List of abstracts submitted to the 2nd NMMP workshop†
9 March - 11 March, 2013
University of South Florida, Tampa, Florida, United States of America
Workshop Sponsors:
The
The
The
The
The
National Science Foundation, USA
Office of Research & Innovation, University of South Florida, USA
College of Arts and Sciences, University of South Florida, USA
Department of Mathematics and Statistics, University of South Florida, USA
Department of Mathematics, University of Central Florida, USA
The Organizing Committee:
Members from University of South Florida:
Dr. Wen-Xiu Ma (Chairman), Email: mawx@cas.usf.edu
Dr. Sherwin Kouchekian, Email: skouchekian@usf.edu
Dr. Razvan Teodorescu, Email: razvan@usf.edu
Members from University of Central Florida:
Dr. S. Roy Choudhury (Co-Chairman), Email: roy.choudhury141@gmail.com
Dr. David J. Kaup, Email: kaup@mail.ucf.edu
Dr. Constance Schober, Email: Constance.Schober@ucf.edu
†
The speakers are flagged with an asterisk (∗ ) in this list.
1
1
Tutorials
Isochronous systems are not rare
Francesco Calogero∗ , University of Rome “La Sapienza”, Italy
[francesco.calogero@roma1.infn.it]
Abstract: A survey will be given of isochronous systems, i. e. systems that oscillate with a fixed
period (for largely arbitrary initial data). It will be shown how to manufacture many such models—
including ”realistic” many-body problems whose time evolution is characterized by Newtonian equations
of motion. In particular a fairly general technique will be described to modify fairly general models
describing a time evolution so that the modified systems are isochronous (with period T ) yet mimic
closely (or even exactly) the behavior of the unmodified system for a time interval T tilde much smaller
(or just smaller) than T .
As a particularly remarkable example (joint work with F. Leyvraz), it will be shown how—given
the (autonomous) Hamiltonian H describing the most general (standard) nonrelativistic many-body
problem (arbitrary number N of particles, arbitrary masses, arbitrary dimensions of ambient space, forces
depending arbitrarily from all the particle coordinates–with the only restriction that the system be overall
translation-invariant, i. e. no external forces)—it is possible to construct another (also autonomous)
Hamiltonian Htilde (in fact, an infinity of such Hamiltonians) featuring the same dynamical variables
and parameters as H and in addition two arbitrary positive parameters T and T tilde with T > T tilde,
and having the following two properties. (i) The new Hamiltonian Htilde yields, over the (arbitrarily
long!) time interval T tilde, a dynamical evolution identical to that yielded by H. (ii) The Hamiltonian
Htilde is isochronous: all its solutions (for arbitrary initial data) are completely periodic with period T .
This finding raises (interesting?) questions about the difference among nonintegrable and integrable
dynamics (all isochronous systems are integrable, indeed more than superintegrable), about the definition
of chaotic behavior (including the apparent need to invent some such notion for a finite time interval),
about the validity (say, for N ≈ 1023 ) of statistical mechanics and of the second principle of thermodynamics, about cosmology (say, for N ≈ 1085 ). It also demonstrates the impossibility to ascertain which
dynamical theory is the correct one, out of an infinity of different theories predicting the same (exactly
the same) evolution over an arbitrarily long time interval, but being qualitatively different (isochronous
versus chaotic, integrable versus nonintegrable).
Main references:
F. Calogero, Isochronous systems, OUP, Oxford, 2008 (paperback, 2012).
F. Calogero and F. Leyvraz, “How to extend any dynamical system so that it becomes isochronous,
asymptotically isochronous or multi-periodic”, J. Nonlinear Math. Phys. 16, 311-338 (2009); “Isochronous
systems, the arrow of time and the definition of deterministic chaos”, Lett. Math. Phys. 96, 37-52 (2011).
2
Introduction to the Painlevé property, test, and analysis
Robert Conte∗ , ENS Cachan et CEA-DAM, France [Robert.Conte@cea.fr]
Micheline Musette, Dienst TENA, Vrije Universiteit Brussel, Belgium
Abstract: Starting from the natural problem “To integrate explicitly a given nonlinear differential
equation”, we first logically arrive at the correct definition of the Painlevé property (PP): singlevalued
dependence of the general solution on the initial conditions. Using examples from Chazy, we explain why
some later definitions of the PP (such as “all solutions can only have poles as movable singularities”)
are incorrect. We insist on the impossibility for the Painlevé test to prove the PP, since the generated
conditions are only necessary: passing the Painlevé test does not imply the PP (example of Picard 1893).
