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Abstracts 报 告 摘 要 1. Ermakov–Ray–Reid Systems in (2+1)-Dimensional Rotating Shallow Water Theory Hongli An(安红利) The Hong Kong Polytechnic University Abstract. A 2+1-dimensional rotating shallow water system with an underlying circular paraboloidal bottom topography is shown to admit a multi -parameter integrable nonlinear subsystem of Ermakov-Ray-Reid type. The latter system, which describes the time evolution of the semi-axes of the elliptical moving shoreline on the paraboidal basin, is also Hamiltonian. The complete solution of the generic eight-dimensional dynamical system governing the reduction is obtained in terms of an elliptic integral representation. 2. A Baecklund transformation on surfaces swept out by moving curves with constant torsion Xifang Cao(曹锡芳) Yangzhou University Abstract. We give a Baecklund transformation on surfaces which are swept out by moving curves with constant torsion. The curvature of the moving curve discussed in this paper is governed by the modified KdV equation. Our result can be regarded as a geometric realization of the well-known Baecklund transformation for the modified KdV equation. As applications, by taking the circular cylinder as a seed surface, we construct some novel surfaces which are swept out by moving periodic curves. This is a joint work with Chuanyou Xu. 3. Immersed Equations of Hyperbolic Surfaces Hsungrow Chan(詹勳國) National Pingtung University of Education Abstract. Several papers on soliton surfaces with negative Gauss curvatures discuss the equivalent quasi-linear hyperbolic systems of Gauss-Codazzi equations. This paper provides other transformations of the system and introduces Hong's construction of the solution into Euclidean three-space. Then, we study the geometry of Hong's immersion by applying the solution and discover a new necessary and sufficient condition for a surface to be a surface of revolution. 4. Differential characteristic set algorithm for symmetry classification of PDEs Lu Chao(朝鲁)Smu College of Arts and Sciences, Shanghai Maritime University Abstract An algorithm for symmetry classification of PDEs based on differential characteristic set method is given; The complete nonclassical symmetry classification of nonlinear wave equation with an arbitrary function is obtained; The open problem mentioned by P. A. Clarkson for the wave equation is solved. 5. From the Neumann type systems to the modified Korteweg-de Vries hierarchy Jinbing Chen(陈金兵) Southeast University Abstract In this talk, a treatment to obtain finite-gap solutions of integrable nonlinear evolution equations (INNLEs) in 1+1 and 2+1 dimensions is presented by using the Neumann type systems through three key steps: (a) Neumann type integrable reduction of INLEEs; (b) straightening out of Neumann type flows; and (c) Jacobi inversion. We take the modified Korteweg-de Vries (mKdV) hierarchy and the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation as the examples to illustrate our constructing scheme. Then, we attain a family of new Neumann type systems and some new finite-gap solutions expressed by Riemann theta functions for the mKdV hierarchy and the 2+1 dimensional CDGKS equation. 6. New additional symmetries of the two-Toda lattice hierarchy Jipeng Cheng(程纪鹏) China University of Mining and Technology Abstract The generalized additional symmetries of the two-Toda lattice hierarchy are investigated in this paper. The new additional symmetries are found, i.e., two coupled Virasoro algebras and a Block type algebra symmetry. To discuss the actions of these algebras on $\tau$-function, we restrict the two-Toda lattice hierarchy to the semi-infinite case, which is related with the two-matrix integrals. A new trace formula is derived. By means of our trace formula, we present the actions of the two coupled Virasoro algebras and Block type algebra on the two-matrix integral $\tau_N$. 7. The tt*-Toda Equations Martin Guest Tokyo Metropolitan Univ. Abstract The tt*-equations are a system of integrable p.d.e. which were introduced by Cecotti and Vafa to describe deformations of supersymmetric quantum field theories. Mathematically, they can be regarded as the equations for real structures of Frobenius manifolds. In differential geometry they can be regarded as equations for certain kinds of pluriharmonic maps. The solutions predicted by physics have very special properties, although the existence of such solutions has not yet been established rigorously. The special solutions are parametrised by asymptotic data, which can be interpreted as holomorphic data (in the "UV limit") or monodromy data (in the "IR limit"). The tt*-Toda equations are a particular case, in which some of the predicted properties can be established. 