De Vries Hierarchy Research Articles

Integrable flows on null curves in the Anti-de Sitter 3-space

Abstract We formulate integrable flows related to the Korteweg–De Vries (KdV) hierarchy on null curves in the anti-de Sitter 3-space ( AdS ). Exploiting the specific properties of the geometry of AdS , we analyze their interrelationships with Pinkall flows in centro-affine geometry. We show that closed stationary solutions of the lower order flow can be explicitly found in terms of periodic solutions of a Lamé equation. In addition, we study the evolution of non-stationary curves arising from a 3-parameter family of periodic solutions of the KdV equation.

Polynomial dynamical systems associated with the KdV hierarchy

In 1974, S.P. Novikov introduced the stationary n-equations of the Korteweg–de Vries hierarchy, namely the n-Novikov equations. These are associated with integrable polynomial dynamical systems, with polynomial 2n integrals, in ℂ3n. In this paper, we construct an infinite-dimensional polynomial dynamical system that is universal for all dynamical systems corresponding to the n-Novikov equations. Thus, we solve the well-known problem of the relationship between the n-Novikov equations for different n.

Non‐auto Darboux transformation for N‐soliton solutions of the non‐isospectral Korteweg–de Vries hierarchy

The usual non‐isospectral Korteweg–de Vries (KdV) hierarchy (see Equation 4) does not have auto Darboux transformations. In this paper, we present a non‐auto Darboux transformation for this hierarchy, from which we are able to construct its ‐soliton solutions. An explicit formula of the solutions with zero background is given, and solutions are illustrated.

On the Quartic Korteweg–de Vries hierarchy of nonlinear Rossby waves and its dynamics

A long-standing challenge in exploring the dynamics of large-scale atmospheric or oceanic motions is to find more appropriate theoretical models. This paper aims at the mechanisms of nonlinear Rossby waves by a new obtained nonlinear evolutionary equation hierarchy approach. The multiple scales method and weak nonlinear perturbation method are adopted based on the quasi-geostrophic potential vorticity conservation law for large scale atmospheric or oceanic motions. It is the first time that a Quartic Korteweg–de Vries model equation is derived to describe the propagation of nonlinear Rossby waves amplitude, which provides a new acceptable theoretical model for the study of atmospheric blocking phenomena. Qualitative and quantitative analysis for the new Quartic KdV model are finished to disclose the evolutionary mechanisms of nonlinear Rossby waves. The corresponding results declares that the β effect, topographic features and detuning parameter are all important factors on the nonlinear Rossby waves.

Pöppe triple systems and integrable equations

We construct the combinatorial Pöppe triple system, or ternary algebra, that underlies the non-commutative nonlinear Schrödinger (NLS) and modified Korteweg–de Vries (mKdV) hierarchy. We demonstrate that the Pöppe triple system provides an effective and systematic procedure for establishing that the NLS and mKdV equations are directly linearisable, which, in principle, extends to the whole hierarchy. This naturally extends the combinatorial Pöppe algebra, recently used to constructively establish integrability and uniqueness for the whole non-commutative potential Korteweg–de Vries hierarchy, to the NLS and mKdV hierarchy.

The algebraic structure of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy

We prove that each member of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine–Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Pöppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Pöppe product—the product rule for Hankel operators pioneered by Ch. Pöppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a ‘Pöppe polynomial’ expansion for basic compositions in terms of ‘linear signature expansions’ representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order.

Integrability of linear structure of components of modified Korteweg–De Vries hierarchy: multiple soliton solutions and breathers solutions

PurposeThe purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation, 5th order and 7th order mKdV equations.Design/methodology/approachThe authors investigate Painlevé integrability of the constructed linear structure.FindingsThe Painlevé analysis demonstrates that established sum of integrable models retains the integrability of each component.Research limitations/implicationsThe research also presents a set of rational schemes of trigonometric and hyperbolic functions to derive breather solutions.Practical implicationsThe authors also furnish a variety of solitonic solutions and complex solutions as well.Social implicationsThe work formally furnishes algorithms for extending integrable equations that consist of components of a hierarchy.Originality/valueThe paper presents an original work for developing Painlevé integrable model via using components of a hierarchy.

Wave solutions to an integrable negative order KdV equation

In this paper, we study the traveling wave solutions to the second representative equation of the negative-order Korteweg–de Vries hierarchy. The corresponding traveling wave system is a singular planar dynamical system with a singular straight line. By using the method of dynamical systems, explicit solutions are given. But this integrable negative order Korteweg–de Vries flow has no traveling solitary wave solution.