In order to prove the PP, one must use the many resources of Painlevé analysis, a set of methods able to
yield a global information (general solution, particular solutions, first integrals) only by the local study
of the movable singularities. Singular (envelope) solutions must be discarded (Chazy 1910). Review of
the method of pole-like expansions (Kowalevski 1889, Gambier 1910, rediscovered by Ablowtiz-RamaniSegur 1978). Identity of Fuchs indices, “Painlevé resonances” and Kowalevski exponents. Two warnings
on the diophantine conditions that all Fuchs indices should be integer: (i) they must be solved for all
families simultaneously; (ii) negative integers must not be discarded. Proof of convergence of the Laurent
series (Chazy 1910, Bureau 1964). Algorithm to compute a very large number of terms. Indecisive cases
of the Kowalevski-Gambier method: absence of dominant behaviour, negative integer Fuchs indices,
insufficient number of Fuchs indices. The previously listed indecisive cases are handled by, respectively,
the α-method (Painlevé 1900), the Fuchsian perturbative method (RC, Fordy, Pickering 1991, 1993),
the nonFuchsian perturbative method (Musette, RC 1995). Negative integer Fuchs indices do not imply
an essential singularity (ex. u′′ + 3uu′ + u3 = 0). Illustrative examples include: u′′ + 4uu′ + 2u3 = 0
2
(α-method, Fuchsian perturbation), u′′′′ + 3uu′′ − 4u′ = 0 (Fuchsian and nonFuchsian perturbations).
Time permitting, application to the determination of all singlevalued solutions of ( i) the six-dimensional
Bianchi IX dynamical system; (ii) a multi-ion electrodiffusion system.
The basics of the inverse scattering transform
David J. Kaup∗ , University of Central Florida, USA [Kaup@ucf.edu]
Abstract: This tutorial will devote itself to discussing the various features of the IST, including
such topics as the Lax Pair, Jost solutions, analytical properties, the Riemann-Hilbert problem, the
inverse scattering equations, soliton solutions, completeness and closures of eigenstates and their adjoints,
perturbations, squared eigenfuctions and their adjoints, and known and new aspects of 3x3 Lax pairs.
Holomorhic PDE and potential theory
Dmitry Khavinson∗ , University of South Florida, USA [dkhavins@usf.edu]
Abstract: We shall try to outline several basic principles involved in studying the behavior of
solutions of linear holomorphic PDEs . The main emphasis will be to illustrate how these ideas can be
applied to examine the dynamics of a large class of nonlinear physical growth processes with moving
boundaries, e.g., oil spills or tumor growths, known as Hele-Shaw processes, or the Laplacian Growth
processes.
3
Ameobas and Ronkin functions of algebraic curves with points
Igor Krichever∗ , Columbia University, USA [krichev@math.columbia.edu]
Abstract: Recently concepts of the amoebas, the Ronkin function and the associated Monge-Ampèr
measure of plane algebraic curves have attracted additional interest and have become a major tool in
the studies of topological types of real algebraic curves and in the study of random surfaces which arise
as height functions of dimer configurations. In the talk, the notions of ameobas and Ronkin functions
would be extended to the case of general (non plane) algebraic curves with punctures.
Constant Gaussian curvature and differential equations
Keti Tenenblat∗ , University of Brasilia, Brazil [K.Tenenblat@mat.unb.br]
Abstract: In this tutorial, I will consider some aspects of the relation between metrics on a 2dimensional manifold with non zero constant Gaussian curvature and differential equations. The notion
of a differential equation which describes a metric of constant curvature was introduced in my joint work
with S. S. Chern. This seminal paper generated a series of articles where a classification of certain types
of differential equations and differential systems was obtained. I will provide a survey on these papers,
including recent results that provide huge classes of differential equations with their respective linear
problems.
4
2
Invited and contributed talks
Explicit solutions to integrable equations
Tuncay Aktosun∗ , University of Texas at Arlington, USA [aktosun@uta.edu]
Abstract: A method is presented to construct exact solutions to certain integrable nonlinear evo-
lution equations that are solvable by the inverse scattering transform method involving a Marchenko
integral equation. An explicit formula and its equivalents are obtained for each integrable equation to
express such exact solutions in a compact form in terms of a triplet of constant matrices and matrix
exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of
exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates.
Geometric curve flows and integrable systems
Stephen Anco∗ , Brock University, Canada [sanco@brocku.ca]
Abstract: The modern theory of integrable soliton equations displays many deep links to
differential geometry, particularly in the study of geometric curve flows by moving-frame methods. In
this talk, I will first review an elegant geometrical derivation of the integrability structure for two
important examples of soliton equations: (1) the nonlinear Schrödinger (NLS) equation; and (2) the
modified Korteweg-de Vries (mKdV) equation. This derivation is based on a moving-frame formulation
of geometric curve flows which model vortex filaments in Euclidean space and vortex-patch boundaries
in the Euclidean plane, arising in ideal fluid flow. Key mathematical tools here are the Cartan structure
equations of Frenet framesand the Hasimoto transformation relating invariants of a curve to soliton
variables, as well as the theory of Poisson brackets for Hamiltonian PDEs. I will then describe a broad
generalization of these results to geometric curve flows in semi-simple Klein geometries M = G/H, giving
a geometrical derivation of group-invariant (multi-component) versions of mKdV, NLS, and sine-Gordon
soliton equations along with their full integrability structure (hierarchies of symmetries and conservation
laws, a bi-Hamiltonian structure and Poisson brackets, a Lax pair and zero-curvature representation, a
linear isospectral eigenfunction problem and inverse scattering transform, etc.).
On the combinatorics of Wronskian-integrable equations
Shabnam Beheshti∗ , Rutgers University, USA [beheshti@math.rutgers.edu]
Amanda Redlich, Massachusetts Institute of Technology, USA
Abstract: Recent works of Kodama and Williams have connected soliton solutions of the wellknown Kadomtsev-Petviashvili (KP) Equation to Grassmann necklaces and more generally to cluster
algebras. We investigate this relationship in the broader context of Wronskian-integrable PDE, using
shallow-water wave equations as guiding examples.
