早稲田可積分系セミナー

 

Waseda Integrable Systems Seminar

 

22

日時:2020年1月17日(金)午後4時30分 − 午後6時

場所:早稲田大学西早稲田キャンパス63号館04−08号室

 

題目:On Geometric Curve Flows and Solitons

講演者:Dr.  Hsiao-Fan Liu (Tamkang University)

概要: Geometric curves flows are curve evolutions whose invariants flow according to some soliton equations. Such correspondences provide a systematic tool to study geometric curve flows via soliton theory,  and vice versa.  In this talk, we discuss certain geometric curve flows and explain how we build relations between curves and soliton equations, how we use the soliton theory to derive B\"acklund transformations for these curve flows, and to study the existence of solutions to the (periodic) Cauchy problems of curve flows. This also provides geometric algorithms to solve periodic Cauchy problems numerically.

 

21

日時:2019年12月13日(金)午後4時 − 午後6時

場所:早稲田大学西早稲田キャンパス63号館04−08号室

 

題目:Discrete Painlevé Equations and Orthogonal Polynomials

 

講演者:Prof.  Anton Dzhamay (University of Northern Colorado)

概要: Over the last decade it became clear that the role of discrete Painlevé equations in applications has been steadily growing. Thus, the question of recognizing a certain non-autonomous recurrence as a discrete Painlevé equation and understanding its position in Sakai’s classification scheme, recognizing whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such example, becomes one of the central ones. Fortunately, Sakai’s geometric theory provides an almost algorithmic procedure of answering this question. In this work we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for orthogonal polynomials can be expressed in terms of solutions of some discrete Painlevé equation. In this work we study orthogonal polynomials with
general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form. This is joint work with Galina Filipuk (University of Warsaw, Poland) and Alexander Stokes (University College, London, UK).

 

20

日時:2019年12月11日(水)午後2時45分 − 午後6時

場所:早稲田大学西早稲田キャンパス51号館17−06号室

 

題目: Geometry of Discrete Painlevé Equations

 

講演者:Prof.  Anton Dzhamay (University of Northern Colorado)

概要: In this talk we give an introduction to some geometric ideas and tools used to study discrete integrable systems. Our main goal is to give an introduction to Sakai’s geometric theory of discrete Painlevé equations. However, we first consider an autonomous example of a dynamical system known as the QRT map. For this example we explain the geometry behind indeterminate (or base)points of birational maps, as well as how to fix such indeterminacies by changing the geometry of the configuration space using the so-called blowup procedure. In the process we also introduce the notions of the Picard lattice of the algebraic surface that is the configuration space of the dynamics, the anti-canonical divisor class, and the linearization of the mapping on the level of the Picard lattice. After that we consider an idea of geometric deautonomzation. Using this approach we introduce discrete Painlevé equations as deautonomizations of QRT maps. We show how such deautonomization results in the decomposition of the Picard lattice into complementary pairs of the surface and symmetry sub-lattices and explain the construction of a binational representation of affine Weyl symmetry groups that gives a complete algebraic description of our non-linear dynamic. We show how to represent a discrete Painlevé equation as a composition of elementary birational transformations (Cremona isometries). We conclude the tutorial by a brief introduction into Sakai’s classification scheme for discrete Painlevé equations.This talk is based on joint work with Stefan Carstea (Bucharest) and Tomoyuki Takenawa (Tokyo).

 

19

日時:2019年6月4日(火) 午後4時30分 - 午後6時

場所:早稲田大学西早稲田キャンパス51号館17−04号室

 

題目: Rational space curves and solitons for the Gelfand-Dickey reductions of the KP hierarchy.

 

講演者:Prof.  Yuji Kodama (Ohio State University)

概要:  It is well known that the algebro-geometric solutions of the KdV hierarchy are

constructed from the Riemann theta functions associated with the hyperelliptic curves,

and that the soliton solutions can be obtained by rational (singular) limits of the hyperelliptic curves.

 

In this talk, I will discuss certain class of KP solitons in the connections with space curves,

which are labeled by certain types of numerical semigroups. In particular, I will show that

the (singular and complex) KP solitons of the Gelfand-Dickey reduction ($l$-reduction)

are associated with the rational space curves of $<l,lm+1,\ldots, lm+k>$ where $m\ge 1$ and

$1\le k\le l-1$. This is a part of the PhD project of my student, Yuancheng Xie.

