召集人：常向科（中国科学院数学与系统科学研究院，副研究员）、Andy Hone（University of Kent，教授）、屈长征（宁波大学，教授）

时间：2025.04.27—2025.05.03

Seminar Schedule

Abstracts

Stephen Anco, Brock University

Title: Dynamics of peakons on kink backgrounds

Abstract: Peakons are peaked exponential functions that go to zero at spatial infinity. In this talk, I present work studying peakons with non-zero boundary conditions at infinity. In particular, explicit peakons propagating on a kink like background will be shown to exist for a class of peakon equations. Included in this class is the FORQ equation with a non-zero convective term. The dynamics of these peakons are very rich. A generalization to multi-peakons, interacting amongst themselves, which propagate on kink like backgrounds will be considered.

Matteo Casati, Ningbo University

Title: Multi-component Hamiltonian difference operators

Abstract: Many integrable differential-difference systems possess bi-Hamiltonian structure in form of difference operators. Until recently, Hamiltonian structures for multi-component systems had been investigated for (-1,1)-shifts operators with nondegenerate leading term and related to Lie bialgebras. In this report, I present the full classification of two-components operators including degenerate case and will present the relation between their Poisson cohomology and the bi-Hamiltonian structure of several known integrable systems. This is a joint work with  D. Valeri (Sapienza University of Rome, Italy)

Anton Dzhamay, Yanqi Lake Beijing Institute of Mathematical Sciences and Applications

Title: Identification problem for Discrete Painlevé Equations and Applications

Abstract: The notion of a discrete Painlevé equation formally appeared over 30 years ago, and since then many examples of such equations were discovered in a variety of applied problems, particularly in the theory of orthogonal polynomials. However, naming a particular discrete Painlevé equation is a non-trivial matter. The famous Sakai classification scheme describes possible types of the equations, but there are still infinitely many non-equivalent equations that have the same type. We also need to pay attention to special values of parameters. In this talk we will describe one possible ``refined" naming scheme and illustrate it by a few examples coming from applications.

Ying Fu, Northwest University

Title: Reducibility of a class of operators induced by the dispersive third order Benjamin-Ono equation

Abstract: TBA

Cheng He, Ningbo University

Title: Algebraic solitons in the massive Thirring model

Abstract: In this talk, I will present the algebraic solitons of the massive Thirring model. First, we derive the double-pole algebraic solitons using the bilinear form, then obtain N-pole solitons via the double-Wronskian technique. Finally, we analyze the stability of these algebraic solitons. Although exact stability problem remain open, I will discuss the challenges each conventional method encounters when dealing with algebraic solitons.

Darryl Holm, Imperial College London and DAMTP, Cambridge

Title: Compound Burgers -- KdV Soliton Behaviour: Refraction, Reflection & Fusion

Abstract: The Burgers–swept KdV (BS-KdV) system governs the interactions between `bore'-like solutions to the Burgers equation and wave-like solutions to the KdV equation in shallow water. We derive the BS-KdV equations and their compound soliton solutions via a variational principle based on the composition of the Burgers and KdV flow maps. We show the Burgers and KdV momenta asymptotically tend toward a balance at which the BS-KdV system reduces to the integrable Gardner equation. Numerical simulations reveal a plethora of nonlinear solution behaviour including refraction, reflection, and soliton fusion, before balance is finally achieved. Joint work with R. Hu, O.D. Street, H. Wang.

Andy Hone, University of Kent

Title: Integrable combinatorics, Diophantine problems and tau functions for integer sequences

Abstract: We consider enumeration problems associated with sequences of positive integers, which can be solved using explicit evaluations of tau functions for integer sequences. In particular, we consider various examples from cluster algebras and Diophantine geometry, and their connections with Somos sequences, which can be understood via the theory of discrete integrable systems.

Xinwei Jin, Zhejiang Normal University

Title: Nonlinear magnetic excitations and their manipulation

Abstract: The competitive balance among multiple interactions in magnetic materials gives rise to a significant class of localized spin structures—magnetic solitons. These solitons exhibit distinct and diverse configurations across different dimensions, making them prime candidates for next-generation magnetic storage units. Moreover, the recent emergence of novel effects in spintronics, including spin transfer torque (STT) and spin-orbit torque (SOT), has provided an extensive toolbox for manipulating and potentially applying magnetic solitons. From both fundamental and technological perspectives, the discovery of novel magnetic solitons continues to attract widespread interest while expanding the family of spin textures. This presentation will explore the fundamental theories and application prospects of various magnetic quasiparticle excitations, with emphasis on our advances in generating and controlling novel magnetic solitons. Key topics include the engineering of chiral magnetic solitons, coupled magnetic solitons, and gap magnetic solitons; manipulation strategies using spin currents, bias voltages, and external magnetic fields; and discussions on their potential applications.

