Integrable Hierarchies Research Articles

Multi-component generalized Gerdjikov–Ivanov integrable hierarchy and its Riemann–Hilbert problem

In this paper, the multi-component generalized Gerdjikov–Ivanov integrable hierarchy is obtained by means of the corresponding spectral problem and the zero curvature equation. With the help of trace identity, the multi-Hamiltonian structures of this integrable hierarchy are investigated. Based on the spectral problem, a specific Riemann–Hilbert problem is formulated for the generalized Gerdjikov–Ivanov integrable hierarchy. When the jump matrix is a unit matrix, through solving this Riemann–Hilbert problem, the N-soliton solutions can be derived.

Coverings and nonlocal symmetries as well as fundamental solutions of nonlinear equations derived from the nonisospectral AKNS hierarchy

In the paper, we apply a new scheme to generate nonisospectral integrable hierarchies of evolution equations. First of all, we work out the positive-order and the negative-order nonisospectral AKNS hierarchies. Next we introduce a kind of loop algebra from which an expanding nonisospectral integrable model is derived. Specially, the expanding integrable model reduces to the well-known bond pricing equation and the interest rate modeling. By using the Lie group theory we investigate the Lie representations and characteristic solutions of a generalized bond pricing equation from the positive-order nonisospectral integrable equation, and with the help of the characteristic solutions and the Laplace transformation, we again obtain the fundamental solutions of the generalized bond pricing equation. Besides, we apply the Lie group analysis to deduce the invariant solutions and the nonlinear self-adjointness as well as conservation laws, non-invariant solutions of the generalized bond pricing equation. Some reductions of the negative-order nonisospectral integrable hierarchy are also produced. Two of them are the Sine-Gordan equation and the Sinh-Gordan equation. Finally, we investigate the coverings and the nonlocal symmetries of the generalized bond pricing equation and a generalized KdV equation, respectively, although the bond equation is linear which has many applications in the aspect of finance field, by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.

On the integrable hierarchy for the resolved conifold

We provide a direct proof of a conjecture of Brini relating the Gromov–Witten (GW) theory of the resolved conifold to the Ablowitz–Ladik (AL) integrable hierarchy at the level of primaries. In doing so, we use a functional representation of the AL hierarchy as well as a difference equation for the GW potential. In particular, we express certain distinguished solutions of the difference equation in terms of an analytic function which is a specialization of a Tau function put forward by Bridgeland in the study of wall-crossing phenomena of Donaldson–Thomas invariants.

The N-soliton solutions to the M-components nonlinear Schrödinger equations by the Riemann–Hilbert approach

In this paper, the main work is to study the N -soliton solutions for the M -component nonlinear Schrödinger equations, the matrix Riemann–Hilbert problem is constructed for this integrable hierarchies by analyzing the block matrix spectral problem of the Lax pair, then the N -soliton solutions for this system are given explicitly by the scattering matrix. Finally, We briefly summarize the work of this article and make prospects for future work.

A new construction of the Drinfeld–Sokolov hierarchies

The Drinfeld–Sokolov hierarchies are integrable hierarchies associated with every affine Lie algebra. We present a new construction of such hierarchies, which only requires the computations of a formal Laurent series.

Several Integrable Properties of the Discrete KP Hierarchy Obtained by Means of Gauge Transformations

The gauge transformation is a powerful method to solve integrable hierarchies. In the discrete KP hierarchy, there are two basic gauge transformations, T D ( q ) = Λ( q ) ∘ Δ ∘ q −1 and T I ( r ) = Λ −1 ( r −1 ) ∘ Δ −1 ∘ r. This paper is concerned with several integrable properties of the discrete KP hierarchy found by means of gauge transformations. The successive applications of T I to the constrained discrete KP hierarchy have been presented. Meanwhile, we derive two novel discrete equations as the orbits of gauge transformations T D and T I for the constrained KP hierarchy. Moreover, with the help of the spectral representations and the gauge transformations, Fay-similar equations of the tau functions for the constrained discrete KP have been investigated. In addition, we also discuss the relation between the gauge transformation and an additional symmetry for the discrete KP hierarchy.