 

 18

日時:2019年3月7日(木) 午後4時 − 午後5時30分

場所:早稲田大学西早稲田キャンパス52号館103号室

 

題目: On the inverse spectral transform for the conservative Camassa-Holm flow

 

講演者:Prof.  Jonathan Eckhardt (Loughborough University)

概要:  The Camassa-Holm equation is a nonlinear partial differential equation that models unidirectional wave propagation on shallow water. I will show how this equation can be integrated by means of the inverse spectral transform method. The global conservative solutions obtained in this way form into a train of solitons (peakons) in the long-time limit.

 

17

日時:2018年11月19日(月) 午後4時30分 − 午後6時

場所:早稲田大学西早稲田キャンパス63号館04―22

 

題目: Detecting and determining preserved measures and integrals of rational maps

 

講演者:Prof.  Reinout Quispel (La Trobe University)

概要:  The search for preserved measures and integrals of ordinary differential equations has been at the forefront of mathematical physics since the time of Galileo and Newton. In this talk our aim will be to develop an analogous theory for the (arguably more general) discrete-time case. This will lead to linear algorithms for detecting and determining preserved measures and integrals of rational maps.

 

16

日時:2018年7月14日(土) 午後3時30分 − 午後4時30分

場所:早稲田大学西早稲田キャンパス62W号館一階大会議室

 

題目:GBDT version of Backlund-Darboux transformation and evolution of Weyl functions

講演者:Prof. Alexander SakhnovichUniversität Wien

概要:  We consider applications of GBDT to dynamical systems and integrable nonlinear equations including nonlocal NLS equation and second harmonic generation equation. The initial-boundary problem for  second harmonic generation equation will be discussed as well.

(本セミナーは講演会を兼ねています)

 

 

15

 

日時:2018年6月23日(土) 午後2時 − 午後3時30分

場所:早稲田大学西早稲田キャンパス52号館102号室

 

題目:On the rational solutions and the solitons of the KP hierarchy

講演者:Prof. Yuji Kodama  (児玉裕治),Ohio State University 

概要: It is well known that the Schur polynomials satisfy the Hirota bilinear equations of the KP hierarchy, and that each Schur polynomial can be parametrized by a unique Young diagram. We also know that the KP solitons (exponential solutions) can be parametrized by certain decomposition of the Grassmannians. In the talk, I will explain the connection between the rational solutions and the KP solitons in terms of the Young diagrams. More explicitly,  I will show how one gets a rational solution from a KP soliton. I will also discuss a connection between quasi-periodic solutions (theta or sigma functions) and the KP solitons.

 

(本セミナーは講演会を兼ねています)

 

 

14

 

日時:2018年5月28日月曜日 午後4時30分 − 午後6時

場所:早稲田大学西早稲田キャンパス62W号館1階大会議室

 

題目:The nonlinear Schroedinger equation and variety of it's nontrivial extensions

講演者:Prof. Nail AkhmedievAustralian National University

概要: The NLSE represents a dynamical system with an infinite number of degrees of freedom and as such it has an infinite number of solutions that includes solitons, breathers, rogue waves, radiation waves and their combinations. Every extension of the NLSE expands dramatically variety of it's solutions. Both conservative and dissipative extensions will be considered in this talk. 

(本セミナーは講演会を兼ねています)

 

 

13

 

日時:2018年5月21日月曜日 午後4時30分 − 午後6時

場所:早稲田大学西早稲田キャンパス62W号館1階大会議室

 

題目: Mach Reflection of a Solitary Wave: Experiments 

講演者:Prof. Harry YehOregon State University

概要: Laboratory and numerical experiments are presented for Mach reflection of an obliquely incident solitary wave at a vertical wall. The numerical model is based on the pseudo-spectral method for the full Euler formulation. With the aid of a laser sheet in the laboratory, the wave profiles are measured optically in sub-millimeter precision. Discrepancies reported in previous works are now substantially improved, partly because of the higher-order KP theory and in part because of the advancement in computational power and laboratory instrumentation. While the theory predicts the maximum of four-fold (4.0) amplification of the Mach stem, the maximum observed in the laboratory was 2.922 (the previous laboratory study had achieved the amplification of 2.4), while our numerical simulation reached the maximum of 3.91 (previously reported amplification was 2.897). Also presented are other laboratory realizations of soliton-soliton.