Huan Liu, Zhengzhou University

Title: From solving integrable equations to extending discrete integrable operators

Abstract: As highlighted in the seminal work by Borodin (IMRN, 2000(9):467-494), there exists a profound connection between discrete integrable operator constructions and soliton solutions of integrable systems. This fundamental relationship naturally motivates an important extension: Can we develop generalized discrete integrable operators that correspond to higher-order pole solutions in integrable partial differential equations? In this talk, we first elucidate the systematic construction of higher-order pole solutions for the nonlinear Schrödinger equation through the framework of Riemann-Hilbert analysis. Building upon this theoretical foundation, we then establish an extension mechanism for discrete integrable operators.

Qingping Liu, China University of Mining and Technology (Beijing)

Title: A two-component Camassa-Holm equation

Abstract: TBA

Kenichi Maruno, Waseda University

Title: Exploring the structure of higher-order recurrence equations

Abstract: A recurrence equation is a discrete dynamical system with the remarkable property that every solution is periodic for arbitrary initial values, making it a special class of integrable mappings within the theory of discrete integrable systems.　In this study, we extend the method introduced by Hirota and Yahagi for constructing recurrence equations, and conduct a systematic search for higher-order recurrence equations.

Frank Nijhoff, University of Leeds

Title: Two bi-elliptic integrable stories

Abstract: This talk comprises two (possibly related) stories in which bi-elliptic structures appear in connection with integrable discrete systems. The first is about bi-elliptic soliton solutions of the Q4 quad-lattice equation, which leans on (novel) bi-elliptic addition formulae (this goes back to work with James Atkinson and Jarmo Hietarinta).

The second story is about a recently proposed generalisation of the QRT map on a bi-elliptic surface (which is work with Nalini Joshi and Allan Steel).  

Vladimir Novikov, Loughborough University

Title: Towards the complete classification of integrable Camassa-Holm type equations

Abstract: After more than 30 years of discovery of the Camassa-Holm equation the complete classification of integrable equations of this type remains an open problem. Camassa-Holm type equations can be viewed as negative flows of hierarchies of integrable evolutionary partial differential equations. There are various approaches to tackle integrability of negative flows. In this talk I will review various integrability tests applicable to such type systems and present the up-to-date classification results.

Maxim Pavlov, Shandong University of Science and Technology

Title: Isomonodromic Deformations and the Tsarev Generalized Hodograph Method

Abstract: More than 45 years ago Vladimir Zakharov introduced the concept of "dispersionless limit" to the theory of Integrable systems. Integrable dispersive systems have plenty particular solutions, known in different applications. For instance: reflectionless potentials (multi-soliton and multi-phase solutions). However all these solutions in a dispersionless limit no longer exist! All these 45 years it was the open question: does exist a class of analytical solutions, which smoothly can be reduced to a class of corresponding solutions for a dispersionless limit? This problem is very important in the Topological Field Theory (WDVV, associativity equations), where one of most important tasks was: how to add infinitely many (in a generic case) higher order correction terms to semi-Hamiltonian hydrodynamic type systems with preservation

of integrability in a dispersionless limit case? Our target is to demonstrate explicit presentation of such particular solutions, known as isomonodromic deformations. They appear in description of a gradient catastrophe known in one-component case as the Gurevich-Pitaevski Problem.

Ruoci Sun, Beijing Normal University

Title: Multi-solitons of the Benjamin-Ono equation

Abstract: This presentation is dedicated to introducing pure multi-soliton solutions of the Benjamin-Ono equation on the real line. I will show that every multi-soliton manifold is invariant under the Benjamin-Ono flow and that we can establish action-angle coordinates to solve the equation by quadrature. Finally, the explicit expressions for all multi-soliton waves are derived.  