5
Conservation laws for a KuramotoSivashinsky equation
with dispersive effects
Maria S. Bruzon∗ , Maria Luz Gandarias, University of Cadiz, Spain [m.bruzon@uca.es]
Abstract: Ibragimov proved a general theorem on conservation laws (2007. J. Math. Anal. Appl.
333, 31128). In order to apply this theorem is necessary to know the symmetries of the equation. We
obtain the classical symmetries of a KuramotoSivashinsky equation with dispersive effects. We determine
conservation laws for this equation.
Lax operators of the exceptional Lie algebras
Paolo Casati∗ , Università di Milano-Bicocca, Italy [paolo.casati@unimib.it]
Abstract: We show how the Lax operator of the exceptional Lie algebras can be directly
obtained as operators which leave invariant particular multilinear forms.
A macroscopic system with undamped periodic compressional oscillations
Francesco Calogero∗ , University of Rome “La Sapienza”, Italy
[francesco.calogero@roma1.infn.it]
Francois Leyvraz, Universidad Nacional Autónoma de México, Mexico
Abstract: A class of macroscopic systems is described which have the remarkable feature that
they can sustain undamped compressional radial oscillations. They consist of an arbitrary number of particles confined by a harmonic potential and interacting among themselves through conservative repulsive
forces scaling as the inverse cube of distances (but being otherwise arbitrary). The radial oscillation
leads to a variation of the thermodynamic quantities characterizing the system. The system therefore
does not approach thermodynamic equilibrium, since the (macroscopic) amplitude of the oscillation does
not decrease as time goes to infinity. The oscillation is harmonic and isochronous, that is, its frequency
is fixed and independent of the initial condition. These results hold independently of the dimension of
the system and are also valid in the quantal context.
This is joint work with F. Leyvraz, to appear in J. Stat. Phys.
Geometry and symmetry of the non-commutative Hirota system
Adam Doliwa∗ , University of Warmia and Mazury, Poland [doliwa@matman.uwm.edu.pl]
Abstract: We present an incidence geometric description of the Hirota system that is valid in
projective geometries over division rings. The four dimensional consistency of the corresponding lattice
maps is equivalent to the celebrated Desargues theorem. From very beginning such a description allows
to uncover the underlying symmetry (the affine Weyl group of type A) of the non-commutative Hirota
system. We will present also basic reductions to lower dimensional systems, which in the commutative
case give some well known integrable 1+1 dimensional equations. In particular we discuss the multidimensional consistency of the corresponding non-commutative systems. We will present also an important
specification of the theory to the case of division ring of quantum rational functions which leads to the
bi-algebra structure of the quantum plane.
6
The symmetry group of Lamé’s system
João Paulo dos Santos∗ , Keti Tenenblat, Universidade de Brassı́lia, Brazil [j.p.santos@mat.unb.br]
Abstract: Solutions of Lamé’s system are related to triply orthogonal system of surfaces and
metrics of conformally flat hypersurfaces in 4-dimensional space forms, when they satisfy the Guichard’s
condition. We show that the symmetry group of Lamé’s system satisfying Guichard’s condition is given
by translations and dilations in the independent variables and dilations in the dependents variables. We
obtain the solutions which are invariant under the action of subgroups of the symmetry group. Moreover,
we prove that there is a type of invariant solution of the Lam e’s system, given in terms of Jacobi elliptic
functions, that corresponds to a new class of conformally flat hypersurfaces. This is a joint work with
Keti Tenenblat.
Discrete Schlesinger transformations and difference Painlevé equations
Anton Dzhamay∗ , University of Northern Colorado/Columbia University, USA [adzham@unco.edu]
Hidetaka Sakai, University of Tokyo, Japan
Tomoyuki Takenawa, Tokyo University of Marine Science and Technology, Japan
Abstract: We study a discrete version of isomonodromic deformations of Fuchsian systems,
called Schlesinger transformations, and their reductions to discrete Painlevé equations. We obtain an
explicit formula for the generating function of elementary Schlesinger transformations in terms of the
coordinates on the so-called decomposition space associated to the Fuchsian system and interpret it as a
discrete Hamiltonian of our dynamic. Using this function we consider two explicit examples of reductions
(1)∗
(1)
of such transformations to discrete Painlevé equations of types d − P (D4 ) and d − P (A2 ). We then
use the birational geometry of rational surfaces associated to these equations to compare the form of
the equations that correspond to the elementary Schlesinger transformations to standard form of the
equations of the same type.
The concept of quasi-integrability
Luiz Agostinho Ferreira∗ , University of Sao Paulo, Brazil [laf@ifsc.usp.br]
Wojtek J. Zakrzewski, University of Durham, England
Gabriel Luchini, University of Sao Paulo, Brazil
Abstract: The concept of quasi-integrability has been proposed recently in the context of
deformations of integrable field theories in 1+1 dimensions. These are theories that possess an infinite
number of charges with intriguing properties. In the scattering of two soliton-like solutions these charges
are asymptotically conserved, i.e. even though they vary considerably during the scattering process their
values in the far past and the far future are the same. In this talk we will discuss the recently performed
application of these ideas to deformations of the sine-Gordon and Nonlinear Schrodinger equations.
