The Center for Nonlinear Studies
25th Annual International Conference

May 16 - 20, 2005
Radisson Santa Fe Hotel
Santa Fe, New Mexico USA

AGENDA

SPEAKER ABSTRACTS (in order of talks)

David Campbell (Boston University)

From FPU to ILMs via the CNLS

The Fermi-Pasta-Ulam (FPU) problem, which was formulated and studied in Los Alamos 50 years ago, produced results initially characterized by Fermi as a "little discovery." In fact, it heralded the beginning of computational and (modern) nonlinear physics, marking the first systematic study of a nonlinear system by digital computers ("experimental mathematics") and leading directly to the discovery of "solitons," as well as to deep insights into deterministic chaos and statistical mechanics.

In this presentation, I introduce briefly the original FPU problem and show how a multiple-scale analysis in the continuum limit leads to the prediction of the stable nonlinear excitations now universally known as "solitons." I next describe how a similar multiple-scale analysis and computational studies carried out in the 1980s at the CNLS led to some seemingly paradoxical results about the existence and stability of “breathers” in continuum nonlinear systems.

The resolution of these paradoxes was the discovery, in the 1990s, of stable "breathers" in discrete nonlinear systems. These discrete
breathers" now more commonly known as Intrinsic Localized Modes (ILMs) remained an appealing theoretical possibility for more than a decade. I review the basic mechanism that allows the existence of ILMs and discuss some of their essential features, including their occurrence in discrete systems in any number of spatial dimensions.

To conclude, I show that the theoretical possibility has become experimental reality by describing recent experiments that have observed ILMs in physical systems as distinct as charge-transfer solids, Josephson junction arrays, photonic structures, and micromechanical oscillator arrays, and indicate possible future directions and applications of these novel nonlinear excitations.

Harvey Segur (Univ. Colorado)

From FPU Recurrence to Frequency Downshifting

The original work of Fermi, Pasta & Ulam (FPU) failed to resolve the question "How is thermal equilibrium achieved in a solid?", but it successfully raised a new question: "Why do we observe the near-recurrence of an initial state after a fairly short time?"  Their work was the first of several generations of subsequent work, often with a similar outcome: the work raised new questions that inspired interesting new research.  A current issue to be resolved is "What causes frequency downshifting?", and in a sense the question is a descendent of the work of F, P & U. 

Roberto Camassa (Univ. North Carolina) 

Integral and Integrable Equations from Water Wave Dynamics

The pioneering investigation of Zabusky and Kruskal on the FPU problem revealed the striking mathematical connections between (near) integrable dynamics in particle lattices and classical models of water waves. Since then, developments in water wave theory have continued to be a source of new ideas in the mathematics of completely integrable systems, while offering an experimental counterpart to  analytical and numerical work. This talk will review some of recent developments in integrable wave models and discuss their connection with particle lattice systems.

I shall recall the emergence of the subject, the methods of "multifractal" analysis that constitute its applications (thermodynamic exponents) and more interesting non-thermodynamic exponents relying upon analyticity.

Randall Hulet (Rice University, Houston)

Matter Wave Soliton Train from a Bose-Einstein Condensate

The atomic interactions in a Bose-Einstein condensate are described by a third-order non-linearity within mean-field theory. Solutions of the non-linear wave equation are solitons in the case of attractive interactions. We have created a train of matter wave solitons from a Bose-Einstein condensate of lithium.  Up to ten solitons form by modulational instability when the interatomic interactions are tuned from repulsive to attractive.  The solitons propagate in a one-dimensional potential formed from a focused laser beam.  Adjacent solitons are observed to strongly repel one another due to destructive wave interference, implying that the solitons in the train have an alternating phase structure.

Sumit Mazumdar (University of Arizona)

From Solitons to Excitons in Conjugated Polymers

In the first part of this talk I will present a historical review of the one-electron theory of conjugated polymers, within which solitons are elementary excitations. Early contributions by CNLS members will be emphasized. Following this I will discuss the consequences of incorporating many-electron Coulomb interactions to the theoretical model on the ground state bond alternation in trans-polyacetylene and on soliton excitations. This part of the talk will be based on work done during my tenure as a postdoctoral fellow at the CNLS. Finally I will review our current understanding of the photophysics of these systems within Coulomb correlated models. I will show that the primary photoexcitations in these systems are strongly bound excitons with large binding energies. Time permitting, I will discuss very recent theoretical and experimental results for semiconducting single-walled carbon nanotubes. 

