Colloquium
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.
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).
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).
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.
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.
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.
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).
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.
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 Sakhnovich，Universität 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.
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 Kodama，Ohio 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.
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.
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.
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
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 Ma，University 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.
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.
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
Abstract：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.
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).
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.
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.
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
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
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
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
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
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 @.