Symmetry Plus Integrability 2010

The First International Conference on Integrable Systems and Nonlinear Waves on the Gulf of Mexico In Honor of Prof. Yuji Kodama’s 60th Birthday

Abstract

June 10, 2010 – June 14, 2010

at South Padre Travelodge, South Padre Island, Texas Plenary Speakers

Mark J. Ablowitz (University of Colorado at Boulder, USA) Anthony Bloch (University of Michigan, USA) Masanobu Kaneko (Kyushu University, Japan) Boris Konopelchenko (University of Salento, Lecce, Italy) Alexander V. Mikhailov (University of Leeds, United Kingdom) Yoshimasa Nakamura (Kyoto University, Japan)

Scientific Advisory Committee

Sarbarish Chakravarty (University of Colorado at Colorado Springs, USA) Kenji Kajiwara (Kyushu University, Japan) William L. Kath (Northwestern University, USA)

Organizing Committee

Bao-Feng Feng (University of Texas – Pan American, USA) Kenji Kajiwara (Kyushu University, Japan) Kenichi Maruno (University of Texas – Pan American, USA) Virgil U. Pierce (University of Texas – Pan American, USA)

Local Arrangements Committee

Paul Bracken (University of Texas – Pan American, USA) Bao-Feng Feng (University of Texas – Pan American, USA) Tim Huber (University of Texas – Pan American, USA) Kenichi Maruno (University of Texas – Pan American, USA) Virgil U. Pierce (University of Texas – Pan American, USA)

Conference Secretary

Maria Lisa Cisneros (University of Texas – Pan American, USA)

Sponsors & Support We acknowledge the support of the following organizations: National Science Foundation, DMS1000037 Global COE Program, Education and Research Hub for Mathematics-for-Industry, Fac- ulty of Mathematics, Kyushu University Department of Mathematics, The University of Texas - Pan American College of Science and Engineering, The University of Texas Pan American Office of Graduate Studies, The University of Texas - Pan American Division of Academic Affairs, The University of Texas - Pan American UTPA University Bookstore

1 Thursday, June 10

8:30am-9:00am Opening Remark: Dr. Cynthia Brown (Vice Provost for Graduate Stud- ies, University of Texas – Pan American) 9:00am-10:00am Plenary Talk: Mark J. Ablowitz (University of Colorado at Boulder, USA) A World of Nonlinear Waves: from Beaches to Lasers

10:00am-10:30am Coffee Break

10:30am-11:00am M. Lakshmanan (Bharathidasan University, India) Integrability of a Class of Nonlinear Ordinary Differential Equations (ODEs) of any Order through Nonlo- cal Transformations 11:00am-11:30am Thiab Taha (University of Georgia, USA) Parallel Numerical Methods for Solving Nonlinear Evolution Equations 11:30am-12:00pm Mark Hoefer (North Carolina State University, USA) The Nonlinear Schr¨odingerequation in the small dispersion regime: applications

12:00pm-2:00pm Lunch & Poster Session. Supported by College of Science and Engineering & Office of Graduate Studies, UTPA

2:00pm-3:00pm Plenary Talk: Anthony Bloch (University of Michigan, USA) Integrable Flows and Asymptotic Stability 3:00pm-3:30pm Barbara Shipman (University of Texas at Arlington, USA) Fixed-point sets of non-maximal torus actions on flag manifolds, and the Toda lattice 3:30pm-4:00pm Michael Gekhtman (University of Notre Dame, USA) Hankel determi- nants, cluster algebras and integrable lattices

4:00pm-4:30pm Coffee Break

4:30pm-5:00pm Peter Miller (University of Michigan, USA) The Benjamin-Ono equation in the zero-dispersion limit 5:00pm-5:30pm Robert Buckingham (Unversity of Cincinnati, USA) Semiclassical spec- tral confinement for the sine-Gordon equation

Friday, June 11

9:00am-10:00am Plenary Talk: Masanobu Kaneko (Kyushu University, Japan) Modular forms satisfying certain differential equation

10:00am-10:30am Coffee Break

2 10:30am-11:00am Tim Huber (University of Texas – Pan American, USA) Differential equations for modular forms on SL(2, Z)-conjugate subgroups of level three 11:00am-11:30am Sarbarish Chakravarty (University of Colorado at Colorado Springs, USA) Nonlinear differential equations for Triangle groups 11:30am-12:00pm Kenji Kajiwara (Kyushu University, Japan) Ultradiscretization of solv- able chaotic systems

12:00pm-2:00pm Lunch & Poster Session, Supported by College of Science and Engineering & Office of Graduate Studies, UTPA

2:00pm-2:30pm Anton Dzhamay (University of Northern Colorado, USA) On Geometric Configurations Related to Matrix Factorizations 2:30pm-3:00pm Teruhisa Tsuda (Kyushu University, Japan) UC hierarchy, monodromy preserving deformations and hypergeometric functions 3:00pm-3:30pm Takeo Kojima (Yamagata University, Japan) The Integrals of Motion for d the Deformed W-algebra Wq,t(glN )

3:30pm-4:00pm Coffee Break

4:00pm-4:30pm William L. Kath (Northwestern University, USA) Methods to determine large deviations and rare events in optical pulses 4:30pm-5:00pm Shinsuke Watanabe (Open University of Japan, Japan) Experiments on shallow water waves 5:10pm-6:00pm Special Talk: Yuji Kodama (Ohio State University, USA)

7:00pm-9:00pm Banquet at Parrot Eyes (in front of Travelodge) Master of Ceremony: Lokenath Debnath (University of Texas – Pan American) Banquet Speakers: Mark J. Ablowitz (University of Colorado at Boulder, USA) Shinsuke Watanabe (Open University of Japan, Japan)

Saturday, June 12

9:00am-10:00am Plenary Talk: Alexander V. Mikhailov (University of Leeds, United Kingdom) Automorphic Lie algebras and corresponding Integrable Systems