8. Rogue wave solutions of the several integrable equations Jingsong He(贺劲松)Ningbo University Abstract In this talk, we would like to introduce the rouge wave of the DNLSI and DNLSIII, which is given by the Darboux transformation from the periodic "seed". If we have enough time, the rouge wave of the variable mass Sine-Gordon is given by a nonlinear transformation from known solutions of the usual Sine-Gordon equation. 9. A new convergence acceleration algorithm related to the lattice Boussinesq equation Yi He(何益) Wuhan Institute of Physics and Mathematics, CAS Abstract The molecule solution of an equation related to the lattice Boussinesq equation is derived by Hirota’s bilinear method. It is shown that this equation can for certain sequences be used as a numerical convergence acceleration algorithm. Numerical examples with applications of this algorithm are presented. 10. Soliton propagation in Doppler-broadened media Guoxiang Huang(黄国翔) East China Normal University Abstract In recent years, much attention has been paid to the study of slow-light propagation in coherent resonant multi-level systems, which has not only fundamental theoretical interest but also many important applications in optical information processing and engineering [1]. For many practical situations, inhomogeneous broadening of quantum emitters induced by Doppler effect, crystal-field, size fluctuations or other physical effects may be significant. In this talk, I shall present new results on the linear and nonlinear light propagations in a Doppler broadened system via electromagnetically induced transparency (EIT), with incoherent population exchange between two lower energy-levels taken into account. Through a careful analysis of base state and linear excitation, we give an EIT condition of the system. Under this condition, the effect due to incoherent population exchange is insignificant, while dephasing dominates the decoherence of the system. Such condition also ensures the validity of a weak nonlinear perturbation theory developed by us for solving Maxwell-Bloch equations with inhomogeneous broadening. By a detailed analysis on nonlinear pulse propagation of the system, we show that it is possible to form spatial-temporal optical solitons in the Doppler broadened medium. Such solitons in such system have ultraslow propagating velocity and can be generated in very low light power [2]. We also give a simple report of our recent study of EIT and pulse propagation in a micro-waveguide filled with Doppler-broadened atomic gas [3]. [1] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005). [2] Liang Li and Guoxiang Huang, Phys. Rev. A 82, 023809 (2010). [3] Liang Li and Guoxiang Huang, to be published. 11. A new (\gamma_n, \sigma_k) KP hierarchy and generalized dressing method Yehui Huang(黄晔辉) Tsinghua University Abstract A new (\gamma_n,\sigma_k)- KP hierarchy with two new time series \gamma_n and \sigma_k, which consists of \gamma_n- flow, \sigma_k- flow and mixed \gamma_n and \sigma_k- evolution equations of eigenfunctions, is proposed. Two reductions and constrained flows of (\gamma_n,\sigma_k)- KP hierarchy are studied. The dressing method is generalized to the (\gamma_n,\sigma_k)- KP hierarchy and some solutions are presented. 12. Symmetry groups and fundamental solutions for systems of parabolic equations Kangjing(康静) Northwest University Abstract In this talk, the relationship between Lie point symmetry and fundamental solution for systems of parabolic equations is explored. It is shown that the fundamental solutions of the systems of parabolic equations admitting certain symmetries can be obtained by inverting the Laplace transformation of the corresponding group-invariant solutions. Several examples are presented to illustrate the approach. Furthermore, the relationship between fundamental solutions for two systems of parabolic equations related by an equivalence transformation is identified. 13. A classification of equivariant constant Gauss curvature and constant mean curvature cylinders Shimpei Kobayashi Hirosaki University Abstract: It is known that constant Gauss curvature (CGC) and constant mean curvature (CMC) are described by (hyperbolic) sine-Gordon equations. Among CGC and CMC surfaces, equivariant ones are simple and non-trivial, and there exist topologically cylinders, the so-called twizzlers. In this talk, we give a classification of CGC and CMC twizzlers using the integrable system technique, in particular, complex CMC (CGC) surfaces and deformation of the spectral parameter 14. Block type symmetry of bigraded Toda Hierarchy Chuanzhong Li(李传忠) Ningbo University Abstract In this talk, we define Orlov-Schulman's operators $M_L$, $M_R$, and then use them to construct the additional symmetries of the bigraded Toda hierarchy (BTH). We further show that these additional symmetries form an interesting infinite dimensional Lie algebra known as a Block type Lie algebra, whose structure theory and representation theory have recently received much attention. By acting on two different spaces under the weak W-constraints we find in particular two representations of this Block type Lie algebra. 