(本セミナーは講演会を兼ねています)

 

 

12

日時:2017年12月27日水曜日 午後1時30分から

場所:早稲田大学西早稲田キャンパス62W号館大会議室

 

題目:直交多項式と古典・量子ランダムウォーク

 

講演者:三木 啓司 (気象大学校)

概要:直交多項式は可積分系・可解な数理モデルとの対応が非常によく知られている。本セミナーでは,1次元の出生死滅過程やスピン鎖モデルにおいて直交多項式がどういう役割を果たしているかについて述べ,これらの高次元拡張の結果を紹介する。

 

 

11

 

日時:2017年11月6日月曜日 午後2時45分 − 午後4時15分

場所:早稲田大学西早稲田キャンパス62W号館1階大会議室

 

題目:  Interaction solutions between lumps and solitons via symbolic computations 

講演者:Prof. Wen-Xiu MaUniversity of South Florida

概要: We will talk about interaction solutions between lump solutions and soliton solutions to integrable equations. A computational algorithm will be discussed, based on the bilinear formulation; and illustrative examples in the cases of the (2+1)-dimensional KP and Ito equations will be presented through Maple symbolic computations.

(本セミナーは講演会を兼ねています)

 

 

10

 

日時:2017年7月11日火曜日 午後5時15分 − 午後6時45分

場所:早稲田大学西早稲田キャンパス62W号館大会議室

 

題目:On nonlocal nonlinear Schrodinger equation and its discrete version

 

講演者:Prof. Zuo-Nong Zhu,   Shanghai Jiao Tong University,  P. R. China

概要: Very recently, Ablowitz and Musslimani introduced reverse space, reverse time, and reverse space-time nonlocal nonlinear integrable equations including the reverse space nonlocal NLS equation, the real and complex reverse space-time nonlocal mKdV, sine-Gordon, Davey-Stewartson equations, et.al. In this talk, we will show that, under the gauge transformations, the nonlocal focusing NLS (and it discrete version) and the nonlocal defocusing NLS (and it discrete version) are, respectively, gauge equivalent to the coupled Heisenberg equation (and it discrete version) and the coupled modified Heisenberg equation (and it discrete version). We will discuss the construction of discrete soliton solutions for the discrete nonlocal focusing NLS. We will demonstrate that the discrete soliton yields soliton of nonlocal focusing NLS under the continuous limit. The relations of these solutions between nonlocal NLS and classical NLS will be given. This is a joint work with Dr. Li-yuan Ma.

(本セミナーは講演会を兼ねています)

 

 

9

 

日時:2017年6月13日火曜日 午後4時30分 − 午後6時

場所:早稲田大学西早稲田キャンパス51号館17−08

 

題目:半古典的波束のハミルトン力学

 

講演者:Prof. Tomoki Ohsawa (大澤知己),(University of Texas at Dallas, USA)

概要: 古典力学と量子力学の基礎方程式はどちらも、適切なシンプレクティック構造を用いて、

ハミルトン力学系として記述されることはよく知られている。この講演では、そういった幾何学的構造を用いて、

量子力学と古典力学の境界である半古典領域の力学をハミルトン力学系として記述する。

特にHagedornによる半古典的波束の力学を対象とする。

 

 

8

 

日時:2017年6月10日土曜日 午後1時 − 午後4時

場所:早稲田大学西早稲田キャンパス54号館101

 

題目:Hypergeometric functions and integrable hydrodynamic systems

 

講演者:Prof. Yuji Kodama (児玉裕治),  Ohio State University, USA

概要:I will show an interesting connection of (generalized) hypergeometric 
functions with integrable hydrodynamic-type systems. The lecture contains the following subjects.
(a) Integrable hydrodynamic systems generated by Lauricella functions.
(b) Confluence of the Lauricella functions and non-diagonalizable hydrodynamic-type systems.

(本セミナーは講演会を兼ねています)

 

 

7

 

日時:2017年6月6日火曜日 午後3時 − 午後6時

場所:早稲田大学西早稲田キャンパス51号館17-08

 

題目:KP solitons

 

講演者:Prof. Yuji Kodama (児玉裕治),  Ohio State University, USA

概要: I will discuss some combinatorial aspects of the KP solitons. This lecture is to explain the following subjects.
(a) Mathematical background of the regular soliton solutions (the totally non-negative Grassmannians and their parametrization).
(b) Applications of the KP solitons to shallow water waves (Mach reflection and rogue waves).

