Special Session 51: Integrable Aspects and Asymptotics of Nonlinear Evolution Equations

Hamiltonian structures for differential-difference equations: classification and cohomology

Matteo Casati Ningbo University Peoples Rep of China Co-Author(s):    D.Valeri, J.P.Wang

Abstract: The Hamiltonian structures of (integrable) differential--difference systems, usually obtained in terms of difference (\emph{shift}) operators, has some remarkable algebraic properties -- some cases have been investigated by Dubrovin many years ago,\footnote{B. A. Dubrovin. Differential-geometric Poisson brackets on a lattice. \emph{Funct. Anal. Appl.}, 23(2):57, 1989.} and more recently it has been shown that they can all be recomprised under the notion of multiplicative Poisson vertex algebras.\footnote{A. De Sole, V. G. Kac, D. Valeri, and M. Wakimoto. Local and Non-local Multiplicative Poisson Vertex Algebras and Differential-Difference Equations. \emph{Commun. Math. Phys.}, 370(3):1019, 2019.} Following a differential-geometrical approach, we introduced the corresponding notion of Poisson bivector and Poisson cohomology.\footnote{MC and J. P. Wang. A Darboux-Getzler theorem for scalar difference Hamiltonian operators. \emph{Commun. Math. Phys.}, 374:1497, 2020.} This allowed us to better understand the classification of bi-Hamiltonian pairs present in De Sole et al., extend the notion to include noncommutative differential-difference systems,\footnote{MC and J. P. Wang. Hamiltonian structures for Integrable Nonabelian Difference Equations. \emph{Commun. Math. Phys.}, 392:219, 2022.} and most recently expand the classification of such structures to multi-component cases.\footnote{MC and D. Valeri, Multi-component Hamiltonian difference operators, \emph{in preparation}}

Dispersive revival phenomena for two-dimensional dispersive evolution equations

Jing Kang Northwest University Peoples Rep of China Co-Author(s):    Changzheng Qu, Zihan Yin

Abstract: In this talk, the dispersive revial phenomenon for two-dimensional linear spatially periodic dispersive evolution equations on a rectangle subject to periodic boundary conditions and discontinuous initial profiles are investigated. We analyze a nonvel revial phenomenon for two-dimensional equations with non-polynomial dispersion relations, in the concrete case of the periodic initial-boundary value problem of the linear Kadomtsev-Petviashvili equation on a square with a step function initial data. Revival in this case exhibits a novel characteristic that there appears radically different qualitative behaviors in x and y directions. We give an analytic description of this dichotomous revial phenomenon, and present illustrative numerical simulations.

Construction and solutions of the semi-discrete Toda and sine-Gordon equations

Chunxia Li Capital Normal University Peoples Rep of China Co-Author(s):

Abstract: This talk aims to explore the relations between the Sylvester equation and semi-discrete integrable systems. Starting from the Sylvester equation $KM+ML=rs^\top$, the master function $S^{(i,j)}=s^\top L^jC(I+MC)^{-1}K^ir$ is introduced. By imposing dispersion relations on $r$ and $s$, the semi-discrete Toda equation, the modified semi-discrete Toda equation and their Miura transformation are established through equations of $S^{(i,j)}$. In addition, Lax pair and solutions are constructed for the semi-discrete Toda equation in a systematic way. Under the symmetric constraint $S^{(i,j)}=S^{(j,i)}$, the semi-discrete sine-Gordon equation, the modified semi-discrete sine-Gordon equation and their Miura transformation are derived. Integrability such as Lax pair, the bilinear form and various types of solutions for the semi-discrete sine-Gordon equation are presented as well.

Theta-function oscillatory solitons of integrable equations

Ruomeng Li Zhengzhou University Peoples Rep of China Co-Author(s):    Xianguo Geng

Abstract: In this talk, a method to construct oscillatory soliton solutions, expressed with theta-functions, is presented, of integrable equations, that is both Lax and Hirota integrable. The Baker-Akhiezer functions in the field of algebraic solutions and the tau-functions in the filed of the direct method are combined to solve the spectral problems, to construct Darboux transformations, and to derive exact solutions.