15. Investigation of multi-soliton andmulti-cuspon solutions and there interaction of the Camassa-Holm equation. Yishen Li(李翊神) University of Science and Technology of China Abstract We study the multi-soliton andmulti-cuspon solutions and there interaction of the Camassa-Holm equation. Especially for n>=3, We also use numerical methods (the so called local discontinuous Galerkin (LDG) method) to simulate the solutions and give the comparison of exact solutions and numerical solutions. 16. Symmetry classifications and dynamical system method for the nonlinear PDEs Hanze Liu(刘汉泽) Binzhou University Abstract We deal with some nonlinear PDEs by the symmetry analysis and the dynamical system method. The complete geometric vector fields of the equations are obtained, and the exact solutions to the equations are presented. 17. Some results on integrable and supersymmetric integrable systems Qingping Liu(刘青平) China University of Mining and Technology, Beijing Abstract In this talk, some recent results will be presented. First the solution constructions will be given for a long-short waves interaction model and NLS. Then the superposition formula will be considered for supersymmetric sinh-Gordon equation. Finally, the supersymmetric reciprocal transformation will be disucssed for the N=2 case. 18. The genus-1 Virasoro conjecture. Xiaobo Liu(刘晓博) University of Notre Dame and Peking University Abstract Virasoro constraints hold for certain tau functions of many integrable systems. It was conjectured by Eguchi-Hori-Xiong and S. Katz that the generating function of Gromov-Witten invariants of every smooth projective variety also satisfies the Virasoro constraints. In case that the target manifold is a point, this conjecture is equivalent to Witten's conjecture, proved by Kontsevich, that the generating fucntion of intersection numbers on the moduli spaces of stable curves is a tau function of the KdV hierarchy. The genus-0 Virasoro conjecture was proved by Tian and myself. Dubrovin and Y. Zhang proved the genus-1 part of this conjecture for manifolds with semisimple quantum cohomology. In this talk, I will explain the current state for the genus-1 Virasoro conjecture for manifolds whose quantum cohomology may not be semisimple. 19. Localization of Non-local symmetry Senyue Lou(楼森岳) Ningbo University Abstract Non-local symmetry of integrable systems can be obtained through Darboux transformation, Backlund transformation and Mobious transformation etc.. We may find many new interesting solutions if we can localize the non-local symmetry. 20. Twisted hierarchies associated with the generalized sine-Gordon equation Hui Ma(马辉) Tsinghua University Abstract Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, we derive explicit interesting first and higher flows of twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies, justify that the $1$-dimensional system of twisted $\frac{O(J,J)}{O(J)\times O(J)}$-hierarchies for $J=I_{q,n-q} (1\leq q \leq n-1)$, called the generalized sinh-Gordon equation, is the Gauss-Codazzi equation for $n$-dimensional timelike submanifolds with constant sectional curvature $1$ and index $q$ in pseudo-Euclidean $(2n-1)$-dimensional space $\mathbb{R}^{2n-1}_q$ with index $q$. Furthermore, a unified treatment of the inverse scattering theory for twisted $\frac{O(J,J)}{O(J)\times O(J)}$-hierarchies is provided. 21. Hamiltonian structures of integrable couplings Wen-Xiu Ma (马文秀) University of South Florida Abstract We discuss how to construct Hamiltonian structures of integrable couplings. The primary tools are classical and super variational identities associated with semi-direct sums of Lie algebras. Applications of component-trace identities furnish Hamiltonian structures of both dark equations and bi-integrable couplings. 