Jacek Szmigielski, University of Saskatchewan

Title: Isospectral Deformations of the Euler-Bernoulli beam problem and a vectorization of the Camassa-Holm equation

Abstract: I will present an integrable system that describes isospectral deformations of the Euler-Bernoulli beam problem and show how one can solve the inverse problem for the special case of a discrete Euler-Bernoulli beam. The solution involves a non-commutative generalization of Stieltjes’ continued fractions, leading to inverse formulas expressed in terms of ratios of Hankel-like determinants. In the second part of the talk, I will explain how this problem is related to a Camassa-Holm type equation and present a generalization of the Euler-Bernoulli beam problem, which leads to a vectorization of the Camassa-Holm equation attached to an arbitrary orthogonal group. The talk is based on earlier work with R. Beals and recent work with A. Hone and V. Novikov.  

Peter Van Der Kamp, La Trobe University

Title: Darboux polynomials and Lotka-Volterra systems

Abstract: I will tell a story about integrable Lotka-Volterra systems from the viewpoint of Darboux polynomials. The story will be based on the results of the following papers:

1. P.H. van der Kamp, T.E. Kouloukas, G.R.W. Quispel, D.T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multi-sums of products, Proc. R. Soc. A 470 (2014) 20140481.

2. P.H. van der Kamp, D.I. McLaren, G.R.W. Quispel, Homogeneous Darboux polynomials and generalising integrable ODE systems, J. Comput. Dyn. 8(1) (2021) 1-8.

3. G.R.W. Quispel, B. Tapley, D.I. McLaren, P.H. van der Kamp, Linear Darboux polynomials for Lotka-Volterra systems, trees and superintegrable families, J. Phys. A: Math. Theor. 56 (2023) 315201.

4. P.H. van der Kamp, R.I. McLachlan, D.I. McLaren, G.R.W. Quispel. Measure preservation and integrals for Lotka–Volterra tree-systems and their Kahan discretisation, J. Comp. Dyn. 11(4) (2024) 468-484.

5. P.H. van der Kamp, G.R.W. Quispel and D.I. McLaren, Trees and superintegrable Lotka-Volterra families, Mathematical Physics, Analysis and Geometry 27:25 (2024).

6. P.H. van der Kamp, D.I. McLaren and G.R.W. Quispel, On a quadratic Poisson algebra and integrable Lotka-Volterra systems with solutions in terms of Lambert's W function, Regul. Chaotic Dyn. (2024) https://doi.org/10.1134/S1560354724580032.

7. P.H. van der Kamp, Hypergraphs and homogeneous Lotka-Volterra systems with linear Darboux polynomials, arXiv:2411.18264 [nlin.SI].

8. Peter H. van der Kamp, David I. McLaren and G.R.W. Quispel, Ian Marquette. Liouville integrable Lotka-Volterra systems, work in progress.

Bao Wang, Ningbo University

Title: On the coupled pentagram maps

Abstract: The pentagram map is a discrete integrable system defined on polygons in the projective plane. We propose a family of novel generalizations of the pentagram map from one polygon to a pair of polygons. Their geometry, integrability, limit points are described. In particular, we find that these maps are related to refactorization maps associated with pairs of non-disjoint progressions. Furthermore, it is shown that some of these maps can be associated with a class of bi-orthogonal polynomials, which generalize the corresponding result for the classical Laurent bi-orthogonal polynomials.

Dong Wang, University of Chinese Academy of Sciences

Title: Local statistics of Muttalib-Borodin ensemble in the hard to soft transitive regime and the related integrability.

Abstract: Muttalib-Borodin ensemble is a typical biorthogonal ensemble. It has a hard edge limit that is expressed by Wright's generalized Bessel functions, or by Meijer G functions if the parameter $\theta$ is an integer. This hard edge limit does not occur in orthogonal polynomial ensembles unless $\theta = 1$. In this talk we consider the transition from the hard edge limit to the soft edge limit of the Muttalib-Borodin ensemble with an integer $\theta$ parameter, and show that the limiting distributions are related to Painleve-type equations. Our result generalizes the known relation between the transitive regime of orthogonal polynomial ensemble and the Painleve XXXIV. This is joint work with Shui-Xia Xu.