(本セミナーは講演会を兼ねています)

 

6

 

日時:2017年5月17日水曜日 午後3時 − 午後6時

場所:早稲田大学西早稲田キャンパス62W号館1階大会議室

 

題目:Riemann-Hilbert Methods in Integrable Systems II

Lecture  3: Asymptotic Analysis of Riemann-Hilbert Problems, Part I

Lecture  4: Asymptotic Analysis of Riemann-Hilbert Problems, Part II

 

講演者:Prof. Peter Miller, University of Michigan, USA

 

Lecture 3, Asymptotic Analysis of Riemann-Hilbert Problems, Part I: 

The Deift-Zhou steepest descent method is a powerful set of techniques applicable to Riemann-Hilbert problems that are generalizations of the classical steepest descent method for the asymptotic expansion of certain contour integrals.  This lecture will focus on the Fokas-Its'-Kitaev Riemann-Hilbert problem characterizing orthogonal polynomials with exponentially varying weights and the asymptotic limit of large degree as an example of the steepest descent method.  The goal of this lecture is to deform the Riemann-Hilbert problem to the point where it appears at a formal level to be asymptotically simple.  

Lecture 4, Asymptotic Analysis of Riemann-Hilbert Problems, Part II: 

This lecture picks up where Lecture 3 left off.  The deformed Riemann-Hilbert problem suggests an approximate solution, known as a parametrix.  The parametrix will be constructed explicitly with the help of elementary and special functions.  Then by comparing the parametrix to the exact solution we will arrive at a Riemann-Hilbert problem of small-norm type (cf., Lecture 2). Estimates on the solution of the latter problem yield explicit leading-order asymptotic formulae for the orthogonal polynomials and related quantities of interest in applications such as random matrix theory.  

 

 (本セミナーは講演会を兼ねています)

 

5

 

日時:2017年5月16日火曜日 午後3時 − 午後6時

場所:早稲田大学西早稲田キャンパス62W号館1階大会議室

 

題目:Riemann-Hilbert Methods in Integrable Systems II

Lecture  3: Asymptotic Analysis of Riemann-Hilbert Problems, Part I

Lecture  4: Asymptotic Analysis of Riemann-Hilbert Problems, Part II

 

講演者:Prof. Peter Miller, University of Michigan, USA

 

Lecture 3, Asymptotic Analysis of Riemann-Hilbert Problems, Part I: 

The Deift-Zhou steepest descent method is a powerful set of techniques applicable to Riemann-Hilbert problems that are generalizations of the classical steepest descent method for the asymptotic expansion of certain contour integrals.  This lecture will focus on the Fokas-Its'-Kitaev Riemann-Hilbert problem characterizing orthogonal polynomials with exponentially varying weights and the asymptotic limit of large degree as an example of the steepest descent method.  The goal of this lecture is to deform the Riemann-Hilbert problem to the point where it appears at a formal level to be asymptotically simple.  

Lecture 4, Asymptotic Analysis of Riemann-Hilbert Problems, Part II: 

This lecture picks up where Lecture 3 left off.  The deformed Riemann-Hilbert problem suggests an approximate solution, known as a parametrix.  The parametrix will be constructed explicitly with the help of elementary and special functions.  Then by comparing the parametrix to the exact solution we will arrive at a Riemann-Hilbert problem of small-norm type (cf., Lecture 2). Estimates on the solution of the latter problem yield explicit leading-order asymptotic formulae for the orthogonal polynomials and related quantities of interest in applications such as random matrix theory.  