Alwyn Scott (University of Arizona)

The Development of Nonlinear Science

The broad structure of modern nonlinear science is sketched and details of developments in several areas of nonlinear research are presented. It is concluded that the emergence of modern nonlinear science as a collective indisciplinary activity was a Kuhnian paradigm shift which has emerged from diverse areas of science in response to two pressures: the steady growth of computing power over the past four decades, and the accumulation of knowledge about nonlinear science, which eventually broke through the traditional barriers of balkanization. Implications of these perspectives for 21st-century research in biophysics and in neuroscience are considered.

George Zaslavsky (New York University)

Field Lines, Topology, and Pseudochaos
 

Alexey Ustinov (University of Erlangen, Germany) 

Observation of 4-pi-kinks in Josephson Junction Arrays

We will report on experimental observation of moving 4-pi-kinks in arrays of parallel-connected small Josephson junctions, which are described by the discrete sine-Gordon model. Such dynamically stable multiple 4-pi-kinks were predicted theoretically more than 20 years ago by Peyrard and Kruskal but never seen experimentally up to now. The observed kinks are superconducting Josephson vortices carrying magnetic flux equal to two magnetic flux quanta. We find that, for a constant value of the driving force, the velocity of kinks with double topological charge is significantly higher than the velocity of ordinary kinks. This behavior is in agreement with theoretical calculations and is explained by reduced radiation losses for the multiple-kink state. We also find a variety of bunched states corresponding to spatially-separated ordinary kinks of the same polarity moving at a constant distance from each other. These metastable bunched states are formed due to interaction between! kinks through their oscillatory tails.

Thierry Dauxois (ENL-Lyon, France)

The Anti-FPU Problem

Several nonlinear physical systems exhibit modulational instability, which is a self-induced modulation of the steady state resulting from a balance between nonlinear and dispersive effects. This phenomenon has been studied in a large variety of physical contexts: fluid dynamics, nonlinear optics and plasma physics. The Fermi-Pasta-Ulam (FPU) lattice is an extremely well--suited model system to study this process. Both the triggering of the instability and its further evolution can be studied in detail, exciting initially high-frequency modes. The original FPU problem was casted instead in the context of long wavelengths. This is why we call the process we analyze in this paper, the Anti--FPU problem because of the analogy with the seminal FPU numerical simulation. At variance with the appearance of (m)KdV-solitons in the FPU original problem, in this process the pathway to equipartition leads to the creation of localized objects that are chaotic breathers. Similar localized structures emerge when cooling the lattice at the edges, starting from thermalized initial states.

Alan Bishop (LANL)

Three Decades of Breathing in Soft Electronic Matter:
Ferroelastics, Conjugated Polymers and DNA

There is growing appreciation in recent years for the essential roles of “complexity” in “soft” matter, including materials traditionally labeled as “hard” (e.g. organic), “soft” (e.g. inorganic) and “biological”. Understanding and learning how to use this complexity is central to designing whole new classes of materials with desired functionalities, as well as to controlling many biological functions. The relevant spatio-temporal complexity appears in many classical and quantum contexts. Interestingly, breathers (intrinsic local modes) are a common concept spanning many of these contexts. I survey some of my own acquaintance with breathers in complex electronic materials over three decades. Breathers indeed span distinct functionalities in hard electronic materials (e.g. ferroelectrics and solid-solid phase-transforming materials), soft electronic materials (e.g. conjugated polymers), and biological materials (e.g. DNA). I trace some of this history, emphasizing common ingredients from nonlinear science, and the important roles that breathers are playing in current multiscale “system” frameworks for complex functional materials.

Miki Wadati (University of Tokyo, Japan)

Matter-Wave Solitons in Spinor Bose-Einstein Condensates

We present a novel integrable system which describes soliton dynamics of an F=1 spinor Bose-Einstein condensate. Using
the inverse scattering method, we obtain soliton solutions and analyze collisional effects between solitons in the same or different spin state(s). As a result, we propose a manupulation of the soliton dynamics by controlling the parameters of colliding solitons.  