7
Nonlinear self-adjointness and conservation laws for
a porous medium equation
Maria Luz Gandarias∗ , Maria S. Bruzon, University of Cadiz, Spain [marialuz.gandarias@uca.es]
Abstract: The concepts of self-adjoint and quasi self-adjoint equations were generalized in
[4] and [9] and the definition of weak self-adjoint equations and nonlinear self-adjoint equations were
introduced. In this paper we find the subclasses of nonlinear self-adjoint porous medium equations. We
show, that the theorem in conservation laws proved in [7] can be applied to construct conservation laws
associated with nonclassical generators. By using the property of nonlinear self-adjointness we construct
some conservation laws associated with some classical and nonclassical generators of the porous medium
equation.
References
[1] Bruzon M S, Gandarias M L and Ibragimov N H 2009 Self-adjoint sub-classes of generalized thin
film equations J. Math. Anal. Appl. 357 307-313.
[2] Gandarias M L 1996 Nonclassical symmetries of a porous medium equation with absorption J. Phys.
A: Math. Gen. 30 608191.
[3] Gandarias M L, Romero J L, and Diaz J M 1999 Nonclassical symmetries of a porous medium
equation with Phys. A: Math. Gen. 32 14611473.
[4] Gandarias M L 2011 Weak self-adjoint differential equations J. Phys. A: Math. Theor. 44 262001.
[5] Ibragimov N H 2006 Integrating factors, adjoint equations and Lagrangians J. Math. Anal. Appl.
318 742-57.
[6] Ibragimov N H 2006 The answer to the question put to me by LV Ovsiannikov 33 years ago Arch.
ALGA 3 53-80.
[7] Ibragimov N H 2007 A new conservation theorem J. Math. Anal. Appl. 333 311-28.
[8] Ibragimov N H 2007 Quasi-self-adjoint differential equations Arch. ALGA 4 55-60.
[9] Ibragimov N H 2011 Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor. 44
432002.
Hirota bilinear equations with linear subspaces of hyperbolic and
trigonometric function solutions
Xiang Gu∗ , University of South Florida, USA [xianggu@mail.usf.edu]
Hongchan Zheng, Northwestern Polytechnical University, P. R. China
Wen-Xiu Ma, University of South Florida, USA
Abstract: Linear superposition principles of hyperbolic and trigonometric function solutions
are analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of N-soliton
solutions formulated by linear combinations of hyperbolic and trigonometric functions. An algorithm
using weights is discussed and a few illustrative application examples are presented.
8
Exact solutions of bidirectional wave equations
Masood Khalique∗ , North-West University, South Africa [Masood.Khalique@nwu.ac.za]
Abstract: The surface water waves in a water tunnel can be described by systems of the form
vt + ux + (uv)x + αuxxx − βvxxt = 0,
ut + vx + uux + γvxxx − δuxxt = 0,
where α, β, γ and δ are real-valued constants [1]. Here x represents the distance along the channel, t is the
elapsed time, the variable v(x, t) is the dimensionless deviation of the water surface from its undisturbed
position and u(x, t) is the dimensionless horizontal velocity. In this talk we present some exact solutions
using the Lie symmetry method along with the simplest equation method, (G′ /G) expansion method
and the Jacobi elliptic function method of the underlying system.
References
[1] J. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves,
Physica D, 116 (1998), 191–224.
Discrete soliton equation hierarchy
Takao Koikawa∗ , Otsuma Women’s University, Japan [koikawa@otsuma.ac.jp]
Abstract: It is well known that the continuous soliton equations such as the KdV equation
have hierarchy structures. For example, higher order KdV equations follow the KdV equation. As for the
disrete soliton equations such as the Toda lattice equation, the hierarchy structure is not well studied.
When the hierarchy exists, the higher order differences might play the role of the higher order derivatives.
We study the Toda lattice equation and try to find the hierarchy structure.
On the nature of large and rogue waves
Mikhail Kovalyov∗ , Sungkyunkwan University, South Korea [mkovalyo@ualberta.ca]
Abstract: In the talk I discuss modeling of large and rogue waves using the KP equation and
compare the predictions of the model with actual observations.
Integrability of differential equations: Double tangent extension method
Lanouar Lazrag∗ , ENS at LYON, France [lanouar.lazrag@gmail.com]
Abstract: We consider systems of ordinary differential equations and present a new method of
finding first integrals called Double tangent extension method. We apply it to study the integrability by
quadratures of differential systems with homogeneous polynomials right hand sides. Some new cases of
integrability with two, three and four degrees of freedom are found.
9
Chaos: a bridge between micro-level inherent physical
uncertainty and macroscopic randomness
Shijun Liao∗ , Shanghai Jiao Tong University, P. R. China [sjliao@sjtu.edu.cn]
Abstract: In this talk, a numerical method based on multiple-precision (MP) data and Taylor
series method (TSM), namely the Clean Numerical Simulation (CNS), is proposed to gain reliable longterm prediction of chaos. By means of the CNS, the numerical noises can be reduced to be so small that
even the micro-level inherent physical uncertainty can be investigated accurately. It is found that, due
to the sensitive dependence on initial conditions (SDIC), the micro-level inherent physical uncertainty
might transfer into the macroscopic randomness. Thus, chaos might be a bridge between the micro-level
uncertainty and macroscopic randomness.