10:00am-10:30am Coffee Break

10:30am-11:00am Harvey Segur (University of Colorado at Boulder, USA) Tsunamis and other shallow water waves 11:00am-11:30am Alfred R. Osborne (University of Torino, Italy) Hyperfast Numerical Codes for Integrable Soliton Equations in 2+1 dimensions using Riemann Theta Func-

3 tions (Cancelled) 11:00am-11:30pm Chiu-Yen Kao (Ohio State University, USA) A pseudo-spectral method with window technique for initial value problems of KP equation 12:00pm- Free Time & Discussion

Sunday, June 13

9:00am-10:00am Plenary Talk: Boris Konopelchenko (University of Salento, Lecce, Italy) Singular sectors of integrable hierarchies, degenerate critical points and Euler- Poisson-Darboux equations

10:00am-10:30am Coffee Break

10:30am-11:00am Takashi Takebe (Higher School of Economics, Russia) hbar-expansion of KP hierarchy: Recursive construction of solutions 11:00am-11:30am Johan van de Leur (Mathematical Institute of Utrecht University, Nether- land) Givental symmetries of Frobenius manifolds and multi-component KP tau-functions

11:30am-12:00pm Wen-Xiu Ma (University of South Florida, USA) Hamiltonian struc- tures of zero curvature equations by variational identities

12:00pm-2:00pm Lunch

2:00pm-2:30pm Jen-Hsu Chang (National Defense University, Taiwan) The σ-flows in the Novikov-Veselov Equation 2:30pm-3:00pm Vladimir Rosenhaus (California State University, Chico, USA) On Differ- ential Equations with Infinite Symmetries and Infinite Conservation Laws

3:00pm-3:30pm Coffee Break

3:30pm-4:00pm Alejandro Aceves (Southern Methodist University, USA) Traveling wave solutions in discrete systems 4:00pm-4:30pm Gregor Kovacic (Rensselaer Polytechnic Institute, USA) Integrable Ran- dom Optical Pulse Dynamics in a Resonant Optical Medium 4:30pm-5:00pm Hayato Chiba (Kyushu University, Japan) Bifurcation theory of the infinite- dimensional Kuramoto model 5:00pm-5:30pm Eleftherios Gkioulekas (University of Texas – Pan American, USA) Can the two-layer QG model explain the Nastrom-Gage energy spectrum of the atmosphere?

4 Monday, June 14

9:00am-10:00am Plenary Talk: Yoshimasa Nakamura (Kyoto University, Japan) Inte- grable Algorithms: from Moser to I-SVD

10:00am-10:30am Coffee Break

10:30am-11:00am Satoshi Tsujimoto (Kyoto University, Japan) Orthogonal polynomials and Toda lattice hierarchy 11:00am-11:30am Yasuhiro Ohta (Kobe University, Japan) Discrete sine-Gordon equa- tions: hyperbolic and elliptic types 11:30am-12:00pm Willy Hereman (Colorado School of Mines, USA) Symbolic Computa- tion of Conservation Laws of Nonlinear Partial Differential Equations in Multiple Space Dimensions

12:00pm-2:00pm Lunch

2:00pm-2:30pm Takao Suzuki (Kobe University, Japan) A class of higher order Painlev´e systems arising from integrable hierarchies of type A 2:30pm-3:00pm Junta Matsukidaira (Ryukoku University, Japan) Constructing two-dimensional integrable mappings that possess invariants of high degree 3:00pm-3:30pm Daisuke Takahashi (Waseda University, Japan) On some ultradiscrete sys- tems 3:30pm-4:00pm Satoru Saito (Tokyo Metropolitan University, Japan) Generations of in- variant varieties of periodic points from the singularity confinement

Tuesday, June 15 Departure

Poster Session Yuhan Jia (Ohio State University, USA) Higher order corrections to the Miles theory of shallow water waves Chuanzhong Li (Ohio State University, USA) The solution of bigraded Toda hierchy Czeslaw Maczka (AGH University of Science and Technology, Krakow, Poland) Traveling wave solutions of the generalized convection-reaction-diffusion equation Hiroshi Miki (Kyoto University, Japan) Skew orthogonal polynomials associated with Askey-Wilson polynomials Nobutaka Nakazono (Kyushu University, Japan) Hypergeometric solutions to the q-Painlev´e IV equation

5 Noriko Saitoh (Yokohama National University, Japan) A behavior of Julia set of higher dimensional maps in the integrable limit Erwin Suazo (University of Puerto Rico, Mayaguez, USA) On Fundamental Solutions for Evolution Equations Kouichi Toda (Toyama Prefectural University, Japan) Axially symmetric soliton solutions in a extended Skyrme-Faddeev model Hidekazu Tsuji (Kyushu University, Japan) Two-dimensional Interaction of solitary waves in a two-layer fluid with large depth

6 Title: A World of Nonlinear Waves: from Beaches to Lasers Name: Mark J. Ablowitz Affiliation: Department of Applied Mathematics, University of Colorado at Boulder, USA

Email: [email protected] Abstract: The study of localized waves has a long history dating back to the discoveries in the 1800s describing water waves in shallow water. In the 1960s researchers found that certain equations, including the Korteweg-de Vries (KdV) and nonlinear Schr¨odinger (NLS) equations, were universal in nature. Subsequent research led to the concept of solitons which are solitary waves which interact “elastically”. In math solitons usually mean the localized waves have the elastic property; but in physics solitons usually don’t have the elastic property–i.e. they are solitary waves. In this lecture a recent nonlocal formulation of water waves will be discussed. The corresponding asymptotic reductions will be related to the Benney-Luke and Kadomstsev-Petviashvili equations in 2+1 dimen- sions, the Boussinesq/KdV and NLS equations in 1+1 dimension and their underlying soliton solutions will also be discussed. Such nonlocal formulations also apply to internal waves. In optics NLS type equations are central. Dispersion-managed systems in com- munications and mode-locked (ML) lasers also lead to interesting NLS systems. In ML lasers one can find mode-locked solitons, strings of ML solitons in both the anomalous and normal dispersive regimes.