Sergej Flach (Max Planck Institute, Dresden, Germany)

From Discrete Breathers to q-Breathers

I will introduce the concept of discrete breathers (DBs) also coined intrinsic localized modes (ILMs). I will discuss recent theoretical results (wave scattering by DBs, measuring statistical properties of DBs in thermal equilibrium, quasi-compact DBs, suppression of tunneling for quantum DBs). Finally I will connect the concept of DBs with the existence of q-breathers (time periodic states localized in reciprocal q-space) in finite FPU chains which are at the heart of the original FPU paradox.

Linn Mollenauer (Lucent)

Use of Dispersion Managed Solitons: Dense WDM, Fiber Optics

Solitons have at long last found their place in telecommunications.  Lucent Technology’s LambdaXtreme is an ultra-long-haul, dense WDM fiber optic transmission system based on dispersion-managed solitons and Raman amplification. It has an advertised reach of >4000 km without electronic regeneration and a capacity of  >100 channels at 10 Gbit/s each. A 20,000 km all-optical network based on LambdaXtreme is already in use by Verizon, and several other service providers intend to purchase it for similar use.  In this talk, I shall sketch the dispersion-managed soliton technology behind LambdaXtreme, and show that the only serious nonlinear penalty stems from interchannel soliton-soliton collisions.  I shall then describe a novel technique of dispersion management, using periodic group delay devices, which very nearly eliminates that penalty as well. With this new technique, experimental results have confirmed a reach of nearly 20,000 km, limited almost solely by the growth of amplifier spontaneous emission noise.

Ildar Gabitov (Univ. Arizona/LANL)

Double Optical Resonance and Left-Handed Nonlinear
Optical Materials with Metallic Nanostructures

The simultaneous resonance of electric and magnetic components of electromagnetic radiation with structures embedded in a dielectric material is capable of inducing effective negative refractive index. It has been recently demonstrated experimentally that simple metallic nanostructures consisting of parallel or U-shaped nanowires can provide negative refractive index in the optical domain. Without the restriction of an envelope approximation, we will derive a system of equations generalizing the classical Maxwell-Lorentz model to describe nonlinear propagation of ultrashort optical pulses in such materials. We will demonstrate the existence of solitary wave solutions and discuss methods of controlling these pulses. We will show that these solitary wave solutions exist for discrete velocities and each velocity corresponds to a unique pulse shape.

Muthusamy Lakshmanan (Trichy, India)

Nonlinear Dynamics of Ferromagnetic Spin Systems in (2+1) Dimensions

After reviewing briefly the integrable cases of classical discrete and continuous Heisenberg ferromagnetic Heisenberg spin chains in (1+1) dimensions, we will consider the nonlinear dynamics underlying the evolution of a 2D nanoscale ferromagnetic film with uniaxial anisotropy in the presence of perpendicular pumping. Considering the associated Landau-Lifshitz spin evolution equation with Gilbert damping together with Maxwell equation for the demagnetization field, we study the dynamics in terms of stereographic variable. We identify explicit novel equatorial and related fixed points of the spin in the plane transverse to the anisotropy axis when the pumping frequency coincides with the amplitude of the static parallel field. We study the spin wave instabilities associated with the fixed points and identify generalized Suhl instability criterion, giving the condition for growth of the so called P-modes. Experimental consequences for ferromagnetic resonance are discussed and possible spatiotemporal patterns indicated.

The work to be reported has been done in collaboration with C. Kosaka, K. Nakamura and S. Murugesh.

Michael Schick (University of Washington)

The Conundrum of Biological Fusion

In order for any biological vesicle to be useful, it must be relatively stable. In particular, its enclosing membrane must be stable to the occurrence of long-lived holes which are thermally activated. Yet in order to undergo fusion, just such long-lived holes must occur at some point along the fusion pathway. It would seem that vesicles could either be stable, or they could undergo fusion, but not both. How they actually manage to exhibit these two conflicting properties is the conundrum. Because of recent work on this problem, my colleagues and I  believe we understand the puzzle's resolution, which will be presented in this talk.