On peaked solitary waves of KdV, BBM and Boussinesq equation
Shijun Liao∗ , Shanghai Jiao Tong University, P. R. China [sjliao@sjtu.edu.cn]
Abstract: In this talk, we reported the closed form solution of peaked solitary waves of the
KdV, BBM and Boussinesq equation. All of them satisfy the the corresponding Rankine-Hogoniot jump
condition. Therefore, the peaked solitary waves might be common for most of shallow water wave models,
no matter whether or not they are integrable and/or admit breaking-wave solutions.
A free boundary problem for Laplace’s equation
Erik Lundberg∗ , Purdue University, USA [elundber@math.purdue.edu]
Dmitry Khavinson, Razvan Teodorescu, University of South Florida, USA
Alexandre Eremenko, Purdue University, USA
Abstract: In this talk, I will discuss recent progress on a problem posed by L. Hauswirth, F.
Hélein, and F. Pacard to characterize all the domains in the plane that admit a ”roof function”, i.e.,
a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data,
and a Neumann problem with constant boundary data. As they suggested, we show, under some a priori
assumptions, that there are only three exceptional domains: the exterior of a disk, a halfplane, and a
nontrivial example. I will also explain connections to quadrature domains and fluid dynamics.
Integrable generalization of the associated Camassa-Holm equation
Lin Luo∗ , Shanghai Second Polytechnic University, P. R. China [luolinmath@yahoo.cn]
Abstract: In this paper, we study a new integrable generalization of the associated CamassaHolm equation. This equation is shown to be completely integrable with Lax pair and bi-Hamiltonian
structure. The bilinear Bäcklund transformations are presented through the Bell polynomial technique.
Meanwhile, its infinite conservation laws are constructed, and conserved densities and fluxes are given
with explicit recursion formulas. Furthermore, A Darboux transformation for this equation is derived
with the help of the gauge transformation between Lax pair. As an application, soliton wave and periodic
wave solutions are given through the Darboux transformation.
10
Integrable couplings and matrix loop algebras
Wen-Xiu Ma∗ , University of South Florida, USA [mawx@cas.usf.edu]
Abstract: We will discuss integrable couplings and their Hamiltonian structures through zero
curvature equations. Matrix loop algebras and variational identities are basic tools. Illustrative examples
include dark equations and bi- and tri-integrable couplings of integrable equations associated with sl(2,R)
and so(3,R).
Soliton solutions of coupled systems by improved-expansion method
Syed Mohyud-Din∗ , HITEC University, Pakistan [syedtauseefs@hotmail.com]
Muhammad Shakeel, COMSATS Institute of Information Technology, Pakistan
Abstract: The paper witnesses the extension of improvedexpansion method to generate travelling wave solutions of coupled systems. The proposed algorithm is extremely effective and is tested on
two very important systems (namely coupled Higgs and Maccari equations) in mathematical physics.
Numerical results reflect complete compatibility of suggested scheme.
Structure-preserving methods for damped Hamiltonian PDEs
Brian Moore∗ , Laura Norena, Constance Schober, University of Central Florida, USA
[brian.moore@ucf.edu]
Abstract: A general framework for constructing numerical methods that exactly preserve dissipative properties of damped Hamiltonian PDEs is presented in detail. These methods are compared
analytically and numerically to standard conservative methods, which generally destroy the actual dissipation rates but do retain other advantages in the dissipative context. Semi-linear wave equations and
nonlinear Schrodinger equations, both with added dissipation, are used as examples to demonstrate the
long-time behavior of the numerical solutions.
Integrability in non-perturbative QFT
Alexei Morozov∗ , Institute of Theoretical and Experimental Physics, Russia, [morozov@itep.ru]
Abstract: Exact non-perturbative partition functions of coupling constants and extrenal fields
exhibit huge hidden symmetry, reflecting the possibility to change integration variables in the functional
integral. In many examples this is reflected in non-linear relations between correlation functions, typcial
for the tau-functions of integrable systems, To many old examples, from matrix models to Seiberg-Witten
theory and AdS/CFT correspondence now adds the Chern-Simons theory of knot invariants. Some knot
polynomials are already shown to combine into tau-functions, the search for entire set of relations is still
in process.
11
Connections between HOMFLY polynomials and tau-functions
Andrey Morozov∗ , Institute of Theoretical and Experimental Physics
and Moscow State University, Russia, [andrey.morozov@itep.ru]
Abstract: HOMFLY polynomials of the knot theory can be expressed as a character expansion
- sum of Schur functions with some coefficients. In principle since we study the topological theory these
Schur functions should be taken in some particular values of variables. But the answer can be generalized
to an arbitrary variables. These generalized HOMFLY polynomials for torus knots possess properties of
KP/Toda tau-functions.