Title: Traveling wave solutions in discrete systems Name: Alejandro Aceves Affiliation: Department of Mathematics, Southern Methodist University, USA Email: [email protected] Abstract: Nonlinear discrete systems arise as models in disciplines such as biology, parti- cle dynamics and nonlinear optics. For the few known integrable models, methods exist that produce traveling wave solitons or fronts. On the other hand, important discrete models lack integrability and in particular this presents and obstacle to the existence of traveling wave solutions. We discuss using particular models the role of nonintegrability and dissipation in the search of traveling wave solutions.

Title: Integrable Flows and Asymptotic Stability Name: Anthony Bloch Affiliation: University of Michigan, USA Email: [email protected]

7 Abstract: In this talk I will discuss the qualitative behavior of various integrable flows which have asymptotically stable equilibria. In particular, I will consider the dynamics of gradient flows, double bracket flows on adjoint orbits, and certain nonholonomic flows which preserve energy but do not preserve measure. In the nonholonomic setting I will discuss both constraints which are linear and nonlinear in the velocities. The latter case occurs in the dynamics of thermostats.

Title: Semiclassical spectral confinement for the sine-Gordon equation Name: Robert Buckingham Affiliation: University of Cincinnati, USA Email: [email protected] Abstract: The small-dispersion or semiclassical sine-Gordon equation is an integrable model of magnetic flux propagation in long Josephson junctions. Solving the sine-Gordon equation via the inverse-scattering method requires computing associated spectral data, including eigenvalues which can lie anywhere in the complex plane. We show that, for a broad class of initial data, the eigenvalues must lie on certain curves in the zero-dispersion limit.

Title: Nonlinear differential equations for Triangle groups Name: Sarbarish Chakravarty Affiliation: University of Colorado at Colorado Springs, USA Email: [email protected] Abstract: A class of third order nonlinear ordinary differential equations on the complex plane is studied. These equations are satisfied by automorphic functions defined on certain triangular fundamental domains in the extended upper half plane. The automorphism group is a subgroup of PSL(2, R), and is a Fuchsian group of the first kind whose fixed points are the vertices of the fundamental triangle. Examples of nonlinear differential equations invariant under the action of such triangle groups can be traced back to the work of Jacobi, Schwarz, Ramanujan, Chazy, Rankin, and many others. A remarkable feature shared by all such equations is the fact that they can be completely described by linear differential equations with 3 regular singular points (hypergeometric equations). This talk will present our work in progress toward a unified approach to derive such equations in a systematic fashion and to characterize them in terms of the underlying hypergeometric functions. This is joint work with Mark Ablowitz (CU Boulder).

Title: The σ-flows in the Novikov-Veselov Equation Name: Jen-Hsu Chang Affiliation: National Defense University, Taiwan Email: [email protected]

8 Abstract: The σ-flows in the Novikov-Veselov equation are used to describe a dynamical system on the n-th elementary symmetric product of roots of the related Gould-Hopper polynomials. We investigate the root dynamics of the related Gould-Hopper polynomials. One can solve the initial value problem of the root dynamics and the Lax equation is established. In some cases, they are the solutions of the Gold-fish Model, a limiting case of the Ruijesenaars-Schneider system. The asymptotic behavior of the root dynamics is also discussed.

Title: Bifurcation theory of the infinite-dimensional Kuramoto model Name: Hayato Chiba Kyushu University, Japan Email: [email protected] Abstract: The Kuramoto model is a system of ordinary differential equations for de- scribing synchronization phenomena. In this talk, a bifurcation structure of the infinite dimensional Kuramoto model is shown. For a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, the spectral theory based on a rigged Hilbert space is developed. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition con- sisting of a countable number of eigenfunctions on a space of generalized functions. The results are applied to the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto. It is proved that there exists a finite dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimensional.

Title: On Geometric Configurations Related to Matrix Factorizations. Name: Anton Dzhamay University of Northern Colorado, USA Email: [email protected] Abstract: An important example of a discrete Lax-Pair representation of an integrable system is a re-factorization transformation of its Lax matrix. We study rational Lax matrices represented as products of Blaschke-Potapov factors. Using residue techniques we show that the exchange rules for the components of such factors can be visualized by labeled atomic trivalent graphs, and that such triples can be glued together following certain strict rules. In particular, the configuration corresponding to the transposition of two such blocks is a cube and both the correct choice of the Lagrangian coordinates and the expression for the Lagrangian itself can be readily obtained from the labeling of the edges and the vertices of this cube.

Title: Hankel determinants, cluster algebras and integrable lattices Name: Michael Gekhtman

9 University of Notre Dame, USA Email: [email protected] Abstract. We use tools form the theory of Toda-type integrable lattices construct a clus- ter algebra structure in the space of rational functions. Joint work with M. Shapiro and A. Vainshtein

Title: Can the two-layer QG model explain the Nastrom-Gage energy spectrum of the atmosphere? Name: Eleftherios Gkioulekas Affiliation: University of Texas – Pan American, USA Email: [email protected] Abstract: An analysis of wind and temperature measurements taken during the Global Atmospheric Sampling Program by Nastrom and Gage showed that there is a robust k−3 energy spectrum extending from approximately 3,000 km to 1,000 km in wavelength and a robust k−5/3 energy spectrum extending from 600 km down to a few kilometers. Tung and Orlando have demonstrated numerically that the two-layer quasi-geostrophic model, forced at large scales by baroclinic instability, can reproduce this energy spectrum. How- ever, their result is considered, by some, to be controversial. More detailed models have been shown to reproduce the Nastrom-Gage energy spectrum as well. However, the ques- tion remains: why do any of these models work, and what is the simplest model that can account for the Nastrom-Gage spectrum? In this talk I will present the recent progress that has been made towards addressing these questions.

