Colloquium

 

Japanese Version

 

 

25

Friday, January 17, 2020, 4:30pm-6pm

04-08, Bld.63, Waseda University, Nishi-Waseda Campus

 

On Geometric Curve Flows and Solitons

 

Dr.  Hsiao-Fan Liu (Tamkang University)

 

Abstract 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.

 

24

Friday, December 13, 2019, 4pm-6pm

04-08, Bld.63, Waseda University, Nishi-Waseda Campus

 

Discrete Painlevé Equations and Orthogonal Polynomials

Prof.  Anton Dzhamay (University of Northern Colorado)

Abstract: 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).

 

 

23

Wednesday, December 11, 2019, 2:45pm-6pm

17-06, Bld.51, Waseda University, Nishi-Waseda Campus

 

Geometry of Discrete Painlevé Equations

 

Prof.  Anton Dzhamay (University of Northern Colorado)

 

Abstract: 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).

 

 

22

Tuesday, June 4, 2019, 4:30pm-6pm

17-04, Bld.51, Waseda University, Nishi-Waseda Campus

 

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

 

Prof.  Yuji Kodama (Ohio State University)

Abstract: 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.

 

21

Thursday, March 7, 2019, 4pm-5:30pm

Room103, Bld.52, Waseda University, Nishi-Waseda Campus

 

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

Prof. Jonathan Eckhardt (Loughborough University)

Abstract:  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.

 

20

Monday, November 19, 2018, 4:30pm-6pm

Room04-22, Bld.63, Waseda University, Nishi-Waseda Campus

 

Detecting and determining preserved measures and integrals of rational maps 

Prof. Reinout Quispel (La Trobe University)

Abstract:  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.

 

19

Friday, November 9, 2018, 4:00pm-5:30pm

05 meeting room, 2nd floor, Bld.63, Waseda University, Nishi-Waseda Campus

 

Symmetry through Geometry

Prof. Nalini Joshi (University of Sydney)

Abstract:  Discrete integrable equations can be considered in two, three or N-dimensions, as equations fitted together in a self-consistent way on a square, a cube or an N-dimensional cube. We show to find their symmetry reductions (and other properties) through a geometric perspective.

 

• N. Joshi and N. Nakazono: Elliptic Painlevé equations from next-nearest-neighbor translations on the E8(1) lattice, Journal of Physics A: Mathematical and Theoretical, 50 (2017), Art. 305205 (17 pp)

• J. Atkinson, P. Howes, N. Joshi and N. Nakazono: Geometry of an elliptic difference equation related to Q4, Journal of the London Mathematical Society, 93 (2016), no. 93, 763–784

• N. Joshi, N. Nakazono, Y. Shi: Reflection groups and discrete integrable systems, Journal of Integrable Systems, (2016), (37 pp).

• N. Joshi and N. Nakazono: Lax pairs of discrete Painlevé equations: (A2+A1)(1) case, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472 (2016), no. 2196 (13 pp).

• N. Joshi, N. Nakazono and Y. Shi (2014). "Geometric reductions of ABS equations on an n-cube to discrete Painlevé systems." Journal of Physics A-Mathematical and Theoretical 47: 505201 (16pp).

 

 

18

Tuesday, July 24, 2018, 3:30pm-4:30pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

Semilinear Klein-Gordon Equation in the Friedmann-Lamaitre-Robertson-Walker spacetime

Prof. Anahit Galtsyan (University of Texas Rio Grande Valley)

Abstract: We present some results on the semilinear massless waves propagating in the Einstein-de Sitter spacetime and semilinear Klein-Gordon Equation in the de Sitter spacetime. We examine the solutions of the semilinear wave equation, and, in particular, of the $\varphi^p$ model of quantum field theory in the curved space-time. More precisely, for $1 < p < 4$ we prove that solution of the massless self-interacting scalar field equation in the Einstein-de Sitter universe has finite lifespan. Furthermore, we present a condition on the self-interaction term that guaranties the existence of the global in time solution of the Cauchy problem for the semilinear Klein-Gordon equation in the FLRW (Friedmann- Lamaitre-Robertson-Walker) model of the contracting universe. For the equation with the Higgs potential we give an estimate for the lifespan of solution.

 

Tuesday, July 24, 2018, 4:45pm-5:45pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

A new integral transform approach to solving equations of the quantum field theory in the curved space-times

 

Prof. Karen Yagdjian (University of Texas Rio Grande Valley)

 

Abstract: In this talk we will present the integral transform that allows us to construct solutions of the hyperbolic partial differential equation with variable coefficients via solutions of a simpler equation. This transform was suggested by the author in the case when the last equation is a wave equation. Then it was used to investigate several well-known equations such as generalized Tricomi equation, the Klein–Gordon equation of the quantum field theory in the de Sitter and Einstein-de Sitter space-times of the expanding universe. In particular it was shown that a field with the mass √2 is huygensian. Moreover, the numbers √2, 0 are the only values of the mass such that equation obeys an incomplete Huygens Principle. Then, it was shown that in the de Sitter space-time the existence of two different scalar fields (with mass 0 and 2), which obey incomplete Huygens' principle, is equivalent to the condition that the spatial dimension of the physical world is 3. In this talk a special attention will be also given to the global in time existence of self-interacting scalar field in the de Sitter universe and to the Higuchi bound of the quantum field theory and equations with the Higgs potential.

 

17

Saturday, July 14, 2018, 3:30pm-4:30pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

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

 

Prof.  Alexander SakhnovichUniversität Wien

 

Abstract: 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.

