Research Projects of Colleges and Universities in Henan Province, China (Grant No. 20A110006).

With the aid of Lenard recursion equations, an integrable hierarchy of nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is proposed, in which the first nontrivial member in the positive flows can be reduced to a new generalization of the Wadati–Konno–Ichikawa (WKI) equation. Further, a new generalization of the Fokas–Lenells (FL) equation is derived from the negative flows. Resorting to these two Lax pairs and Riccati-type equations, the infinite conservation laws of these two corresponding equations are obtained.

Abelian versus non-Abelian Bäcklund charts: Some remarks

Connections via Baecklund transformations among different non-linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Baecklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Baecklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The Baecklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.

KdV-type equations linked via Bäcklund transformations: Remarks and perspectives

Third order nonlinear evolution equations, that is the Korteweg–de Vries (KdV), modified Korteweg–de Vries (mKdV) equation and other ones are considered: they all are connected via Bäcklund transformations. These links can be depicted in a wide Bäcklund Chart which further extends the previous one constructed in [22]. In particular, the Bäcklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Bäcklund Chart. That is, such a Bäcklund Chart is expanded to encompass the nonlinear equation for the KdV eigenfunctions [30], which finds its origin in the early days of the study of Inverse scattering Transform method, when the Lax pair for the KdV equation was constructed. The nonlinear equation for the KdV eigenfunctions is proved to enjoy a nontrivial invariance property. Furthermore, the hereditary recursion operator it admits [30] is recovered via a different method. Then, the results are extended to the whole hierarchy of nonlinear evolution equations it generates. Notably, the established links allow to show that also the nonlinear equation for the KdV eigenfunction is connected to the Dym equation since both such equations appear in the same Bäcklund chart.

Nonlinear bi-integrable couplings of a generalized Kaup–Newell type soliton hierarchy

Based on the three-dimensional special linear Lie algebra sl(2,ℝ), a generalized Kaup–Newell type soliton hierarchy is derived, the nonlinear bi-integrable coupling of which is constructed through the semi-direct sum of Lie algebra. The variational identity is used to furnish Hamiltonian structures of the resulting nonlinear evolution equations hierarchies.

Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions

This paper is a continuation of our previous work in which we studied a sl(3,ℂ) Zakharov-Shabat type auxiliary linear problem with reductions of Mikhailov type and the corresponding integrable hierarchy of nonlinear evolution equations. Now, we shall demonstrate how one can construct special solutions over constant background through Zakharov-Shabat’s dressing technique. That approach will be illustrated on the example of the generalized Heisenberg ferromagnet equation related to the linear problem for sl(3,ℂ). In doing this, we shall discuss the differences between the Hermitian and pseudo-Hermitian cases.

Conservation laws and Hamiltonian structures of the generalized sine-Gordon hierarchy

By introducing a 2 × 2 matrix spectral problem, a new hierarchy of nonlinear evolution equations is proposed. A typical equation in this hierarchy is the generalization of sine-Gordon equation. With the aid of trace identity, the Hamiltonian structures of the hierarchy are constructed. In addition, the infinite sequence of conserved quantities of the generalized sine-Gordon equation are obtained.

A Hierarchy of New Nonlinear Evolution Equations and Their Bi-Hamiltonian Structures

A hierarchy of new nonlinear evolution equations associated with a 3 × 3 matrix spectral problem with four potentials is proposed, in which two typical members are a new coupled Burgers equation and a new coupled KdV equation. The bi-Hamiltonian structures for the hierarchy of nonlinear evolution equations are established by using the trace identity.

A novel hierarchy of differential—integral equations and their generalized bi-Hamiltonian structures

With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 × 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.

Darboux transformation and Hamiltonian structure for the Jaulent–Miodek hierarchy

Starting from the Jaulent–Miodek spectral problem, we derive the associated hierarchy of nonlinear evolution equations in this paper. It is shown that this hierarchy is completely integrable in the Liouville sense and possesses the Hamiltonian structure. Moreover, by virtue of symbolic computation, two types of Darboux transformations for the whole hierarchy are explicitly constructed, which enables us to find the new soliton-like, shock and anti-shock solutions for the Jaulent–Miodek hierarchy. Figures are presented to discuss the properties of the new soliton-like solutions.

Nonlinear Evolution Equations: Bäcklund Transformations and Bäcklund Charts

Backlund Charts have been introduced to depict links, via Backlund Transformations, which relate different Nonlinear Evolution Equations. Notably, the links established among different Nonlinear Evolution Equations can be extended to the whole Hierarchies of Nonlinear Evolution Equations. This approach proved to be very fruitful, as well known, when applied to scalar Nonlinear Evolution Equations since it induces very many interesting results. Some of them are here reviewed and further new research perspectives are shown. Specifically, when the non commutative analogue Nonlinear Evolution Equations are introduced new problems arise. Here, hierarchies of non-commutative Nonlinear Evolution Equations together with their links via Backlund Transformations are considered. Specifically, a non-commutative analogues of Cole-Hopf and of Miura transformation are shown, respectively, to connect the operator versions of Burgers equation to heat equation and the operator version of KdV equation to the corresponding operator version of modified KdV equation. Again, the links are connecting not only the base member equations but all the corresponding equations in the hierarchy generated by the recursion operator it admits. Finally, it is pointed out how the method here presented allows to construct new non-commutative operator equation.

The Multi-Component Jaulent-Miodek Hierarchy and its Multi-Component Integrable Coupling System with Two Arbitrary Functions

We devise a new simple loop algebra GM and an isospectral problem. By making use of Tu scheme, the multi-component Jaulent-Miodek (JM) hierarchy is obtained. Furthermore, an expanding loop algebra FM of the loop algebra GM is presented. Based on FM the multi-component integrable couplings system with two arbitrary functions of the multi-component Jaulent-Miodek (JM) hierarchy are worked out. The method can be applied to other nonlinear evolution equations hierarchies.

Quasi‐Periodic Solutions of Nonlinear Evolution Equations Associated with a 3 × 3 Matrix Spectral Problem

A hierarchy of new nonlinear evolution equations associated with a 3 × 3 matrix spectral problem with three potentials is proposed. With the aid of the characteristic polynomial of Lax matrix for the hierarchy, we introduce an algebraic curve of arithmetic genus m − 1, and discuss in detail the properties of the associated Baker–Akhiezer function and meromorphic function. On the basis of the theory of algebraic curves, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic function, and, in particular, that of solutions for the entire hierarchy of nonlinear evolution equations.

A new eight-dimensional Lie superalgebra and two corresponding hierarchies of evolution equations

A new eight-dimensional Lie superalgebra is constructed and two isospectral problems with six potentials are designed. Corresponding hierarchies of nonlinear evolution equations, as well as super-AKNS and super-Levi, are derived. Their super-Hamiltonian structures are established by making use of the supertrace identity, and they are integrable in the sense of Liouville.

A LIE SUPERALGEBRA AND CORRESPONDING HIERARCHY OF EVOLUTION EQUATIONS

A Lie super algebra G is constructed and an isospectral problem with five potentials is designed. From there, a corresponding hierarchy of nonlinear evolution equations is derived and super-AKNS, super-TD, super-D-AKNS are deduced. Their super-Hamiltonian structure is established by making use of the supertrace identity, and they are intergrable in sense of Liouville.