New integrable multi-Lévy-index and mixed fractional nonlinear soliton hierarchies

In this letter, we present a simple and new idea to generate two types of novel integrable multi-Lévy-index and mix-Lévy-index (mixed) fractional nonlinear soliton hierarchies, containing multi-index and mixed fractional higher-order nonlinear Schrödinger (NLS) hierarchy, fractional complex modified Korteweg–de Vries (cmKdV) hierarchy, and fractional mKdV hierarchy. Their explicit forms can be given using the completeness of squared eigenfunctions. Moreover, we present their anomalous dispersion relations via their linearizations, and fractional multi-soliton solutions via the inverse scattering transform with matrix Riemann–Hilbert problems. These obtained fractional multi-soliton solutions may be useful to understand the related super-dispersion transports of nonlinear waves in multi-index fractional nonlinear media.

New semi-discrete AKNS hierarchy and its reductions

In the paper we consider the new semi-discrete Ablowitz–Kaup–Newell–Segur (sdAKNS) hierarchies and their reduction cases. In terms of a new linear combination of the two-potential Ablowitz–Ladik (AL) recursion operator and inverse recursion operator, the new sdAKNS isospectral and non-isospectral flows, as well as corresponding hierarchies are derived. By means of the continuous limit theory, both these flows and hierarchies go to the continuous AKNS flows and hierarchies, and the recursion operator goes to the AKNS recursion operator. Moreover, as the reduction systems of the new sdAKNS system, the vector form of the new semi-discrete nonlinear Schrödinger (sdNLS) hierarchies and the new semi-discrete modified Korteweg–de Vries (sdmKdV) hierarchies can be deduced under the different reductions. Furthermore, a new sdNLS equation and a new sdmKdV equation are derived. The above equations go to the counterparts of the continuous AKNS system and its reduction cases under continuous limit.

W algebra and eigenvalues of the low-order string constraint operators for the discrete KP hierarchy

Firstly, for any p ∈ N , the string constraint operators without the redundant variables are calculated, which are imposed by the string equation on the τ -function of the p -reduced discrete Kadomtsev–Petviashvili (dKP) hierarchy. Furthermore, the commutators of the string constraint operators without the redundant variables are calculated. It is found that although the string constraint operators without the redundant variables are different from the original string constraint operators with the redundant variables, they also form a Virasoro algebra and a W algebra after a transformation. Moreover, using the algebraic structure, eigenvalues of the low-order string equation constraint operators are calculated. Based on the obtained eigenvalues, it is also proved that the τ -function of the p -reduced dKP hierarchy constrained by the string equation is a vacuum vector for a Witt algebra. When p = 2 , it is coincident with the classical fact that the τ -function of the Korteweg–de Vries hierarchy constrained by the string equation is a vacuum vector for a Virasoro algebra.

Nonlocal symmetries and Darboux transformations of the Camassa–Holm equation and modified Camassa–Holm equation revisited

By the nonlocal symmetry approach, Hernández-Heredero and Reyes [J. Phys. A: Math. Theor. 42, 182002 (2009)] and Bies et al. [J. Math. Phys. 53, 073710 (2012)] obtained Darboux Transformations (DTs) of the Camassa–Holm equation and the modified Camassa–Holm equation. However, the wave function does not appear in their DTs explicitly. We introduce wave functions to the DTs and show how they are related to the binary DT of the first negative flow in the Korteweg–de Vries (KdV) hierarchy. Furthermore, we connect nonlocal symmetries of the Camassa–Holm equation and the modified Camassa–Holm equation with those of the negative KdV equation and the negative modified KdV equation, respectively.

The non-commutative Korteweg–de Vries hierarchy and combinatorial Pöppe algebra

We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg–de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearisable. Indeed we prove this for any linear combination of fields from this hierarchy. That each member of the hierarchy is linearisable, and integrable in this sense, means that the time evolving solution can be generated from the solution to the corresponding linear dispersion equation in the hierarchy, combined with solving an associated linear Fredholm equation representing the Marchenko equation. Further, we show that within the class of polynomial partial differential fields, at every order, each member of the non-commutative potential Korteweg–de Vries hierarchy is unique. Indeed, we prove to all orders, that each such member matches the non-commutative Lax hierarchy field, which is therefore a polynomial partial differential field. We achieve this by constructing the abstract combinatorial algebra that underlies the non-commutative potential Korteweg–de Vries hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by the set of all compositions endowed with the Pöppe product. This product is the abstract representation of the product rule for Hankel operators pioneered by Ch. Pöppe for integrable equations such as the Sine–Gordon and Korteweg–de Vries equations. Integrability of the hierarchy members translates, in the combinatorial algebra, to proving the existence of a ‘Pöppe polynomial’ expansion for basic compositions in terms of ‘linear signature expansions’. Proving the existence of such Pöppe polynomial expansions boils down to solving a linear algebraic problem for the expansion coefficients, which we solve constructively to all orders.