Title: Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations in Multiple Space Dimensions Name: Willy Hereman Affiliation: Department of Mathematical and Computer Sciences, Colorado School of Mines, USA Email: [email protected] Abstract: A method will be presented for the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs) involving multiple space variables and time. Using the scaling symmetries of the PDE, the conserved densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative is used to compute the undetermined coefficients. The homotopy operator is used to invert the divergence operator, leading to the analytic expression of the flux vector. The method is algorithmic and has been implemented in the syntax of the computer algebra system MATHEMATICA. The software is being used to compute conservation laws of nonlinear PDEs occuring in the applied sciences and engineering.

10 The software package will be demonstrated for PDEs that model shallow water waves, ion-acoustic waves in plasmas, sound waves in nonlinear media, and transonic gas flow. The featured equations include the Korteweg-de Vries and Boussinesq equations, the Navier and Kadomtsev-Petviashvili equations, and the Zakharov-Kuznetsov and Khoklov- Zabolotskaya equations.

Title: The Nonlinear Schr¨odingerequation in the small dispersion regime: applications Name: Mark A. Hoefer Affiliation: North Carolina State University, USA Email: [email protected] Abstract: Recent experimental and theoretical research in Bose-Einstein condensation and nonlinear optics have demonstrated novel supersonic, fluid-like phenomena. Super- sonic dynamics in these systems can be modeled by the Nonlinear Schr¨odingerequation in the small disperison regime. The Whitham averaging technique will be used to ap- proximately solve several fundamental problems of dispersive fluid dynamics leading to descriptions of nonlinear wave interactions in the small dispersion regime. Numerical simulations and connections to experiment will be discussed.

Title: Differential equations for modular forms on SL(2, Z)-conjugate subgroups of level three Name: Tim Huber Affiliation: University of Texas – Pan American, USA Email: [email protected] In 1916, S. Ramanujan showed that the Eisenstein series satisfy a coupled system of nonlinear differential equations. We extend Ramanujan’s elementary technique to derive analogous differential equations for modular forms on SL(2, Z)-conjugate subgroups of level three. These forms arise as series coefficients in relevant decompositions for the Weierstrass ℘-function at points on the period lattice of finite order. The series appearing in the differential systems were first studied by Ramanujan in his second notebook and are closely related to the cubic analogues of the Jacobi theta functions.

Title: Higher order corrections to the Miles theory of shallow water waves Name: Yuhan Jia Affiliation: Ohio State University, USA Email: [email protected] Abstract: We reconsider the Miles theory of solitary wave interactions in shallow water, and confirm that the leading order of the theory is indeed equivalent to the KP theory. We then consider the higher order corrections to the Miles theory. There were several

11 numerical simulations, in which all the authors concluded that the results indicated dis- agreements with the Miles theory. Contrary to those previous numerical studies, we found that their results are in good agreements with the theory when the higher order correc- tions are included.

Title: Ultradiscretization of solvable chaotic systems Name: Kenji Kajiwara Affiliation: Faculty of Mathematics, Kyushu University, Japan Email: [email protected] Abstract: We consider some solvable chaotic systems, which lie on the border of inte- grable and non-integrable systems. Applying the ultradiscretization, we establish a new relashionship between certain class of piecewise linear maps and rational maps, together with their general solutions. A typical example is the tent map and the Sch¨odermap. The general solutions of piecewise linear maps can be constructed in terms of the ultradiscrete (tropical) theta functions with the aid of the tropical geometry.

Title: Modular forms satisfying certain differential equation Name: Masanobu Kaneko Affiliation: Faculty of Mathematics, Kyushu University, Japan Email: [email protected] Abstract: We discuss several aspects of modular and quasimodular form solutions of a single, one-parameter differential equation in the complex upper-half plane. We also men- tion the arithmetic nature of this differential equation, which first emerged from our work with D. Zagier on supersingular j-invariants.

Title: A pseudo-spectral method with window technique for initial value problems of KP equation Name: Chiu-Yen Kao Affiliation: Ohio State University, USA Email: [email protected] Abstract: In this talk, we will present a numerical method to study the initial value problem of the KP equation with certain initial waves. The numerical approach is based on the peuso-spectral method with a window technique to take care of the non-periodic condition at the computational boundary. We show that for those initial waves, the so- lutions asymptotically converge to some of the exact solutions. The convergence is in a locally defined L2-sense that the main part of the interaction pattern of the solution agrees with that of the exact solution. The corresponding exact solution is then identified via a minimization process. Finally, based on the present numerical study, we will discuss

12 the stability of the exact soliton solutions of the KP equation.

Title: Methods to determine large deviations and rare events in optical pulses Name: William L. Kath and Jinglai Li Affiliation: Engineering Sciences and Applied Mathematics, Northwestern University Email: [email protected] Abstract: In optical fiber systems, amplified spontaneous emission noise causes signal fluctuations that lead to errors in those rare cases when the noise-induced changes are large. We discuss methods capable of determining large deviations induced by noise in such systems. First, large deviations are found using a constrained optimization problem that exploits the mathematical structure of the governing equations and a numerical im- plementation of the singular value decomposition. The results of the optimization problem then guide importance-sampled Monte-Carlo simulations to determine the events’ prob- abilities. We show that the method works for a general class of intensity-based optical detectors and for arbitrarily shaped pulses.

d Title: The Integrals of Motion for the deformed W-algebra Wq,t(glN ) Name: Takeo Kojima Affiliation: Yamagata University, Japan Email: [email protected] Abstract: We construct infinitely many commutative operators in terms of the deformed d W-algebra Wq,t(glN ), which give elliptic quantization of the integrals of motion of the KdV equation.