22. Geometry of Lagrangian Submanifolds and Isoparametric Hypersurfaces Yoshihiro Ohnita (Osaka City University & OCAMI) Abstract: In this talk I shall provide a breif survey on my recent works and their environs on differential geometry of Lagrangian submanifolds in specific symplectic Kahler manifolds, such as complex projective spaces, complex space forms, Hermitian symmetric spaces and generalized flag manifolds. This talk is mainly based on my joint work with Associate Professor Hui Ma (Tsinghua University, Beijing). First I shall discuss the Hamiltonian minimality and Hamiltonian stability of Lagrangian submanifolds, the classification problem of homogeneous Lagrangian submanifolds, the tightness problem of Lagrangian submanifolds, and so on. The relationship between certain minimal Lagrangian submanifold in complex hyperquadrics and isoparametric hypersurfaces in spheres will be emphasized. Recently we gave a complete classification of compact homogeneous Lagrangian submanifolds in complex hyperquadrics and we determined the (strictly)Hamiltonian stability of ALL compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as the Gauss images of homogeneous isoparametric hypersurfaces in spheres. We shall also mention related problems on cohomogeneity one special Lagrangian submanifolds in the tangent bundle over a sphere and the Gauss maps of isoparametric hypersurfaces in the Semi-Riemannian cases. 23. Invariant geometric flows and CH-DP types equations in certain geometries Changzheng Qu (屈长征) Northwest University Abstract In this talk, it is shown that the Camassa-Holm (CH), Degasperis-Procesi (DP), Olver-Rosenau equation, μ-CH, μ-DP and multi-component CH equations arise from invariant curve flows in certain geometries. Blow up and wave breaking phenomena of μ-CH and μ-DP equations are also studied. 24. On the local well posedness and blow-up solution to a new coupled Camassa-Holm equations in Besov Space Lixin Tian(田立新) Jiangsu University Abstract In this paper, we establish the local well-posedness for a new coupled Camassa-Holm system in a range of the Besov spaces by employing a series of norm estimates. A wave-breaking mechanism for solutions is described in detail and a result of blow-up solutions with certain profile is established. 25. Bright N-soliton solutions of the matrix nonlinear Schrodinger system Dengshan Wang(王灯山) The Institute of Physics, CAS Abstract The matrix nonlinear Schrodinger system includes several important physical models such as the integrable spin-1 Bose-Einstein condensates model, the three-component Manakov equation, the coherently coupled nonlinear Schrodinger equation in the Kerr medium, and so on. The rank-one, rank-two and rank-N soliton solutions of the matrix nonlinear Schrodinger system are obtained by the Riemann-Hilbert formulation which is a simply version of inverse scattering transform. The collision dynamics between the two solitons is also analyzed. 26. Lagrangian submanifolds in complex space forms Xianfeng Wang(王险峰) Tsinghua University Abstract I will talk about some classification results for the Lagrangian submanifolds in complex space forms, including the isotropic Lagrangian submanifolds and Lagrangian submanifolds with parallel second fundamental form, etc. 27. R-matrices and Hamiltonian Structures for Certain Lax Equations Chaozhong Wu (吴朝中) SISSA Abstract One of the most efficient methods for introducing Hamiltonian structures of evolutionary equations in Lax form is the classical R-matrix formalism. Here we derive R-matrices on a certain so-called coupled Lie algebra by solving the modified Yang-Baxeter equation. One of these R-matrices are employed to construct Hamiltonian structures for certain Lax equation with negative flows, which include the two-component BKP hierarchy and the Toda lattice hierarchy. Moreover, such Hamiltonian structures have corresponding reductions when the hierarchies of Lax equations are reduced to their subhierarchies,. 28. The dispersionless BKP equation with self-consistent sources and its hodograph solution Hongxia Wu(吴红霞) Jimei University Abstract The symmetry constraint of dispersionless BKP hierarchy is firstly derived by taking the dispersionless limit of the counterpart of BKP hierarchy. In addition, the first type of the dispersionless BKP equation with self-consistent sources is constructed and its hodograph solution is also given. 29. ON Lie supper algebras and super integrable system Tiecheng Xia(夏铁成) Shanghai University Abstract Based on Lie supper algebra and supper super trace identity,we set up some supper integrable and their super-Hamiltonian structure and obtain integrable couplings. These results state that fermion and boson satisfies the same equations. 30. Fermionic covariant prolongation structure theory for high dimensional super nonlinear evolution equation Zhaowen Yan(颜昭雯) Capital Normal University Abstract The fermionic covariant prolongation structure theory for the super nonlinear evolution equations is generalized for supersymmetric equations in two spatial dimensions for which an inverse scattering formulation exists. The fermionic covariant fundamental equations determining the prolongation structure are presented. As an example, we investigate a super nonlinear evolution equation in the framework of this fermionic covariant prolongation structure theory and give its Lax pairs and Backlund transformation. 