Localization in linear and nonlinear complex media
Ziad Musslimani∗ , Florida State University, USA [musliman@math.fsu.edu]
Abstract: When waves propagate in a linear and perfectly periodic lattice, the wave functions
in these structures are the well-known Bloch functions, which are extended all over the lattice. However,
if the lattice is not perfectly periodic (say due to the presence of disorder or random impurities) it is
known that any random potential of arbitrary strength (for low-dimensional systems) causes the wave
function to localize in space. This phenomenon is known in the literature as Anderson localization
(for this seminal result, Anderson was awarded the physics Nobel prize in 1977). The mathematical
model that describes Anderson localization is the random linear Schrödinger equation. Experimental
observation of Anderson localization in atomic lattices is a formidable task. However, a random photonic
microstructure superimposed on top of a periodic photonic lattice provide an excellent experimental
setting to observe Anderson localization. In this talk we shall discuss linear and nonlinear transport
properties of waves propagation in periodic and random photonic lattices. Some of the topics include
Anderson localization, formation of two-dimensional coherent structures and lattice vortices. We shall
highlight universal behavior and discuss future directions.
On the inverse scattering transform for the defocusing nonlinear Schrödinger
equation with non-vanishing boundary conditions
Barbara Prinari∗ , University of Colorado at Colorado Springs, USA [bprinari@uccs.edu]
Federica Vitale, Università del Salento, Italy
Francesco Demontis, Cornelis van der Mee, Università di Cagliari, Italy
Abstract: This talk will report on a recent work aimed at developing a rigorous theory of the
inverse scattering transform for the defocusing nonlinear Schrödinger equation: iqt = qxx − 2|q|2 , q with
nonvanishing boundary conditions q(x, t) ∼ q± as x → ±∞ (same amplitude |q|2 ≡ q0 > 0 is assumed
at both space infinities).
The direct scattering problem is shown to be well-posed for potentials q such that q −q± ∈ L1,2 (R± ).
As to the inverse problem, we formulated and solved it both via Marchenko integral equations, and as a
12
Riemann-Hilbert problem in terms of a suitable uniform variable, determined the asymptotic behavior
of the scattering data and showed that the linear system solving the inverse problem is well-defined.
An important open issue is whether an “area” theorem can be established, to relate the existence
and location of discrete eigenvalues of the scattering problem to the area of the initial profile of the
solution, suitably defined to take into account the boundary conditions. In this regard, we proved that
discrete eigenvalues of the scattering problem, if they exist, are confined to two real semiintervals of ±q0
∫0
∫ +∞
whose sizes decrease to zero with the area: −∞ dx|q(x) − q− | + 0 dx|q(x) − q+ |.
References
[1] V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP, 37 (1973).
[2] T. Kawata and H. Inoue, J. Phys. Soc. Japan, 43 (1977) and 44 (1978).
[3] V. S. Gerdjikov and P. P. Kulish, Bulgar. J. Phys., 5(4) (1978).
[4] N. Asano and Y. Kato, J. Math. Phys., 22 (1980) and 25 (1984).
[5] M. Boiti and F. Pempinelli, Nuovo Cimento A, 69 (1982).
[6] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer,
Berlin and New York, 1987).
[7] F. Demontis, B. Prinari, C. van der Mee and F. Vitale, Stud. App. Math., DOI: 10.1111/j.14679590.2012.00572.x (2012).
Dynamical criteria for rogue waves in NLS models
Constance Schober∗ , University of Central Florida, USA [Constance.Schober@ucf.edu]
Abstract: In this talk we investigate rogue waves in deep water in the framework of the nonlinear
Schrödinger (NLS) and Dysthe equations. Amongst the homoclinic orbits of unstable NLS Stokes waves,
we seek good candidates to model actual rogue waves. In this article we propose two selection criteria:
stability under perturbations of initial data, and persistence under perturbations of the NLS model.
We find that requiring stability selects homoclinic orbits of maximal dimension. Persistence under (a
particular) perturbation selects a homoclinic orbit of maximal dimension all of whose spatial modes are
coalesced. These results suggest that more realistic sea states, described by JONSWAP power spectra,
may be analyzed in terms of proximity to NLS homoclinic data. In fact, using the NLS spectral theory, we
find that rogue wave events in random oceanic sea states are well predicted by proximity to homoclinic
data of the NLS equation.
13
Positive solutions for a classes of elliptic bi-variate systems with
combined nonlinear effects
Jaffar Ali Shahul Hameed∗ , Florida Gulf Coast University, USA [jahameed@fgcu.edu]
Ratnasingham Shivaji, Mississippi State University, USA
Abstract: Consider the system
−∆u = λf (v, w)
in Ω
−∆v = λg(w, u)
in Ω
−∆w = λh(u, v)
in Ω
u =v = w = 0
on ∂Ω
where ∆ is the Laplacian operator, λ is a non-negative parameter, Ω is a bounded domain in Rn with
smooth boundary ∂Ω and f, g, h ∈ C 1 ([0, ∞) × [0, ∞), R) are monotonically increasing functions that
satisfy certain combined sublinear conditions at ∞. In this talk, we discuss the existence and multiplicity
results for positive solutions via the method of sub-super solutions.