Title: Singular sectors of integrable hierarchies, degenerate critical points and Euler- Poisson-Darboux equations. Name: Boris Konopelchenko Affiliation: Department of Physics, University of Salento, Lecce, Italy. E-mail: [email protected] Abstract: Singular sectors (of gradient catastrophe) for the Burgers-Hopf, 1-layer Ben- ney system (classical long wave equation) and the dispersionless integrable coupled KdV hierarchies are discussed. Hodograph solutions of these hierarchies describe critical and degenerate critical points of scalar functions which obey the Euler-Poisson-Darboux equa- tions. Singular solutions for these hierarchies define a stratification of the affine space of flow parameters. This stratification is associated with the Birkhoff stratification of the Sato Grassmannian. Each stratum contains a family of hyperelliptic curves. Deforma- tions of these curves are described by the hierarchies of integrable hydrodynamical type systems. A connection with the classical theory of gradient catastrophe (Thom, Arnold

13 etc) is discussed.

Title: Integrable Random Optical Pulse Dynamics in a Resonant Optical Medium Name: Gregor Kovacic Affiliation: Rensselaer Polytechnic Institute, USA Email: [email protected] Abstract: In a resonant interaction, light of specific wavelengths excites electron transi- tions between atomic energy levels in an active optical medium such as gas or crystal. For instance, in the lambda-configuration, light interacts with a medium via a pair of elec- tron transitions between an energetically higher and two energetically lower atomic levels, which involve light of opposite circular polarizations or two different colors. We have iden- tified a switching mechanism in this interaction: The polarization of the light will switch so that it will interact with the medium only through the transition between the higher level and the lower level less populated with electrons. If the initial occupation of the two lower levels varies randomly, an optical pulse passing through this material will switch randomly between the two polarizations/colors. Mathematically, this phenomenon is de- scribed by exact solutions of a completely integrable random partial differential equation, thus combining the opposing concepts of integrability and disorder. Exact probability distributions of the parameters describing the light polarization will be presented and their properties discussed.

Title: Integrability of a Class of Nonlinear Ordinary Differential Equations (ODEs) of any Order through Nonlocal Transformations Name: M. Lakshmanan Affiliation: Centre for Nonlinear Dynamics, Bharathidasan University, India Email: [email protected] Abstract: Integrability of many nonlinear dynamical systems is intimately related to the existence of some kind of linearizing transformations. For example, even the inverse scattering transform of soliton equations can be interpreted as linearizing canonical trans- formations to action angle variables. Considering nonlinear ODEs of finite order either single or coupled, we point out that classes of linearizing transformations can be iden- tified which include point, contact, Sundman, generalized linearizing and their various combination and nonlocal transformations, when the systems are integrable. Specifically considering modified Emden type equations and its two coupled versions, we identify non- local transformations which connect them to linear equations via Bernoulli and Ricatti type equations. The procedure is extended to third order and nth order coupled ODEs. References [1] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Proc. R. Soc. London Series A 461, 2451 (2005);462, 1831 (2006);465, 585 (2009); 465, 609 (2009);465, 2369 (2009).

14 [2] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, J. Phys. A: Math.Gen. 39, L69 (2006). [3] R. Gladwin Pradeep, V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, J. Phys. A: Math.Gen. 42, 135206 (2009); submitted (2010).

Title: Givental symmetries of Frobenius manifolds and multi-component KP tau-functions

Name: Johan van de Leur Affiliation: University of Utrecht Email: [email protected] Abstract: We establish a link between two different constructions of the action of the twisted loop group on the space of Frobenius structures. The first construction (due to Givental) describes the action of the twisted loop group on the partition functions of formal (axiomatic) Gromov-Witten theories. The explicit formulas for the corresponding tangent action were computed by Y.-P. Lee. The second construction describes the action of the same group on the space of Frobenius structures via the multi-component KP hierarchies. Our main theorem states that the genus zero restriction of the Y.-P. Lee formulas coincides with the tangent multi-component KP action. Based on joint work with Evgeny Feigin and Sergey Shadrin.

Title: The solution of bigraded Toda hierchy Name: Chuanzhong Li Affiliation: Ohio State University, USA Email: [email protected] Abstract: In this paper, we firstly prove BTH (bigraded toda hierarchy) has a symme- try between (N,M) BTH and (M,N) BTH. Because BTH is equivalent to (N,M) band infinite-matrix formed Toda hierarchy(MBTH), so we consider its reduction, i.e. the semi- infinite and finite matrix form of MBTH. Then we give all the primary Hirota equations of BTH and the general solutions of MBTH using orthogonal polynomials in matrix form. We also give some rational solutions of BTH. After that, a geometric explanation using polytope will be given.

Title: Hamiltonian structures of zero curvature equations by variational identities Name: Wen-Xiu Ma Affiliation: University of South Florida, USA Email: [email protected] Abstract: We will discuss Hamiltonian structures of zero curvature equations, both con- tinuous and discrete. The basic tools are variational identities, including component-trace

15 identities, associated with matrix spectral problems. Illustrative examples are integrable couplings associated with non-semisimple Lie algebras. Applications of bi-trace identities will furnish Hamiltonian structures of dark equations, particularly the first-order pertur- bation equations.

Title: Traveling wave solutions of the generalized convection-reaction-diffusion equation Name: V. Vladimirov , Cz. Maczka Affiliation: AGH University of Science and Technology, Krakow, Poland Email: [email protected] Abstract: We consider a hyperbolic generalization of the convection-reaction-diffusion equation − n ≥ α utt + ut + µ u ux κ (u ux)x = f(u), n 1, (1) where f(u) is a polynomial function. Let us note, that the term α utt appears when the memory effects are taken into account. We show that Eqn. (1) possess a rich set of phys- ically meaningful traveling wave (TW) solutions: kinks, periodic, soliton-like solutions, shock fronts, peakons and compactons. Using the knowledge of the analytical form of some of the above mentioned solutions, we give a sufficient condition of their stability. Qualitative and analytical studies are backed by the numerical experiments, based on the Godunov algorithm. Numerical simulations confirm the results of the qualitative analysis. Besides, they reveal, that some of the TW solutions, for which the results of the qualitative studies do not apply, evolve in a stable self-similar mode.