Giovanni Zocchi (Univ. California, Los Angeles)

Spring-Loaded Proteins

Enzymes in the living cell are turned on and off through conformational changes induced by binding of regulatory molecules. We have created an artificial mechanism at the nanoscale to similarly control the function of virtually any protein. The strategy is to attach a "molecular spring" to the protein, and control the protein's conformation through the tension of the spring. These spring-loaded molecules allow to probe in unprecedented detail the dynamical properties of a protein's molecular architecture. Near and distant future applications range from amplified molecular probes to developing "smart drugs".

Kim Rasmussen (LANL)

DNA Denaturation

It has long been known that double-stranded DNA is subject to temporary, localized openings of its two strands. Particular regions along a DNA polymer are destabilized structurally by available thermal energy in the system. The localized sequence of DNA determines the physical properties of a stretch of DNA, and that in turn determines the opening profile of that DNA fragment. We show that the Peyrard-Bishop nonlinear dynamical model of DNA, which has been used to simulate denaturation of short DNA fragments, gives an accurate representation of the instability profile of a defined sequence of DNA, as verified using S1 nuclease cleavage assays. We show that the predicted openings correlate almost exactly with the promoter transcriptional start sites and major regulatory sites. Physicists have speculated that localized melting of DNA might play a role in gene transcription and other processes. Our data link sequence-dependent opening behavior in DNA to transcriptional activity for the first time. Finally, we suggest that studying the opening profile of DNA may be a way to gain insight into the location of promoters and genes in the genome.

Vitali Nesterenko (University of California, San Diego)

Title (Poster): Solitary Waves in "Sonic Vacuum": Theory, Experiment and Metamaterials.

Abstract: Famous Fermi-Pasta-Ulam paper was an inspiration to start at the end of 70th and the beginning of 80th a research on wave dynamics of strongly nonlinear granular chains.  The unusual feature of this system is a negligible linear range of the interaction force between a neighboring particles resulting in zero or very small sound speed in uncompressed or weakly compressed case ("sonic vacuum" - systems without phonon spectrum).  At the same time granular chain has a unique property of tuning into weakly nonlinear regime (the behavior of such chain is the subject of FPU paper) or even into linear regime by initial precompression.  The practical motivation for this research was an intention to understand (and possibly optimize) the performance of granular beds composed from iron shots serving to mitigate a shock wave caused by contact explosion in explosive chambers used for industrial application and for research purposes.
 
Theoretical and experimental results on strongly nonlinear wave dynamics in elastic granular media will be presented with emphasis
on the properties of a new type of solitary waves and shock waves. Examples of materials with this unusual behavior include not only
initially unstressed granular materials but also unstressed chains of particles or molecules in transverse motion and other examples.  Periodic waves, compression solitary and shock waves for these materials are qualitatively different from weakly nonlinear KdV case. They have unique features: the spatial extent of compression solitons does not depend on amplitude, initial sound speed does not determine the soliton parameters if strain in the wave is much greater than its initial value, and the initial impulse is split into a soliton train quickly on very short distances from the entrance.  Additionally "sonic vacuum" based systems allows outstanding tunability of wave properties impossible in linear elastic media.

Assembled metamaterials with typical  “sonic vacuum” properties will be demonstrated as well as recent experimental results on wave
propagation and wave interaction with contact of different “sonic vacuums”. 

This work was supported by NSF (Grant No. DCMS03013220).



Bedros Afeyan (Polymath Research Inc., President)

Title (Poster): KEEN Waves: New Kinetic Nonlinear Coherent Structures Living in the Spectral Gap of Linear Plasma Theory.

Abstract: We will describe theory and simulations of KEEN (kinetic electrostatic electron nonlinear) waves. These are coherent nonlinear structures in phase space obeying the Vlasov-Poisson system of equations.  The chaotic particles which nonadiabatically cross  separatrices maintain the stability of these waves while making the entire process non stationary. These long lived states straddle the world between BGK modes which are stationary equilibria and models where chaotic orbits lead to diffusion and the damping of waves at the other end of the spectrum of models. KEEN waves live with spatially nonlocal interactions between potential wells and particles which are in turns trapped and untrapped only to exchange energy back and forth with the wave and sustain its long range order.

KEEN-KEEN interactions as well as KEEN-EPW (electron plasma wave) interactions will be describeed as well. These objects are reminiscent of solitons, now in phase space, and resist equipartition of energy just as the PFU computer experiments showed in that innocent looking FPU nonlinearly coupled oscillator model. 