 

 (本セミナーは講演会を兼ねています)

 

 

4

 

日時:2016年12月27日火曜日 午後1時00分 −

場所:早稲田大学西早稲田キャンパス62W号館1階大会議室

 

題目:可積分なコマの方程式の差分化の現状

 

講演者:木村欣司(京都大学)

概要:オイラー、ラグランジュ、コワレフスカヤの場合については、コマの方程式は可積分であることが分かっており、そのうち、オイラー、ラグランジュの場合については、その離散アナログとして、保存量ならびに楕円関数解を持つものが知られている。しかし、それらの離散アナログに対するLax対は、不明であった。今回、連続時間のオイラーのコマを拡張したものからPainleve-6方程式を導出できるという事実を、差分方程式の世界で実現することを試みた。その結果、楕円差分Painleve方程式のパラメータが少ない場合の方程式を得られることがわかった。そのような計算結果を考察することで、離散オイラーのコマのLax対を構成できた。離散オイラーのコマのLax対からの類推として、より一般化された高階の離散オイラーのコマを手に入れることができる。しかし、その可積分性は、わかっていない。一方で、連続時間では、すでに可積分と証明されている高階のオイラーのコマが存在する。その離散アナログについての予想も存在する。こちらの方程式についても、その可積分性は、わかっていない。以上のように、この分野は、未整理な状態にある。講演では、この分野の研究の現状について、詳細に報告する。

 

 

 

3

 

日時:2015年3月16日月曜日 午後3時00分 − 午後5時00分

場所:早稲田大学西早稲田キャンパス62W号館1階大会議室

 

題目:From Schlesinger Transformations to Difference Painlevé Equations

 

講演者:Prof. Anton Dzhamay, School of Mathematical Sciences, University of Northern Colorado, USA

 

概要:The goal of the talk is to explain some recent joint work with T.Takenawa (Tokyo University of Marine Science and Technology) on reduction

from elementary Schlesinger transformations, including higher-rank transformations, to difference Painlevé equations. This is a continuation of an earlier project with 

T.Takenawa and H. Sakai (the University of Tokyo). In the first part of the talk we will consider the general formalism of discrete evolution equations that describe

elementary Schlesinger transformations and see how this equations can be written in a discrete Hamiltonian form. In the second part we consider some examples of 

reductions from these discrete evolutionary Schlesinger equations to difference Painlevé equations. We will particularly emphasize the role played by the geometry 

of the Okamoto space of initial conditions in understanding the structure of these reductions.

   (本セミナーは早稲田大学理工学術院講演会を兼ねています)

 

 

2

 

日時:2014年725日金曜日午後2時45分 – 午後6時

場所:早稲田大学西早稲田キャンパス63号館1階数学応数会議室

 

講演1:午後2時45分 午後4時15分

題目: Beach waves and line-solitons of the KP equation

講演者: Sarbarish Chakravarty, Department of Mathematics, The University of Colorado at Colorado Springs, USA

Nonlinear interactions among small amplitude, long wavelength,  obliquely propagating waves on the surface of shallow water often generate web-like patterns.

In this talk, we discuss how line-soliton solutions of the Kadomtsev-Petviashvili (KP) equation can approximate such web-pattern

in shallow water wave. We describe an ``inverse problem" which maps a certain set of measurable data from the solitary waves in the given

pattern to the parameters required to construct an exact KP soliton that describes the non-stationary dynamics of the pattern.

We illustrate the inverse problem using explicit examples of shallow water wave pattern.

 

講演2:午後4時30分 午後6時

題目: Triangulations of convex polygon and solitons in 2-dimension

講演者: Yuji Kodama, Department of Mathematics, Ohio State University, USA

We give an explicit connection between the patterns (soliton graphs) generated by soliton solutions (also obtained by hyperplane arrangements) and triangulations (subdivisions)

of polygons inscribed in conic curves.  Two dimensional integrable systems admitting those soliton solutions include the KP equation (for parabola),

two-dimensional Toda lattice (for hyperbola) and the Davey-Stewartson systems (for ellipse).

 

   (本セミナーは講演会を兼ねています)

  

 

 

1

 

日時:2014年6月2日月曜日午後2時45分 – 午後4時15分

場所:早稲田大学西早稲田キャンパス51号館17−06

 

題目: q-Painleve 方程式の特殊解とリーマン・ヒルベルト問題

講演者: 大山陽介(大阪大学大学院情報科学研究科情報基礎数学専攻)

 

アブストラクト
q-Painleve
方程式は、q-線型方程式の接続係数保存変形として得られる。 q-Painleve 方程式の特殊解の中で、接続係数を決定可能な例を探るとともに、接続係数に関するバーコフのリーマン・ヒルベルト問題について解説する。

 

講演会はこちら

数物コロキウムはこちら

 

セミナー世話人:丸野健一
(可積分系、数理物理の周辺の話題についてのセミナーを不定期に開催します。話題を提供して下さる方は大歓迎です。)