Title: Constructing two-dimensional integrable mappings that possess invariants of high degree Name: Junta Matsukidaira Affiliation: Ryukoku University Email: [email protected] Abstract: We propose a method for constructing two-dimensional integrable mappings that possess invariants with degree higher than two. Such integrable mappings are ob- tained by making a composition of a QRT mapping and a mapping that preserves the invariant curve of the QRT mapping except for changing the integration constant involved. We show several concrete examples whose integration constants change with period 2 and 3.

Title: Recursion operators, conservation laws and integrability conditions for difference equations Name: Alexander V. Mikhailov Affiliation: University of Leeds, United Kingdom

16 Email: [email protected] Abstract: In this work we try to give a consistent background and definitions suitable for the theory of integrable difference equations. We adapt a concept of recursion opera- tor to difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. Similar to the case of partial differential equations these canonical densities can serve as integrability conditions for difference equations. We have found the recursion operators for the Viallet and all of the Adler–Bobenko–Suris equations.

Title: Skew orthogonal polynomials associated with Askey-Wilson polynomials Name: Hiroshi Miki Affiliation: Kyoto University, Japan Email: [email protected] Abstract: Skew orthogonal polynomials (SOPs) are a set of polynomials defined on an antisymmetric bilinear form called skew-symmetric inner product, which are introduced by Dyson in the context of random matrix theory. In this presentation, a skew-symmetric inner product with some symmetry is considered. Several properties of the corresponding SOPs are shown. The relation between the SOPs and the Askey-Wilson polynomials is also discussed.

Title: The Benjamin-Ono equation in the zero-dispersion limit Name: Peter Miller Affiliation: University of Michigan, USA Email: [email protected] Abstract: The Benjamin-Ono equation is a model for several physical phenomena, includ- ing gravity-driven internal waves in certain density-stratified fluids. It has the features of being a nonlocal equation (the dispersion term involves the Hilbert transform of the disturbance profile) and also of having a Lax pair and an associated inverse-scattering algorithm for the solution of the Cauchy initial-value problem. We will review known phenomena associated with this equation in the limit when the dispersive effects are nominally small, and compare with the better-known Korteweg-de Vries equation. Then we will present a new result (joint with Zhengjie Xu) establishing the zero-dispersion limit of the solution of the Benjamin-Ono Cauchy problem for certain initial data, in the topol- ogy of weak convergence. Our methodology is a novel analogue of the Lax-Levermore method in which the equilibrium measure is given more-or-less explicitly rather than via the solution of a variational problem. The proof relies on aspects of the method of mo- ments from probability theory.

Title: Integrable Algorithms: from Moser to I-SVD

17 Name: Yoshimasa Nakamura Affiliation: Kyoto University, Japan Email: [email protected] Abstract: From the pioneer works by Moser, Symes, and Papageorigou-Gramaticos- Ramani and so on an intimate relationship between integrable systems and known numer- ical algorithms has been known. One step of an algorithm is just a time evolution of an integrable system in some case. Recurrence relation of an algorithm can be regarded as a discrete time integrable system in other case. For example the discrete time Toda equa- tion for semi-infinite chain is the recurrence relation of Rutishauser’s quotient-difference (qd) algorithm for computing poles of meromorphic functions and solving a matrix eigen- value problem. However no one succeeded to design a new numerical algorithm whose recurrence relation is a discrete integrable system and is superior than the existing nu- merical algorithms. In this talk I will give a brief review of this area. An integrable discrete Lotka-Volterre (dLV) equation helps us to formulate a new algorithm named dLV for matrix singular values. By introducing shifts the dLV algorithm is developed to the modified dLV with shift (mdLVs) algorithm which has a better relative accuracy as well as speed. Singular vectors are also computed by using dLV type transformations. Then the I-SVD algorithm for matrix singular value decomposition is now complete.

Title: Hypergeometric solutions to the q-Painlev´eIV equation Name: Nobutaka Nakazono Affiliation: Graduate School of Mathematics, Kyushu University, Japan Email: [email protected] Abstract: I will show two hypergeometric solutions to a q-Painlev´eIV equation arising (1) from the birational representation of the affine Weyl group of type A4 . One is the hy- pergeometric solution of lattice type and the other is that of molecule type.

Title: Discrete sine-Gordon equations: hyperbolic and elliptic types Name: Yasuhiro Ohta Affiliation: Department of Mathematics, Kobe University, Japan Email: [email protected] Abstract: Discrete sine-Gordon equations of hyperbolic and elliptic types are studied by using direct method. In continuous case, the two types are transformed each other through independent variable transformation, while in discrete case, they are not simply related by variable transformations and have different structures. Soliton solutions for elliptic type are given in Pfaffian form.

Title: Hyperfast Numerical Codes for Integrable Soliton Equations in 2+1 dimensions using Riemann Theta Functions

18 Name: A.R. Osborne Affiliation: University of Torino, Italy Email: [email protected] Abstract: I show how to develop numerical codes for the space/time evolution of nonlin- ear, integrable soliton equations for periodic boundary conditions using Riemann theta functions. To accelerate the computation of the theta functions (1) I carry out a modular transformation of the Riemann spectrum, (2) sum the theta function on an N-ellipsoid and (3) reduce the theta function to an ordinary Fourier series with time varying coeffi- cients. The codes are perfectly parallelizable and so on a modern multi-core computer of N cores one can obtain a speed gain of a factor of N. Relative to standard spectral codes, an additional factor of about 1000 is found for a single core. I give as an example the KP equation where on an 8000 core AMD computer the speed increase relative to a standard spectral code on a single core is about 8,000,000. This reduces a 10,000 hour run to about 5 sec. On an 8 core MacIntel the speedup is about a factor of 8000, or about 75 min. A typical application is for redesigning the dikes in New Orleans, where Boussinesq codes are estimated to run for about 120 years of computer time. Details are given in the new book: Nonlinear Ocean Waves and the Inverse Scattering Transform, by A. R. Osborne (Academic Press, 977 pages, 2010).