 

16

Saturday, June 23, 2018, 2pm-3:30pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

On the rational solutions and the solitons of the KP hierarchy

Prof. Yuji KodamaOhio State University

Abstract 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.

 

15

Monday, May 28, 2018, 4:30pm-6:00pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

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

 

Prof. Nail Akhmediev,   Australian National University

Abstract: 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. 

 

14

Monday, May 21, 2018, 4:30pm-6:00pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

Mach Reflection of a Solitary Wave: Experiments

Prof. Harry Yeh,  Oregon State University

Abstract: 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 interaction predicted by the KP theory.

 

13

Thursday, November 16, 2017, 5pm-6:30pm

Meeting room 05, 2nd floor, Building#63, Waseda University, Nishi-Waseda Campus

 

Syncronization of the Kuramoto Model on Network

 

Dr. Hayato Chiba,  Kyushu University

 

12

Monday, November 6, 2017, 2:45pm-4:15pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

Interaction solutions between lumps and solitons via symbolic computations 

 

Prof. Wen-Xiu MaUniversity of South Florida, USA

Abstract: 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.

 

11

Tuesday,  July 11, 2017,  5:15pm-6:45pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

On nonlocal nonlinear Schrodinger equation and its discrete version

 

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

 

Abstract:  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.

 

10

Saturday, June 10, 2017,  1:00pm-4:00pm

54-101, Waseda University, Nishi-Waseda Campus

 

Hypergeometric functions and integrable hydrodynamic systems

 

Prof. Yuji Kodama, Ohio State University, USA

AbstractI 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.

 

9

Tuesday, June 6, 2017,  3:00pm-6:00pm

51-17-08, Waseda University, Nishi-Waseda Campus

 

KP solitons

 

Prof. Yuji Kodama, Ohio State University, USA

Abstract 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).

 

8

Wednesday, May 17, 2017,  3:00pm-6:00pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

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

 

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.  

 

7

Tuesday, May 16, 2017,  3:00pm-6:00pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

Riemann-Hilbert Methods in Integrable Systems I

Lecture  1: Riemann-Hilbert Problems and Lax Pairs

Lecture  2: Some Theory of Riemann-Hilbert Problems

 

Prof. Peter Miller, University of Michigan, USA

 

Lecture 1, Riemann-Hilbert Problems and Lax Pairs: 

The inverse-scattering transform can be used to study the initial-value problem for certain nonlinear wave equations, and the most important part of this analysis frequently leads to a Riemann-Hilbert problem of complex function theory.  This lecture will explain how Riemann-Hilbert problems arise in this setting, and will then reveal why Riemann-Hilbert problems are fundamentally related to integrability by means of Lax pairs arising from the dressing construction. 

 

Lecture 2, Some Theory of Riemann-Hilbert Problems: 

A Riemann-Hilbert problem is fundamentally a problem of complex analysis, a kind of boundary-value problem for the Cauchy-Riemann equations.  However, as with many problems of elliptic partial differential equations, a Riemann-Hilbert problem can be recast as a singular integral equation.  This lecture will highlight some of the key ideas of the connection between Riemann-Hilbert problems and integral equations, with emphasis on the small-norm setting and how to achieve it by deformation techniques.

 

6

Tuesday, May 17 2016,  1:00pm-2:30pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

Unidirectional wave propagation and integrable models in shallow water

 

Prof. Roberto Camassa, University of North Carolina, USA

 

Tuesday, May 17, 2016,  2:45pm-4:15pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

Periodic waves and stability in deep water

 

Prof. Wooyoung Choi,  New Jersey Institute of Technology, USA

 

5

Monday, May 16, 2016,  1:00pm-2:30pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 Fundamentals of fluid mechanics and free surface flows

 

Prof. Roberto Camassa, University of North Carolina, USA

 

Monday, May 16, 2016,  2:45pm-4:15pm

Large meeting room, 62W 1st floor, Waseda University, Nishi-Waseda Campus

 

Asymptotic theories for nonlinear water waves

 

Prof. Wooyoung Choi,  New Jersey Institute of Technology, USA

 

4

Monday, March 16, 2015

3:00pm-5:00pm

Large meeting room, 62W 1st floor

Waseda University, Nishi-Waseda Campus

 

From Schlesinger Transformations to Difference Painlevé Equations

 

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

 

3

Friday, July 25, 2014

2:45pm-4:15pm

Mathematics & Applied Mathematics meeting room, 63 1st floor

Waseda University, Nishi-Waseda Campus

 

Triangulations of convex polygon and solitons in 2-dimension

Prof. Yuji Kodama, Department of Mathematics, Ohio State University, USA

 

2

Friday, July 25, 2014

2:45pm-4:15pm

Mathematics & Applied Mathematics meeting room, 63 1st floor

Waseda University, Nishi-Waseda Campus

 

Beach waves and line-solitons of the KP equation

Prof. Sarbarish Chakravarty, Department of Mathematics, The University of Colorado at Colorado Springs, USA

1

Thursday, May 22, 2014

5:00pm-6:30pm

 

Large meeting room, 62W 1st floor

Waseda University, Nishi-Waseda Campus

 

 

The complex and coupled complex short pulse equations, their integrable discretizations and novel numerical simulations

 

Prof. Bao-Feng Feng, Department of Mathematics, The University of Texas - Pan American

 

Organizers: Dr. Daisuke Takahashi (daisuket atmark waseda.jp) or Dr. Kenichi Maruno(kmaruno atmark waseda.jp). atmark means @.