Symbolic and computational tools to study bifurcations in steady state
solutions for a class on nonlinear partial differential equations
Muhammad Usman∗ , University of Dayton, USA [musman1@udayton.edu]
Abstract: In this talk bifurcations in steady state solutions of a class of nonlinear dispersive wave
equations and forced Kuramoto-Sivashinsky equation is discussed. Using an asymptotic perturbation
method stability of solutions will be discussed. By defining the detuning parameter, primary resonance
will be considered. External excitation and frequency-response curves are shown to exhibit jump and
hysteresis phenomena for the models. Mathematica and MATLAB were used to perform the symbolic
and numerical computations, respectively.
Iterative methods for a forward-backward equation
Venkataram Vanaja∗ , University of South Florida, USA [vvanaja2@usf.edu]
Abstract: We consider a boundary value problem involving a simple Fokker-Planck equation
∂
2 ∂u
of the form x ∂u
∂t = α ∂x ((1 − x ) ∂x ), where α is a positive constant. This equation is forward parabolic
when x > 0, and backward parabolic when x < 0. Such an equation arises in electron scattering theory.
An iterative method to solve the equation is given and some numerical results are presented.
Empirical study on the relationship between China’s grain international
trade and grain domestic and foreign prices based on the perspective
of food security
Rui Wang∗ , Wuhan Polytechnic University, P. R. China [xmchx111@sina.com]
Abstract: Based on monthly time series from January 2003 to August 2011, using empirical
methods such as Co integration analysis, Granger Causality Test and Error Correction Model, we analyze
14
the relationship between Chinas international grain trade and the grain prices of domestic market and
international market from the perspective of food security. The results show that: First, there is a link
between China’s grain market and the international grain market, but not very closely linked; Second,
China’s grain imports and exports do not affect the international grain prices, and there are not “big
country effects” existing in Chinas imports and exports; Third, the impact of China’s grain imports
and exports on the domestic grain prices is very limited; Fourth, both the changes of international
and domestic grain prices affect Chinas grain exports. Therefore, China may lower the domestic food
self-sufficiency rate appropriately, increase grain imports, and make use of the international market to
protect domestic grain security.
Boundedness of nonlinear Volterra integro-differential equations
Tingxiu Wang∗ , Texas A & M University-Commerce, USA [tingxiu.wang@tamuc.edu]
Abstract: By Lyapunov’s Second Method and a method of linearization, we obtain lower and
upper bounds of nonlinear Volterra integro-differential equations. In the linear case, our results become
Wazewskis inequality.
Symmetry preserving discretization of differential equations and
invariant difference schemes
Pavel Winternitz∗ , Universite de Montreal, Canada [wintern@CRM.Umontreal.CA]
Abstract: We show how one can approximate an Ordinary Differential Equation by a Difference
System that has the same Lie point symmetry group as the original ODE. Such a discretization has
many advantages over standard discretizations. In particular it provides numerical solutions that are
qualitatively better, specially in the neighborhood of singularities.
Dissipative solitons in a generalized coupled cubic-quintic
Ginzburg-Landau equation
Emmanuel Yomba∗ , Gholam-Ali Zakeri, California State University at Northridge, USA
[eyomba@csun.edu]
Abstract: Three families of exact analytical solutions of a generalized nonlinear coupled system
of cubic-quintic complex Ginzburg-Landau equations are obtained. These families of solitary waves which
describe the evolution of progressive bright-bright, front-front, dark-dark and other families of solitary
waves are investigated. Using total energy of pulse modes, a loss-gain analysis due to nonlinearity in
comparison to a linear gain is done. The stability of the solitary waves is examined using linear stability
analysis and also a complete numerical stability of these families of solutions is provided. The results
reveal that the most of these solitary waves obtained here can propagate in a stable way under slight
perturbation of white noise and the disturbance of parameters of the system.
15
Growth problems of Laplacian type and Hurwitz numbers
Anton Zabrodin∗ , Institute of Theoretical and Experimantal Physics, Russia [zabrodin@itep.ru]
Abstract: We report on the integrable structure of the 2D growth problems of Laplacian type
with zero surface tension. The most familiar examples are the growth problems in the plane and in an
infinite channel with periodic boundary conditions in the transverse direction. These problems can be
embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. We characterize the corresponding solutions of the hierarchy by the string equations and construct dispersionless tau-functions
for these solutions. The Taylor coefficients of the tau-functions are shown to be given by double Hurwitz
numbers counting connected ramified coverings of the 2D sphere of a certain ramification type.
Blowup solutions for systems of equations in two dimensional spaces
Lei Zhang∗ , University of Florida, USA [leizhang@ufl.edu]
Chang-shou Lin, National Chung Cheng University, Taiwan
Juncheng Wei, Chinese University of Hong Kong, Hong Kong
Abstract: In this talk I will describe the asymptotic behavior of blowup solutions to Liouville
systems and Toda systems that come from various disciplines of sciences.
The effects of singular lines in nonlinear differential equations
Lijun Zhang∗ , Zhejiang Sci-Tech University, P. R. China [li-jun0608@163.com]
Li-Qun Chen, Shanghai University, P. R. China
Jibin Li, Kunming University of Science and Technology, P. R. China
Abstract: In this talk, by using bifurcation theory of planar dynamical systems, we investigate
bounded traveling wave solutions of some nonlinear differential equations to study the effects of singular
lines in nonlinear wave equations. We focus on the relationships between the smooth traveling wave solutions and the non-smooth ones with the orbits of the corresponding traveling wave systems, especially
those close to the singular lines of the systems. We find that some kinds of new bounded singular traveling wave solutions which are different from the singular traveling wave solutions such as compactons,
cuspons, peakons appear if their corresponding traveling wave systems have horizontal straight lines.