Variable Separation Approach and Abundant Nondegenerate Solitons

Ji Lin Zhejiang Normal University Peoples Rep of China Co-Author(s):    Xueping Cheng

Abstract: We propose to combine the bilinear method with the variable separation approach to study the nondegenerate multi-soliton solution of the two-component long-wave-short-wave resonance interaction system in (2+1) dimension. We successfully obtain N nondegenerate soliton general solutions for each shortwave component containing N arbitrary functions of the independent variable y. By containing arbitrary functions, rich soliton forms can be easily obtained.

On the progresses on some open problems related to infinitely many symmetries

SY Lou Ningbo University Peoples Rep of China Co-Author(s):

Abstract: The quest to reveal the physical essence of the infinitely many symmetries and/or conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of $n$-wave solutions, such as the $n$-soliton solutions, multiple breathers, complexitons, and the $n$-periodic wave solutions (the algebro-geometric solutions with genus $n$), wherein $n$ denotes an arbitrary integer that can potentially approach infinity. Each sub-wave comprising the $n$-wave solution may possess free parameters, including center parameters $c_i$, width parameters (wave number) $k_i$, and periodic parameters (the Riemann parameters) $m_i$. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinitely many symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. This conjecture appears to hold true for all integrable systems with $n$-wave solutions. The conjecture intimates that the currently known infinitely many symmetries is not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned Korteweg-de Vries (KdV) equation and the Burgers equation as simple examples, the conjecture is substantiated for the $n$-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework. Notably, a ren-symmetric integrable Burgers hierarchy is explicitly derived. Both the supersymmetric and the classical integrable hierarchies are encompassed within the ren-symmetric integrable hierarchy.

The higher-order $\mu$-Camassa-Holm equations

Changzheng Qu Ningbo University Peoples Rep of China Co-Author(s):    Ying Fu, Hao Wang, Kexin Yan

Abstract: It is well-known that the Camassa-Holm (CH)-type equations admit peaked solitons. In this talk, we are mainly concerned with the nonlocal $\mu$-CH-type equations. First, we review the known results and properties of $\mu$-CH and modified CH equation. Second, we introduce the higher-order $\mu$-CH equations. Third, stability and instability of periodic peaked solitons to the higher-order CH equations will be studied.

The hidden algebra in the dispersionless cKP hierarchy

Kelei Tian Hefei University of Technology Peoples Rep of China Co-Author(s):    Ge Yi, Song Li, Ying Xu

Abstract: In the talk, we will show some results on the hidden algebra in the dispersionless constrained KP hierarchy. The additional symmetries of the dispersionless constrained KP hierarchy are given by introducing vital formal Laurent series. The additional flows form a subalgebra of the Virasoro algebra.

Nonlinear dispersive wave phenomenon for two (3+1)-dimensional system

Lixin Tian Jiangsu University Peoples Rep of China Co-Author(s):

Abstract: Two different nonlinear evolution equations (NLEEs) have been investigated employing Hirota`s bilinear technique in this work, and both of them describe breather waves in complex media and bubbles in liquid fluctuations, respectively. For the extended (3+1)-dimensional JML equation, using the generalized bilinear method and a trial function, breather waves and multiwave solutions were constructed, revealing various dynamic features and the influence of parameter changes on the behavior of these solutions. For the Kudryashov-Sinelshchikov equation, by the virtue of binary Bell polynomials, the bilinear representation, bilinear Baecklund transformation, and the associated Lax pair were obtained. Furthermore, four new lump solutions were constructed utilizing Hirota`s bilinear representation, and the interaction between lump and periodic solutions was thoroughly analyzed. The three-dimensional surface, two-dimensional density, and contour plots of the solutions mentioned above were generated using Maple software, illustrating intriguing dynamic behaviors. This study provides new solutions for (3+1)-dimensional nonlinear equations and uncovers rich physical characteristics.