Title: On differential equations with infinite symmetries and infinite conservation laws Name: V. Rosenhaus Affiliation: Department of Mathematics and Statistics, California State University, Chico, USA Email: [email protected] Abstract: We study second order variational PDE possessing an infinite symmetry alge- bra parametrized by an arbitrary function(s) of dependent variables and their derivatives. It was shown earlier, see V.Rosenhaus, Theor. Math. Phys. 160 (2009) 1042-1049, ibid. 151 (2007) 869-878, that these symmetries lead to an infinite number of (essential) conser- vation laws, unlike infinite symmetries with arbitrary functions of independent variables. We will discuss the derivation and classification of Lagrangian equations of the second order possessing an infinite set of conservation laws with an arbitrary function of the de- pendent variable and its first and second derivatives. We will show that the equations of this class include a subclass of equations of Liouville-type (Darboux-integrable systems).

Title: Generations of invariant varieties of periodic points from the singularity confine- ment Name: Satoru Saito Affiliation: Department of Physics, Tokyo Metropolitan University, Japan Email: [email protected] or saito [email protected]

19 Abstract: We have shown, in our previous works, that periodic points of higher dimen- sional integrable maps form a variety different for each period, if the maps have sufficient number of invariants. This is in contrast with nonintegrable cases where periodic points are isolated from each other. In this talk we would like to show that the series of invariant varieties of periodic points (IVPP) can be generated iteratively by the mapping recovered from the singularity confinement. Moreover we discuss that the generation of the IVPP’s is associated with a B¨acklund transformation, which can be naturally understood as a derived category.

Title: A behavior of Julia set of higher dimensional maps in the integrable limit Name: Noriko Saitoh Affiliation: Department of Applied Mathematics, Yokohama National University, Japan Email: [email protected] Abstract: We study analytically the behavior of the Julia set of nonintegrable maps where it approaches and disappears in the integrable limit.

Title: Tsunamis and other shallow water waves Name: Harvey Segur Affiliation: University of Colorado at Boulder, USA Email: [email protected] Abstract: The tsunami of December, 2004 killed more than 200,000 people along the shores of the Indian Ocean. But other underwater earthquakes since 2004 have been ap- proximately as strong as the 2004 quake, and yet generated no significant tsunami. Why not? This talk describes the dynamics of tsunamis, how they work, what makes them dangerous, and what mathematical models apply to them. If time permits, we can also discuss some other interesting (and sometimes dangerous) phenomena in shallow water, like rip currents.

Title: Fixed-point sets of non-maximal torus actions on flag manifolds, and the Toda lattice Name: Barbara Shipman Affiliation: The University of Texas at Arlington, USA Email: [email protected] Abstract: This talk will describe the structure of fixed-point sets of non-maximal torus actions on flag manifolds G/B, where G is a complex semisimple Lie group and B is a Borel subgroup, from the perspective of moment maps and length functions on Weyl groups. This approach is motivated by singular flows of the Toda lattice that generate an action of a group A on the flag manifold, where A is a product of a non-maximal torus

20 and a unipotent group.

Title: Fundamental Solutions Name: Erwin Suazo Affiliation: University of Puerto Rico, Mayaguez, USA Email: [email protected] Abstract: The Shroedinger equation (S.E) is one of the most important equations of math- ematical physics. It has been studied extensively. It plays an analogous role in Quantum mechanics as Newton Law does in classical mechanics. One of the useful characteristics for S.E is that for certain Hamiltonians the Cauchy initial value problem can be expressed as an integral form. In this poster we consider a S.E with quadratic Hamiltonian depending on time and using separation variables we can solve it explicitly. Particular cases include the free particle, simple harmonic oscillator and constant magnetic fields propagators.

Title: A class of higher order Painlev´esystems arising from integrable hierarchies of type A Name: Takao Suzuki Affiliation: Department of Mathematics, Kobe University, Japan Email: [email protected] Abstract: The connection between the second Painlev´eequation and the KdV equation was clarified by Ablowitz and Segur. Since their result, a relationship between (higher order) Painlev´esystems and infinite-dimensional integrable hierarchies has been studied. In a recent work, we investigate the Drinfeld-Sokolov hierarchies and derive a class of higher order Painlev´esystems by similarity reductions. In this talk, we present a new higher order Painlev´esystem, which is expressed as a 2n-th order Hamiltonian system with coupled sixth Painlev´eHamiltonian. Our new system has some important proper- (1) −1 ties, affine Weyl group symmetry of type A2n+1, Lax form associated with sl2n+2[z, z ] and particular solution in terms of the hypergeometric function n+1Fn.

Title: Parallel Numerical Methods for Solving Nonlinear Evolution Equations Name: Thiab R. Taha Affiliation: Department of Computer Science, University of Georgia, Athens, GA 30602 USA Email: [email protected] Abstract: Nonlinear evolution equations are of tremendous interest in both theory and applications. In this talk we introduce parallel algorithms for numerical simulations of CMKdV, NLS and and CNLS equations in 1+1 and 1+2 dimensions. The parallel meth- ods are implemented on multiprocessor system. Numerical experiments have shown that

21 these methods give accurate results and considerable speedup.

Title: On some ultradiscrete systems Name: Daisuke Takahashi Affiliation: Faculty of Science and Engineering, Waseda University, Japan Email: [email protected] Abstract: Some comtinuous systems are directly related to particle systems through ultra- discretization. For example, we obtain the elementary cellular automaton of rule 184, the box and ball system and the ultradiscrete Toda equation by ultradiscretizing the Burgers equation, the KdV equation and the Toda equation respectively. In my talk, I will report some particle systems of cellular automaton type. The space and time variables are both discrete and the dependent variable is binary in the systems. Their common features are the conservation of the number of particles and the direct relation to the continuous systems, for example, a system of ordinary differential equations or a partial differential equation. Moreover, behavior of solutions, mathematical features and physical aspect of each system will also be discussed.