 
Avinash Khare (Institute of Physics, Bhubaneswar, India)  

Title (Poster): Exact Elliptic Solutions for A Class of FPU Like Chains.

Abstract (Poster) (A. Khare and A. Saxena): We report exact solutions for different discretizations of the Fermi-Pasta-Ulam problem in terms of Jacobi elliptic functions invoking recently discovered identities relating elliptic functions. These are different standing-wave-like solutions which in the infinite lattice limit reduce to localized soliton-like solutions.


Ioana Bena (University of Geneva, Switzerland) 

Title (Poster): Moving Discrete Breathers in Inhomogeneous Fermi-Pasta-Ulam Chains: An Illustration of Possible Behaviors

Abstract (Poster) (I. Bena and A. Saxena): We investigate numerically the scattering of moving discrete breathers (DBs) on a junction (or a pair of junctions) in a Fermi-Pasta-Ulam chain that consists of two (or three) segments with different characteristics: either different masses of the particles or different interaction parameters.  Depending on the "engineering" (i.e., the parameters and the spatial extent) of the imhomogeneity region in the chain, as well as on the characteristics of the DBs (frequency and velocity), several behaviors of the DBs can be generated at the level of the inhomogeneity -- reflection, transmission, splitting, trapping, focussing, capture of several DBs, etc.  These results can be rationalized by evaluating the change in the Peierls-Nabarro barrier for the various situations and point to interesting practical applications.


Rong Fan (New York University) 

Title (Poster): Pseudochaotic Dynamics Near Global Periodicity.

Abstract: We study a piecewise linear version of kicked oscillator model: saw-tooth map. A special case of global periodicity, in which
every phase point belongs to a periodic orbit, is presented. With few analytic results known for the corresponding map on torus, we
numerically investigate transport properties and statistical behavior of Poincar\'e recurrence time in two cases of deviation from global
periodicty. A non-KAM behavior of the system, as well as subdiffusion and superdiffusion, is oberseved through numerical simulations.
Statistics of Poincar\'e recurrences shows Kac lemma is valid in the system and there is a relation between the transport exponent and the
Poincar\'e recurrence exponent. We also perform careful numerical computation of  capacity, information and correlation dimensions of
the so-called exceptional set in both cases. Our results show that the fractal dimension is strictly less than 2 and that the fractal structures are unifractal rather than multifractal.


Bruce Miller (Texas Christian University)

Title (Poster): Exactly Integrable Analogue of a One-dimensional Gravitating System.

Abstract: The astrophysical analogue of the Fermi-Pasta-Ulam system is a one dimensional model consisting of N planar, parallel mass
sheets interacting solely through gravitational forces that are constrained to move in the perpendicular direction to their surfaces (OGS). In common with FPU, it was the first N-body gravitational system studied with numerical simulation and it also failed to convincingly exhibit ergodic behavior and a clear cut  approach to equilibrium. Exchange symmetry in acceleration partitions the configuration space of the OGS into N! equivalent cells. We take advantage of the resulting small angular extent of each cell to demonstrate the existence of a nearby, exactly integrable, version of the system.  It takes the form of a central force problem in N-1 dimensions and may explain the resistance of the OGS to attaining equilibrium. Its properties, including the construction of trajectories, as well as several continuum limits, are developed. Dynamical simulation is employed to compare the two models. For a class of initial conditions, excellent agreement is observed.


Joshua Soneson (Univ. Arizona)

Title (Poster): Polariton Dynamics in Nanocomposite Media.

Abstract: We consider the problem of optical pulse propagation in media embedded with metallic nanoparticles.  In such media resonance between the optical carrier wave and the plasmonic oscillations in the nanoparticles induces a strong nonlinear response.  Solitary wave (polariton) solutions describing the propagation of the optical field coupled with the material excitation are presented.  Numerical simulations reveal that (1) collision dynamics are highly sensitive to initial polariton parameters and (2) the system exhibits
self-induced transparency.  
 
 
Maxim Shkarayev (Univ. Arizona)

Title (Poster): Large Fluctuation of Error Rates in High Speed Optical Fiber Links: Theoretical and Experimental Study.