Zhiwei Wu, Sun Yat-sen University

Title: Darboux transformations for the KdV-type hierarchies

Abstract: The KdV-type hierarchies can be related to affine Kac-Moody algebras, and some of them have natural geometric correspondence. In this talk, we will discuss the Darboux transformations for KdV-type hierarchies in terms of loop group factorization.They are associated to different types of affine Kac-Moody algebra and related to curve flows under certain group actions. Permutability formulas are derived to construct explicit solutions.

Bo Yang, Ningbo University

Title: Concentric-ring patterns in higher-order KP-I lumps

Abstract: In this talk, we will review some new results on large-time wave patterns of general higher-order lump solutions in the Kadomtsev–Petviashvili I (KP-I) equation. This is joint work with Prof. Jianke Yang.

Yiling Yang, Chongqing University

Title: Soliton gas for the modified Camassa-Holm equation and its transient asymptotics

Abstract: In this paper, we investigate the soliton gas for the modified Camassa-Holm (mCH) equation with the eigenvalues of the Lax pair of mCH equation accumulating symmetrically on unit circle and imaginary axis. Based on the Riemann-Hilbert problem characterizing the mCH soliton gas, we provide a complete analysis on the long-distance and long-time asymptotic behavior of the mCH soliton gas. Especially the long-time behavior of the mCH soliton gas is described by genus-3, genus-7 or genus-11 hyperelliptic wave under different velocity. It is the first time that we give the long-time asymptotics for the soliton gas solution in three kinds of transitional regions.

Zihan Yin, Northwest University

Title: Talbot effects for dispersive evolution equations

Abstract: Talbot effect was first discovered by M.V. Berry in quantum mechanics, optics and analytic number theory. P. J. Olver found that the Talbot effect is universal in the dispersive equation. And he called this dichotomy phenomenon dispersive quantization/fractalisation. In this talk, we focus on the periodic initial boundary problem of two-component dispersive equations. Firstly, we prove that Talbot effect exists in two-component dispersive equations, and obtain the corresponding dispersive quantization condition. Then We perform a numerical simulation of Manakov system which is a classical Schrodinger-type system. We find that the Talbot effect still exists in nonlinear systems. Finally, we focus on the periodic initial boundary problem of the bidirectional hyperbolic dispersive equations. We find the new revival phenomena of the bidirectional hyperbolic dispersive equations.

Guofu Yu, Shanghai Jiao Tong University

Title: Cauchy bi-orthogonal polynomials and integrable lattices

Abstract: In this talk, we first consider the generalized two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. In the second part, we construct discrete spectral transformations for Cauchy orthogonal polynomials, and find its corresponding discrete integrable systems. It turns out that the normalization factor of Cauchy orthogonal polynomials acts as the tau-function of the discrete CKP equation with a discrete Gram determinant structure.

Dajun Zhang, Shanghai University

Title: Direct linearization scheme and Marchenko equation associated with the Lamé function: KP and reductions.

Abstract: We will establish an elliptic direct linearization (DL) scheme for the KP equation. The scheme consists of an integral equation involving the Lamé function and a formula for elliptic soliton solutions, which is confirmed by checking Lax pair of the KP. Based on analysis of real-valuedness of the Weierstrass functions, we are able to construct a Marchenko equation for elliptic solitons and nonsingular real solutions from this elliptic DL scheme. By utilizing elliptic Nth roots of unity and reductions, the elliptic DL schemes, Marchenko equations and nonsingular real solutions can be obtained for the KdV equation and Boussinesq equation.

Yingnan Zhang, Nanjing Normal University

Title: Two classes of Benjamin–Ono-type equations with the Hilbert operator related to the Calogero–Moser system and the classical orthogonal polynomials

Abstract: We investigate two distinct classes of Benjamin–Ono(BO)-type equations with the Hilbert operator. The first class consists of equations with constant coefficients, derived from linear differential equations, with a specific focus on the Mikhailov–Novikov equation and Satsuma–Mimura equation. The second class involves BO-type equations with variable coefficients linked to orthogonal polynomials, including Hermite, Jacobi, and Laguerre polynomials. A key aspect of transforming these differential equations into BO-type equations is that the zeros of the polynomial or periodic solutions must lie in the upper half-plane. For linear and quadratic polynomials, we directly analyze their zeros to determine the solutions of corresponding BO-type equations. For higher-order polynomials, we use the pole expansion method to derive the governing many-body systems of the zeros. This is a joint work with L-J Yan, Y-J Liu and X-B Hu.