Title: hbar-expansion of KP hierarchy: Recursive construction of solutions Name: Takashi Takebe Affiliation: Faculty of Mathematics, Higher School of Economics, Moscow, Russia Email: tktk [email protected], [email protected] Abstract: The ~-dependent KP hierarchy is a formulation of the KP hierarchy that de- pends on the Planck constant ~ and reduces to the dispersionless KP hierarchy as ~ → 0. A recursive construction of its solutions on the basis of a Riemann-Hilbert problem for the pair (L, M) of Lax and Orlov-Schulman operators is presented. The Riemann-Hilbert problem is converted to a set of recursion relations for the coefficients Xn of an ~-expansion 2 of the operator X = X0 + ~X1 + ~ X2 + ... for which the dressing operator W is expressed in the exponential form W = exp(X/~). Given the lowest order term X0, one can solve the recursion relations to obtain the higher order terms. The wave function Ψ associated with W turns out to have the WKB form Ψ = exp(S/~), and the coefficients Sn of the 2 ~-expansion S = S0 + ~S1 + ~ S2 + ..., too, are determined by a set of recursion relations. This WKB form is used to show that the associated tau function has an ~-expansion of −2 2 the form log τ = ~ (F0 + ~F1 + ~ F2 + ...).

Title: Axially symmetric soliton solutions in a extended Skyrme-Faddeev model Name: Kouichi TODA Affiliation: Department of Mathematical Physics, Toyama Prefectural University, Japan Email: [email protected]

22 Abstract: In this talk, we construct static soliton solutions with non-zero Hopf topolog- ical charges, for a field theory that has found interesting applications in many areas of Physics. It is a (3 + 1)-dimensional Lorentz invariant field theory for a triplet of scalar fields ~n, living on the two-sphere S2, ~n2 = 1, and defined by the Lagrangian density[1]:

1 β L = M 2∂ ~n · ∂µ~n − (∂ ~n ∧ ∂ ~n)2 + (∂ ~n · ∂µ~n)2 , (2) µ e2 µ ν 2 µ where the coupling constants e2 and β are dimensionless, and M has dimension of mass. The first two terms correspond to the so- called Skyrme-Faddeev model, as the gener- alization to 3 + 1 dimensions of the CP 1 model in 2 + 1 dimensions. In a Minkowski space-time the static Hamiltonian associated to (2) is

1 β H = M 2∂ ~n · ∂ ~n + (∂ ~n ∧ ∂ ~n)2 − (∂ ~n · ∂ ~n)2 , (3) static i i e2 i j 2 i i with i, j = 1, 2, 3. Therefore, it is positive definite for M 2 > 0, e2 > 0 and β < 0. By an axially symmetric ansatz based on toroidal coordinates, we construct numerical solutions with Hopf charge up to four, and calculate their analytical behavior in some limiting cases. The solutions present an interesting behavior under the changes of a special combination of the coupling constants of the quartic terms. Their energies and sizes tend to zero as that combination approaches a particular special value. In addition, the model presents an integrable sector with an infinite number of local conserved currents which apparently are not related to symmetries of the action. In the intersection of those two special sectors the theory possesses exact vortex solutions (static and time dependent) which were constructed by one of the authors[2]. It is believed that such model describes some aspects of the low energy limit of the pure SU(2) Yang-Mills theory, and our results may be important in identifying important structures in that strong coupling regime. This is a collaboration work with L. A. Ferreira and Nobuyuki Sawado.

Title: Two-dimensional Interaction of the solitary waves in two-layer fluid with large depth Name: Hidekazu Tsuji Affiliation: Research Institute for Applied Mechanics, Kyushu University, Japan Email: [email protected] Abstract: Propagation and interaction of solitary waves in stratified fluid have been stud- ied by many researchers. Unlike surface wave, the character of the internal wave varies with the depth of the fluid. The author have investigated two-dimensional interactions of the one-dimensional solitary waves by solving corresponding model equations — two- dimensional Benjamin-Ono (2dBO) equation for infinitely deep case and two-dimensional intermediate long wave (2dILW) equation for intermediate depth. These analysis were limited to the symmetrical interaction with respect to the propagation direction. In this study the characters of the asymmetrical interaction are discussed. (Joint work with Prof. Oikawa)

23 Title: Orthogonal polynomials and Toda lattice hierarchy Name: Satoshi Tsujimoto Affiliation: Kyoto University, Japan Email: [email protected] Abstract: We give some reviews on the theory of the orthogonal polynomials from the point of view of the theory of the Toda lattice hierarchy. Then it is planned to discuss the role and importance of the Darboux transformations which lead us to the classical orthogonal polynomial solutions of the discrete Toda lattice.

Title: UC hierarchy, monodromy preserving deformations and hypergeometric functions Name: Teruhisa Tsuda Affiliation: Kyushu University, Japan Email: [email protected] Abstract: The UC hierarchy is an extension of the KP hierarchy, which possesses not only an infinite set of positive time evolutions but also that of negative ones. Through a sim- ilarity reduction, the UC hierarchy yields a broad class of Schlesinger systems including (higher order) Painlev´eVI and Garnier systems, which describe monodromy preserv- ing deformations of Fuchsian ODE with certain spectral types. We also show several properties of the above Schlesinger systems: polynomial Hamiltonian structure, algebraic solutions, hypergeometric solutions, Weyl group symmetry, etc.

Title: Experiments on shallow water waves Name: Shinsuke Watanabe Affiliation: Open University of Japan, Japan Email: [email protected] Abstract: One dimensional and two dimensional propagations of a shallow water wave have been investigated experimentally. In the one dimensional propagation, the maxi- mum amplitude for a stable wave and the instability of the wave are examined. In the two dimensional case, the interaction of water waves are investigated.

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