31. Nonautonomous discrete rogue wave solutions and their interaction in the nonlinear lattice model with variable coefficients Zhenya Yan(闫振亚) Academy of Mathematics and Systems Science, CAS 32. Shocks, explosions and vortices in two-dimensional homogeneous quantum magnetoplasma Jianrong Yang(杨建荣) Ningbo University Abstract Using the quantum hydrodynamic (QHD) model for a uniform quantum magnetoplasma, and considering that the collision between ions and neutrals is dominant, a two-dimensional nonlinear system is derived. The linear dispersion relation is obtained and thus the variations of the dispersion relation with the obliqueness angle and density are discussed in detail. Shock, explosion and vortex solutions of the nonlinear system are obtained. It is found that increasing the plasma density may enhance the strength of the shock and the width of the explosion. However, the higher the collision frequency is, the weaker the shock and the narrower the explosion is. The temporal and spatial distributions for the vortex potential are studied. Spatially, it forms a periodic vortex street. Temporally, the vortex street may evolve in various ways owing to the arbitrary function of time. 33. Some generalizations for discrete integrable systems Da-jun Zhang(张大军)Department of Mathematics, Shanghai University Abstract I would like to talk about three aspects of generalizations in discrete integrable systems. The first is a generalized Cauchy matrix approach. This will enlarge solution set for the lattice equations obtained in this approach. The second is to derive rational solutions for some multidimensionally consistent systems by bilinear approach. Note that Cauchy matrix approach usually becomes trivial when deriving rational solutions. For some systems such as H1 and lattice Boussinesq equation, parameter-extension is necessary to get rational solutions. Finally, we will introduce some embedding relations appearing when we investigated multidimensional consistent lattices of Boussinesq-type [J Hietarinta, JPA, 2011]. In this part we hope to see possible relationship of lattice equations coming from different dispersion relations in terms of direct linearization approach or other approaches. This talk is partly based on joint work with Prof. J. Hietarinta, F. Nijhoff, et al. 34. Generalized solutions for ABS Lattice Equation: Cauchy Matrix Approach Songlin Zhao(赵松林) Shanghai University Abstract In this paper we discuss solutions of KdV-type lattice equations through Cauchy matrix approach and use those results to construct solutions for ABS lattice equation except for the elliptic case of Q4. Starting from condition equation set, we get KdV-type lattice equations, including lattice potential KdV equation, lattice potential mKdV equation, lattice Schwarzian KdV equation and NQC equation. By taking coefficient matrix K in condition equation set in the form of canonical matrices, then we get many kinds of solutions which are different from solitons. 35. Integrable reductions of restricted AKNS flows Ruguang Zhou(周汝光) Xuzhou Normal University Abstract The restricted AKNS flows are reviewed and their integrable reductions are considered. We show how to get the cerebrated C. Neumann system ($G_0$-constraint KdV flow), the Garnier system (the $G_2$-constraint KdV flow), the H\'enon-Heiles system (the $G_4$-constraint KdV flow), the $G_2$-constraint mKdV flow and $G_1$-constraint NLS flow from the restricted AKNS flows by allowable reductions of variables and parameters. 36. Finite dimensional Hamiltonian system related to Lax pair with symplectic and cyclic symmetries Zixiang Zhou(周子翔)Fudan University Abstract For the 1+1 dimensional Lax pair with a symplectic symmetry and cyclic symmetries, it is shown that there is a natural finite dimensional Hamiltonian system related to it by presenting a unified Lax matrix. The Liouville integrability of the derived finite dimensional Hamiltonian systems is proved in a unified way. 37. Euler equations related to the generalized Neveu-Schwarz algebra Dafeng Zuo(左达峰) University of Science and Technology of China Abstract In this talk, we will report Euler equations related to the generalized Neveu-Schwarz algebra, including supersymmetric Euler equations and bi-superhamiltonian Euler equations. Especially, we obtained several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-component Camassa-Holm equation and the 2-component Hunter-Saxton equation. To our knowledge, most of them are new.