可 積 系 統 研 討 會
Workshop on Integrable Systems

時 間:2008年3月20日

Time: March 20th, 2008

地 點:中研院數學所會議室

Place: Institute of Mathematics, Academia Sinica

主 題:Soliton Theory, Integrable Systems and Related Topics

 

Invited Speakers:

 

 

Leonid Bogdanov (Landau Inst. of Theoretical Physics, RAS, Russia)
Oleg I. Mokhov (Landau Inst. of Theoretical Physics, RAS, Russia)
Sergey Tsarev (Krasnoyarsk St. Pedagogical Univ., Russia)

Dmitry Demskoy (Institute of Mathematics, Academia Sinica, Taiwan)

 

Program

  9:30

Registration

Mar. 20th

10:00~11:00

Sergey Tsarev (Krasnoyarsk St. Pedagogical Univ., Russia), The Monge Problem: quadrature-free integration of nonlinear ODEs, Pfaffian forms and efficient car parking

 

11:00~11:15

Tea Break

 

11:15~12:15

Oleg Mokhov (Landau Inst. of Theoretical Physics, RAS, Russia), Integrability of the class of all flat torsionless submanifolds in pseudo-Euclidean spaces, series of integrable
hydrodynamic type hierarchies associated with an arbitrary
submanifold of this class, and Frobenius manifolds

 

12:15

Lunch

 

14:00~15:00

Leonid Bogdanov (Landau Inst. of Theoretical Physics, RAS, Russia), On Dunajski equation hierarchy and related hierarchies

 

15:00~15:30

 Tea Break

 

15:30~16:30 

 Dmitry Demskoy, (Academia Sinica) On recursion operators for elliptic integrable models

 

16:30~

Discussion

 

 17:30

 Dinner

 

Abstracts

Bogdanov:
"On Dunajski equation hierarchy and related hierarchies"

Abstract:
Dunajski equation, which is an integrable generalization of Plebansky
second heavenly equation, is considered as a characteristic example
of multidimensional integrable equations connected with commutation
of vector fields. A dressing scheme applicable to Dunajski
equation is developed, an example of constructing solutions
in terms of implicit functions is considered. Dunajski equation hierarchy
is described, its Lax-Sato form is presented. Dunajski equation
hierarchy is characterized by means of three-dimensional volume form, in
which a spectral variable is taken into account.Related hierarchies and some
reductions are considered.

Demskoy:
"On recursion operators for elliptic integrable models"

Abstract: The Krichever-Novikov and the Landau-Lifshitz
equations play a role of the universal models for the classes
of KdV and NLS-type equations. It is shown that the associative
algebra of quasilocal recursion operators for these models is generated by a
couple of operators related by an elliptic curve equation. A
theoretical explanation of this fact for the Landau-Lifshitz
equation is given in terms of multiplicators of the corresponding
Lax structure. New quasilocal Hamiltonian operators for these models
are found as well.

Mokhov:
"Integrability of the class of all flat torsionless
submanifolds in pseudo-Euclidean spaces, series of integrable
hydrodynamic type hierarchies associated with an arbitrary
submanifold of this class, and Frobenius manifolds"

Abstract: We consider the nonlinear system describing all flat torsionless
submanifolds in pseudo-Euclidean spaces and prove that this system is
a very natural generalization of the associativity equations of
two-dimensional topological quantum field theories and that this system is
integrable by the inverse scattering method. We prove that each flat
torsionless submanifold in a pseudo-Euclidean space gives a nonlocal
Hamiltonian operator of hydrodynamic type with flat metric, a special
pencil of compatible Poisson structures, a recursion operator and a
natural class of integrable dispersionless hierarchies, which are all
directly associated with this flat torsionless submanifold. We prove that
Frobenius manifolds are a special class of flat torsionless submanifolds
in pseudo-Euclidean spaces defined by the associativity equations and
consider the corresponding reductions.

Tsarev:
"The Monge Problem: quadrature-free integration of nonlinear ODEs,
Pfaffian forms and efficient car parking"

Abstract: This is about an old topic of finding closed-form solutions
of UNDERDETERMINED systems of nonlinear ordinary differential
equations, started by G.~Monge in 1784 and later
followed by Goursat (1905), Hilbert (1913) and Cartan (1914).
It has a classical counterpart in the theory of Pfaffian systems
in $R^n$, namely: can we find a reasonably large class
of Pfaffian systams in $R^n$, which admit a simple normal form?
In the last decades of the XX century these problems draw attention
of specialists in nonlinear control. In particular,
the technique of this problem was used in developing
motion algorithms for nonholonomic mechanical systems,
a typical example being a car with N trailers.
Parking such a "car train" moving back is a popular difficult task!
Modern results based on the old investigations of Goursat
make automatic control of such vehicles possible.
For our interests in the problem of integration of ODEs and PDEs one
can often remove (unnecessary) quadratures
in the final expressions for the complete solution of a C-integrable
nonlinear PDEs using the results described here.




報名表[Plain text]

下載[MS Word 7.0格式]
 
一般資訊(交通、住宿...)
 
中研院位置圖及院區圖
 

聯絡處:

115)台北市南港研究院路二段128號中央研究院數學所

聯絡人:

李志豪

Dr. Jyh-Hao Lee
E-mail:leejh@math.sinica.edu.tw

 

 焦源鳴

Tel:02-27851211
Ext.341(Mr. Y-M Chiao)

傳 真:

02-27827432

網 址:

http://www.math.sinica.edu.tw/workshop/workshop_c.php