Abstract: Polarization mode dispersion (PMD) is major limiting factor for high-speed optical communication systems. The performance of these systems is described by the Bit Error Rate (BER) parameter defined as the ratio of erroneous bits to total bits. In this work we demonstrated that BER is a fluctuating parameter if the spacial disorder of fiber birefringence is slowly varying in time (compared to pulse width). We proposed to use statistical characteristics of birefringence to charaterize these BER fluctuations. We showed the probability
distribution function (PDF) for the BER is lognormal. Consequently, the tails of the PDF are longer than those of a Gaussian distribution.
This means that the likelihood of outages is higher than previously thought. We verified this theory using experimental result.
 

Vadim Zharnitsky (University of Illinois at U-C)

Title (Poster):  Dispersion Managed Solitons  in Higher Order DM NLS in the Absence of Residual Dispersion.

Abstract: Ground states are found in higher order averaged dispersion managed NLS.  The ground states are quasi-stationary solutions to dispersive equations with nonlocal nonlinearity, which arise as averaging approximations in the context of strong dispersion management in optical communications. It is shown  that the averaged equation possesses ground state solutions in the case of a single higher order dispersion term and in the mixed case of the 2nd and the 3rd order dispersion terms.

This is joint work with Markus Kunze and Jamison Moeser.


Boris Gershgorin (RPI)

Title (Poster): Breathers and Renormalized Waves in Beta-FPU.   

Abstract: (Boris Gershgorin, David Cai, Yuri Lvov)  We demonstrate via numerical experiments that (i) even in a very strongly nonlinear limit, beta-FPU system in thermal equilibrium behaves surprisingly like weakly nonlinear waves in properly renomalized normal variables. This happens because the collective effect of the strongly nonlinear  interaction  effectively renormalizes linear dispersion frequency and (ii) thermalized beta-FPU chain is characterized by the coexistence of breather excitations and wave excitations --- spatially highly localized  discrete breathers ride chaotically on spatially extended, renormalized waves.
 

Andrei Piryatinski (LANL)

Title (Poster): Semiclassical Scattering of Photoexcited Wavepackets on Conical Intersections.

Abstract: The problem of nonadiabatic vibrational dynamics in the vicinity of the electronic energy surface crossing is a key to understanding of variety of fundamental processes in photophysics and photochemistry including radiativeless energy relaxation and
photoisomerization in (bio)molecules. To address the problem, advanced theoretical methods have been developed and implemented as numerical techniques. In this contribution we focus on the photoexcited wavepacket scattering problem in the vicinity of conical intersection, and demonstrate that simple analytical expressions for the scattering matrix can be obtained in the semiclassical approximation. Simplicity of the latter expressions allow us to develop a clear quantitative picture of the photochemical processes taking place near the level crossing surface. This picture is verified using the numerical simulations, and good agreement is found for the realistic set of parameters. Therefore, it is now feasible to implement of our computational method into the large scale molecular dynamics simulations capable of modeling the photoexcited dynamics and related spectroscopic observables.


Serguei Goupalov (LANL)

Title (Poster): Chirality Dependence of Raman Cross-Section of Carbon Nanotubes.

Abstract: A continuum model for long wave-length phonons in carbon nanotubes is developed. A transparent analytical description of exciton coupling with the radial breathing mode in carbon nanotubes is presented. It is shown that exciton coupling with the radial breathing mode in carbon nanotubes is determined by two terms whose relative contributions strongly depend on the nanotube chirality. The ratio between the two terms is experimentally determined by means of Raman spectroscopy.

Alwin Scott (University of Arizona)

Title (Poster): The Encyclopedia of Nonlinear Science 

Abstract:  Comprised of 438 essays arranged alphabetically in one large volume, this Encyclopedia covers subjects such as chaos and turbulence in addition to the formation (emergence) and dynamics of coherent structure (solitons, nerve impulses, shock waves, tornados, and so on). Entries describe basic phenomena that arise in mathematics; theoretical and applied physics; chemistry; physical chemistry; electrical, chemical, and mechanical engineering; atmospheric and earth sciences; biology; economics; and neuroscience; among several others. Some of the entries are theoretical in nature, while others present phenomena in intuitive terms, but all are introductory, leading the reader toward further insights in the area of interest.