A discrete KdV equation hierarchy: continuous limit, diverse exact solutions and their asymptotic state analysis

In this paper, a discrete KdV equation that is related to the famous continuous KdV equation is studied. First, an integrable discrete KdV hierarchy is constructed, from which several new discrete KdV equations are obtained. Second, we correspond the first several discrete equations of this hierarchy to the continuous KdV equation through the continuous limit. Third, the generalized (m, 2N − m)-fold Darboux transformation of the discrete KdV equation is established based on its known Lax pair. Finally, the diverse exact solutions including soliton solutions, rational solutions and mixed solutions on non-zero seed background are obtained by applying the resulting Darboux transformation, and their asymptotic states and physical properties such as amplitude, velocity, phase and energy are analyzed. At the same time, some soliton solutions are numerically simulated to show their dynamic behaviors. The properties and results obtained in this paper may be helpful to understand some physical phenomena described by KdV equations.

Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals

We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy which controls the special cubic Hodge integrals.

Virasoro constraints for moduli of weighted pointed stable curves

We formulate Virasoro constraints for the generating functions of the intersection numbers on Hassett's moduli of weighted pointed curves and show that they are governed by the KdV integrable hierarchy.

Squared eigenfunction symmetries of the constrained integrable hierarchies

In this paper, we mainly investigate the constrained Kadomtsev–Petviashvili (KP) and modified KP (mKP) hierarchies. Squared eigenfunction (SE) symmetry is an important symmetry defined by eigenfunctions and adjoint eigenfunctions, which can be viewed as the generator of the additional symmetries. Due to special constraints on the Lax operators, it is found that there are three different cases for the generating eigenfunctions and adjoint eigenfunctions of the SE symmetries. Then we discuss the changes of the SE symmetries under the Miura links between the KP and mKP hierarchies. Next, the exponential of the SE symmetries is shown to be the binary Darboux transformations. At last based upon the results above, new additional symmetries for the constrained KP and mKP hierarchies are constructed, different from the additional Virasoro symmetries defined by Aratyn.

Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity

We discuss integrable aspects of the logarithmic contribution of the partition function of cosmological critical topologically massive gravity. On one hand, written in terms of Bell polynomials which describe the statistics of set partitions, the partition function of the logarithmic fields is a generating function of the potential Burgers hierarchy. On the other hand, the polynomial variables are solutions of the Kadomtsev-Petviashvili equation, and the partition function is a KP \tauτ function, making more precise the solitonic nature of the logarithmic fields being counted. We show that the partition function is a generating function of Hurwitz numbers, and derive its expression. The fact that the partition function is the generating function of branched coverings gives insight on the orbifold target space. We show that the logarithmic field \psi^{new}_{\mu \nu}ψμνnew can be regarded as a branch point field associated to the branch point \mu l =1μl=1.

A typology of northwestern Bantu gender systems

AbstractNorthwestern Bantu is the most linguistically diverse area of the Bantu-speaking world. Several unusual grammatical gender systems are reported for this area, but there has been a lack of comprehensive comparative studies. This article is a typological investigation of northwestern Bantu gender systems based on a sample of 179 languages. We study the distribution of various patterns of animacy-based agreement in the languages of the sample and in relationship with the Agreement Hierarchy. We find that animacy-based agreement is widespread in northwestern Bantu. If restricted to animate nouns, it tends to coexist in stable variation with syntactic agreement. When generalized to both animate and inanimate nouns, animacy-based agreement appears to contribute to the erosion of gender marking. In line with the prediction of the Agreement Hierarchy, we find that animacy-based agreement is prevalent with verbs and pronouns. Within the noun phrase, it spreads in ways that are suggestive of a hierarchy of syntactic integration between nouns and adnominal modifiers, which had gone unnoticed in the existing literature. These results have important implications for current models of Bantu gender systems and shed new light on animacy effects in the diachrony of gender more generally.

Novel computational technique; the second positive member in a new completely integrable hierarchy

This paper investigates novel solitary wave solutions of the second positive member in a new utterly integrable hierarchy by employing a novel computational technique. This new method is called the extended Khater (Ext Khat.) method, and it is the first time to apply it to any nonlinear evolution equation. Besides its large number of solutions but it also covers six different analytical schemes’ solutions. Consequently, these methods can be considered as special cases of our new technique. Many distinct solutions are obtained, and no one of them is singular. It is also considered an additional advantage of our new technique, where many of previous employed computational methods get some singular solutions among their solutions. The singular solution does not have any physical meaning, which makes it meaningless. Our new technique is applied here to a Lax representation and bi–Hamiltonian structure. Many solutions are obtained in different formulas such as hyperbolic, trigonometric, rational, and so on.Additionally, some solutions are represented through different sketch styles. Moreover, the stability property of these solutions is also investigated based on the properties of the Hamiltonian system. Finally, our paper’s originality is investigated by comparing our solutions, and those have been obtained in previous work.