On the long-time asymptotic of the modified Camassa-Holm equation with nonzero boundary conditions in space-time solitonic regions

Shoufu Tian China University of Mining and Technology Peoples Rep of China Co-Author(s):    Jin-JIe Yang and Zhi-Qiang Li

Abstract: In this talk, we report the long-time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with finite density initial data in different regions. We prove that the soliton resolution conjecture holds, that is, the solution of the mCH equation can be expressed as the soliton solution on the discrete spectrum, the leading term on the continuous spectrum, and the residual error. This work is joint with Jin-JIe Yang and Zhi-Qiang Li.

Self-similar Painlev\`{e} regions in long-time asymptotics of good Boussinesq equation and Sawada-Kotera equation

Deng-Shan Wang Beijing Normal University Peoples Rep of China Co-Author(s):    Deng-Shan Wang, Xiaodong Zhu

Abstract: In this talk, we report our recent work on the long-time asymptotics of good Boussinesq equation and Sawada-Kotera equation with decaying initial data. Especially, the self-similar Painlev\`{e} regions in the two integrable systems are investigated in detail. For the good Boussinesq equation, the self-similar region is described by the Painlev\`{e} IV equation, while for the Sawada-Kotera equation, the self-similar region is described by the fourth-order analogues of Painlev\`{e} transcendent. The Miura transformations along with the modified Boussinesq equation and modified Sawada-Kotera equation are used in the asymptotic analysis.

Superintegrability of matrix models

Rui Wang China University of Mining and Technology, Beijing Peoples Rep of China Co-Author(s):    Weizhong Zhao, Fan Liu, etc.

Abstract: Matrix models have wide applications in mathematics and physics. In the study of matrix models, the superintegrability means that the average of a properly chosen symmetric function is proportional to ratios of symmetric functions on a proper locus, i.e., $\langle$ character $\rangle$~character. $W$-representations of the matrix models realize the partition functions by acting on elementary functions with exponents of the given $W$-operators. In this talk, I will introduce our recent works on how to derive the superintegrability of several matrix models from their $W$-representations. Meanwhile, we construct the partition function hierarchies with $W$-representations, and present their character expansions with respect to the Schur (Jack) polynomials. For the negative branch of hierarchies, it gives the $\tau$-functions of the KP hierarchy. The $W$-operators in the positive branch of hierarchies can be related to the many-body systems.

Numerical Computation for long time behavior for derivative nonlinear schrodinger eqution

Zhen Wang Beihang University Peoples Rep of China Co-Author(s):

Abstract: Decay properties of water waves is related to the long time behavior of governing equation. Its leading order asymptotic expression may be given by phase stationary method for linear and by Riemann Hilbert method for nonlinear integrable system. Numerical solution for long time behavior of derivative schrodinger equation is explored, it has advantages on avoiding the error cumulative of long time evolution.

Nonlinear localized excitation on the elliptic periodic wave background

Yunqing Yang Zhejiang University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we first introduce two types of elliptic functions, namely Jacobi and Weierstrass elliptic functions, and their corresponding properties. Secondly, two types of nonlinear wave solutions on the periodic wave background of elliptic functions have been constructed by using the solution of linear spectral problems and Darboux transformation technique, and the corresponding dynamic properties are also studied. Finally, the relationship between nonlinear wave solutions on constant background and periodic wave background are discussed.