Riemann–Hilbert Problems and Soliton Solutions of Type (λ∗, −λ∗) Reduced Nonlocal Integrable mKdV Hierarchies

Reduced nonlocal matrix integrable modified Korteweg–de Vries (mKdV) hierarchies are presented via taking two transpose-type group reductions in the matrix Ablowitz–Kaup–Newell–Segur (AKNS) spectral problems. One reduction is local, which replaces the spectral parameter λ with its complex conjugate λ∗, and the other one is nonlocal, which replaces the spectral parameter λ with its negative complex conjugate −λ∗. Riemann–Hilbert problems and thus inverse scattering transforms are formulated from the reduced matrix spectral problems. In view of the specific distribution of eigenvalues and adjoint eigenvalues, soliton solutions are constructed from the reflectionless Riemann–Hilbert problems.

Lagrangian Multiforms for Kadomtsev–Petviashvili (KP) and the Gelfand–Dickey Hierarchy

Abstract We present, for the first time, a Lagrangian multiform for the complete Kadomtsev–Petviashvili hierarchy—a single variational object that generates the whole hierarchy and encapsulates its integrability. By performing a reduction on this Lagrangian multiform, we also obtain Lagrangian multiforms for the Gelfand–Dickey hierarchy of hierarchies, comprising, among others, the Korteweg–de Vries and Boussinesq hierarchies.

On the spectral problem of the quantum KdV hierarchy

The spectral problem for the quantum dispersionless Korteweg–de Vries (KdV) hierarchy, aka the quantum Hopf hierarchy, is solved by Dubrovin. In this article, following Dubrovin, we study Buryak–Rossi’s quantum KdV hierarchy. In particular, we prove a symmetry property and a non-degeneracy property for the quantum KdV Hamiltonians. On the basis of this we construct a complete set of common eigenvectors. The analysis underlying this spectral problem implies certain vanishing identities for combinations of characters of the symmetric group. We also comment on the geometry of the spectral curves of the quantum KdV hierarchy and we give a representation of the quantum dispersionless KdV Hamiltonians in terms of multiplication operators in the class algebra of the symmetric group.

Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves

This survey is devoted to integrable polynomial Hamiltonian systems associated with symmetric powers of plane algebraic curves. We focus our attention on the relations (discovered by the authors) between the Stäckel systems, Novikov’s equations for the th stationary Korteweg– de Vries hierarchy, the Dubrovin–Novikov coordinates on the universal bundle of Jacobians of hyperelliptic curves, and new systems obtained by considering the symmetric powers of curves when the power is not equal to the genus of the curve. Bibliography: 52 titles.

Nonlocal symmetries and molecule structures of the KdV hierarchy

The nonlocal symmetries are important because they carry information about the existence of Darboux–Backlund and linearizing transformations, and they also allow us to construct nontrivial solutions to systems. In this paper, we focus on investigating three sets of nonlocal symmetries which are realized as appropriate local symmetries of related auxiliary systems for the Korteweg–de Vries hierarchy. Specifically, we construct infinitely many nonlocal symmetries from three different aspects using residues, Lax pairs and quadratic pseudopotentials for each member of the Korteweg–de Vries hierarchy and show how these nonlocal symmetries connect each other. These infinitely many nonlocal symmetries enable us to construct multiple soliton solutions. Moreover, we adapt the multiple soliton solutions derived from nonlocal symmetries and describe a procedure by using velocity resonance mechanism to find molecule solutions, which can be found not only in the optical systems, but also in fluid systems, for the combined third-fifth-order Korteweg–de Vries equation. Several types of molecule structures including soliton molecules, breather molecules and soliton–breather molecules are illustrated.

Hamiltonian multiform description of an integrable hierarchy

Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by the results of the authors on the role of covariant Hamiltonian formalism for integrable field theories, we propose the notion of Hamiltonian multiforms for integrable 1 + 1-dimensional field theories. They provide the Hamiltonian counterpart of Lagrangian multiforms and encapsulate in a single object an arbitrary number of flows within an integrable hierarchy. For a given hierarchy, taking a Lagrangian multiform as the starting point, we provide a systematic construction of a Hamiltonian multiform based on a generalization of techniques of covariant Hamiltonian field theory. This also produces two other important objects: a symplectic multiform and the related multi-time Poisson bracket. They reduce to a multisymplectic form and the related covariant Poisson bracket if we restrict our attention to a single flow in the hierarchy. Our framework offers an alternative approach to define and derive conservation laws for a hierarchy. We illustrate our results on three examples: the potential Korteweg–de Vries hierarchy, the sine-Gordon hierarchy (in light-cone coordinates), and the Ablowitz–Kaup–Newell–Segur hierarchy.

On non-steady planar motions of fibre-reinforced fluids. Geometry and integrable structure

It is shown that the kinematic system describing planar non-steady motions of ideal fibre-reinforced fluids may be reduced to a single two-dimensional third-order partial differential equation in which time enters parametrically. A procedure is established which maps steady motions to non-steady motions. The resulting motions inherit their hidden integrable structure from the steady case. The formalism presented here also readily recovers the connection with the scattering problem of the modified Korteweg–de Vries (mKdV) hierarchy established in previous work.