Two new integrable differential-difference hierarchies related to the relativistic Toda lattice equation and their soliton fusion or fission

Starting from a discrete spectral problem, two new integrable differential-difference hierarchies are presented and their Lax pairs are given, respectively. The first hierarchy is related to the relativistic Toda lattice equation. Darboux transformations for the first systems in the two hierarchies are derived, respectively, accordingly their explicit solutions are obtained. By selecting appropriate parameters, we can see the phenomena of soliton fission or soliton fusion.

Constrained Toda hierarchy and turning points of the Ruijsenaars–Schneider model

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^{\dag}$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities.

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.

Realization of integrative hierarchy by in-situ solidification of ‘semi-cured’ microcilia array in candle flame for robust and flexible superhydrophobicity

Flexible superhydrophobic surfaces have recently attracted extensive interest owing to the potential for diverse curvatures and emerging flexible electronics. However, the simultaneous realization of mechanical robustness, chemical resistance, and maintained superhydrophobicity during deformation is still challenging. Herein, we introduced an environmental-friendly method to produce the integrative multi-level structures by generating the micro-cilia array in a ‘template-free’ manner, followed with in-situ thermal curing in the candle flame before the complete solidification of the micro-structured surface. Instead of weak physical adhesion (van der Waals forces), the candle soot nanoparticles penetrated into the ‘semi-cured’ micro-cilia surface, along with the high temperature to rapidly cure and generate the PDMS-connected integrative architectures, which effectively avoid the interface between the micro/nano-scaled components to eliminate the mechanical fragility. Comparative investigations confirm that the elevated robustness is attributed by the synergistic effect from the ‘semi-cured’ surface and the formation mechanism of subsequent hierarchical structures. The flexible film thus exhibits significantly improved stability against mechanical damage (ultrasonic processing, adhesive pressing, and linear abrasion), chemical corrosion, and water-jet impalement (∼10.4 m s−1 at ∼ 0.17 MPa). With the realization of co-existent features and real-life applications, the green methodology should be promising to open up an avenue that moves superhydrophobicity further to our world.

Multivariate Toda hierarchies and biorthogonal polynomials

A new multivariate Toda hierarchy of nonlinear partial differential equations adapted to multivariate biorthogonal polynomials is discussed. This integrable hierarchy is associated with non-standard multivariate biorthogonality. Wave and Baker functions, linear equations, Lax and Zakharov–Shabat equations, KP type equations, appropriate reductions, Darboux or linear spectral transformations, and bilinear equations involving linear spectral transformations are presented.

Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations

By introducing a 3×3 matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral–nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the 3×3 matrix Lie algebra into a 2×2 matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral–nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schrödinger integrable system and a nonlocal nonlinear Schrödinger equation, which indicates that we unite the KdV equation and the nonlinear Schrödinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.

Multivariate Orthogonal Laurent Polynomials and Integrable Systems

An ordering for Laurent polynomials in the algebraic torus (\mathbb{C}^*)^D , inspired by the Cantero–Moral–Vel´azquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus \mathbb{T}^D . The Gauss–Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus, which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second-kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second-kind functions are found, as well as Plemelj-type formulae. Spectral matrices give string equations for the moment matrix, which model the three-term relations as well as the Christoffel–Darboux formulae. Christoffel-type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used to find a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for prepared Laurent polynomials, which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of prepared Laurent polynomial perturbation and poised sets is given; full-column-rankness of the corresponding multivariate Laurent–Vandermonde matrices and a product of different prime prepared Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue–Haar measure. Discrete and continuous deformations of the measure lead to a Toda-type integrable hierarchy, being the corresponding flows described through Lax and Zakharov–Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of two-dimensional Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss–Borel factorization of the Jacobi-type matrices and its quasi-determinants allow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted \tau -functions in the one-dimensional scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomtsev–Petviashvili-type system.