Symmetries of Supersymmetric Fermionic Partial Differential Equation with Arbitrary Function

Ruoxia Yao Shaanxi Normal University Peoples Rep of China Co-Author(s):    Sen-Yue Lou

Abstract: A supersymmetric nonlinear partial differential equation (NPDE) with fermionic fields combines supersymmetric field theory with the integrable system theory. The fermionic fields describe the spin$-1/2$ particles (fermions, particles with half-integer spin) in supersymmetric theories and are crucial for understanding particle interactions and symmetries in field theory. In this talk, we consider the NPDE $\begin{equation} \frac{{\rm d} \Phi}{{\rm d} t} = {\Phi} \left(\frac{{\rm d} {\Phi}}{{\rm d} x}\right) \left(\frac{{\rm d}^{2}{\Phi}}{{\rm d} x^{2}}\right) F \! \left({\mathcal{D}}{\Phi}, \frac{{\rm d} {\mathcal{D}}{\Phi}}{{\rm d} x}, \frac{{\rm d}^{2}{\mathcal{D}}{\Phi}}{{\rm d} x^{2}}, \frac{{\rm d}^{3}{\mathcal{D}}{\Phi}}{{\rm d} x^{3}}\right) \end{equation}$ with $F$ being an arbitrary function of the bosonic fields using the dot product first to combine the fermions appearing in it and then unearth its higher order supersymmetries with arbitrary functions that only related to the bosonic fields and their derivatives. It is interesting that the dot product of two fermions can be related to symmetry properties, and the study of supersymmetry sets up a bridge between the fermionic and bosonic fields.

The self-dual Yang-Mills equation: New solutions and related integrable structure

Da-jun Zhang Shanghai University Peoples Rep of China Co-Author(s):

Abstract: It is well-known that the self-dual Yang-Mills (SDYM) equation is a fundamental equation in conformal field theory as well as a general 4D equation in integrable systems. In Yang`s formulation, one can first solve the unreduced SDYM equation in a general complex 4D space and then implement reductions so that solutions meet the reality conditions and gauge conditions in the real 4D spaces. In this talk, I will show that the unreduced SDYM equation can be formulated from the matrix KP hierarchy and the matrix AKNS hierarchy. Such formulations are based on the Cauchy matrix scheme and solutions for the unreduced SDYM equation can be constructed by solving the Sylvester equations. These new structures enable us to obtain new solutions for the SU(N) SDYM equation in the different real 4D spaces (with different signatures) as well as to get some integrable equations arising from reductions of the SDYM equation. I will also introduce the reductions of solutions and connections (with other equations, e.g. the Fokas-Lenells equation). This talk is mainly based on joint works with Shangshuai Li and Changzheng Qu.

Classification of solutions for the (2+1)-dimensional Fokas-Lenells equations based on bilinear method and Wronskian technique

Qiulan Zhao Shandong University of Science and Technology Peoples Rep of China Co-Author(s):    Qiulan Zhao, Xuejie Zhang, Xinyue Li

Abstract: In this paper, we apply the bilinear method and Wronskian technique to the (2+1)-dimensional Fokas-Lenells (FL) equations for the first time, which simulate the propagation of richer pulses in the fibers. Specifically, based on the bilinear form of the above equations with parameters in the zero background, the double Wronskian solutions are provided and proved, and then various types of solutions for the local and nonlocal (2+1)-dimensional FL equations are obtained by using the reduction technique. This enables us to have a relatively complete classification of the solutions of the above two reduced equations as much as possible. Notably, we compare the solutions of these two reduced equations in detail, and find that the nonlocal equation has new characteristics that are different from the local ones, such as the $N$-order solutions of the nonlocal equation have $\frac{(N+2)!}{N!2!}$ combinations in the cases of complex eigenvalues, which are much more complex than the local ones. In addition, the physical properties of the one-soliton and one-periodic solutions are investigated, the asymptotic behavior of the two-soliton solutions at the infinite time limit is analyzed, and then the coefficients of the equations controlling the rotation, separation and density of the solutions are discovered. Finally, we also talk about the periodic solutions, algebraically decayed solitary waves and mixed interaction solutions of the local (2+1)-dimensional FL equation that are not studied previously, which belong to real eigenvalues cases.