These singular traveling wave solutions are characterized by discontinuous second-order derivatives at
some points, even though their first-order derivatives are continuous. It is worth mentioning that nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of
traveling wave solutions.
On exact solutions in the continuous limits for soliton equations
Yi Zhang∗ , Zhejiang Normal University, P. R. China [zy2836@163.com]
Abstract: In this talk, by using the Darboux transformation and Hirota bilinear method, we will
discuss some exact solutions in the continuous limits for soliton equations. Motivated by the continuous
limits theory, we find that many exact solutions may be constructed by performing an appropriate
limiting procedure on the classical soliton solutions.
16
A method for generating Lie algebras and some applications
Yufeng Zhang∗ , China University of Mining and Technology, P. R. China [mathzhang@126.com]
Binlu Feng, Weifang University, P. R. China
Abstract: A method for generating Lie algebras is introduced. An integrable hierarchy of
evolution equations is obtained. Specially, we derive its Hamiltonian structures by the variational identity.
Binary nonlinearization of a super NLS-mKdV equation
Qiulan Zhao∗ , Yu-Xia Li, Xin-Yue Li, Shandong University
of Science and Technology, P. R. China [ql zhao@yahoo.cn]
Ye-Peng Sun, Shandong University of Finance and Economics, P. R. China
Abstract: Based on the constructed Lie superalgebra, the super Hamiltonian structure of a
NLS-mKdV hierarchy is obtained by making use of super-trace identity. Moreover, an explicit super
Bargmann symmetry constraint and its associated binary nonlinearization of Lax pairs are carried out
for the super NLS-mKdV system.
On nonlinear Schrödinger equations with magnetic fields
Shijun Zheng∗ , Georgia Southern University, USA [szheng@georgiasouthern.edu]
Abstract: The dissipative mechanism of Schrödinger equation is mathematically described by
the energy decay of the solution. We mainly address how an electromagnetic field affects the local and
global in time existence for certain nonlinear Schrödinger equations. In the focusing case we show that
the construction of solitons contributes to the profile decomposition of the solution.
Shock creation and Painlevé property of colliding peakons
in the Degasperis-Procesi equation
Lingjun Zhou∗ , Tongji University, P. R. China [zhoulj@tongji.edu.cn]
Jacek Szmigielski, University of Saskatchewan, Canada
Abstract: The Degasperis-Procesi equation (DP) is one of several equations known to model
important nonlinear effects such as wave breaking and shock creation. It is, however, a special property of
the DP equation that these two effects can be studied in an explicit way with the help of the multipeakon
ansatz. In essence this ansatz allows one to model wave breaking as a collision of hypothetical particles
(peakons and antipeakons), called henceforth collectively multipeakons. It is shown that DP multipeakons
have Painlevé property which implies a universal wave breaking behaviour, that multipeakons can collide
only in pairs, and that there are no multiple collisions other than, possibly simultaneous, collisions of
peakon-antipeakon pairs at different locations. Moreover, it is demonstrated that each peakon-antipeakon
collision results in creation of a shock thus making possible a multi shock phenomenon.
17
Speaker and author index
Aktosun T., 5
Anco S., 5
Beheshti S., 5
Bruzon M. S., 6, 8
Calogero F., 2, 6
Casati P., 6
Chen L. Q., 16
Conte R., 3
Demontis F., 12
Doliwa A., 6
dos Santos J. P., 7
Dzhamay A., 7
Eremenko A., 10
Feng B. L., 17
Ferreira L. A., 7
Gandarias M. L., 6, 8
Gu X., 8
Kaup D. J., 3
Khalique M., 9
Khavinson D., 3, 10
Koikawa T., 9
Kovalyov M., 9
Krichever I., 4
Lazrag L., 9
Leyvraz F., 6
Li J. B., 16
Li X. Y., 17
Li Y. X., 17
Liao S. J., 10
Lin C. S., 16
Luchini G., 7
Lundberg E., 10
Luo L., 10
Ma W. X., 8, 11
Mohyud-Din S., 11
Moore B., 11
Morozov Al., 11
Morozov An., 12
Musette M., 3
Musslimani Z., 12
Norena L., 11
Prinari B., 12
Redlich A., 5
Sakai H., 7
Schober C., 11, 13
Shahul Hameed J. A., 14
Shakeel M., 11
Shivaji R., 14
Sun Y. P., 17
Szmigielski J., 17
Takenawa T., 7
Tenenblat K., 4, 7
Teodorescu R., 10
Usman M., 14
van der Mee C., 12
Vanaja V., 14
Vitale F., 12
Wang R., 14
Wang T. X., 15
Wei J. C., 16
Winternitz P., 15
Yomba E., 15
Zabrodin A., 16
Zakeri G.-A., 15
Zakrzewski W. J., 7
Zhang L., 16
Zhang L. J., 16
Zhang Y., 16
Zhang Y. F., 17
Zhao Q. L., 17
Zheng H. C., 8
Zheng S. J., 17
Zhou L. J., 17
18