Download Abstracts 报 告 摘 要 Ermakov–Ray–Reid Systems in (2+1

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.