At this workshop the speakers will present recent work and outline current problems in the field of nonlinear waves and solitons on lattices. As well as mathematical analysis, the speakers cover a wide range of disciplines including both pure and applied approaches to science and engineering. It will thus provide an excellent introduction to the subject area for younger researchers. In addition, time will be made available for discussions and to allow the formation of new collaborations.

PDFs of some speakers' talks are now available, follow the link at the end of the title.

**Juan
Archilla** (Madrid, Spain)
**Kramer's reaction rate theory in the presence of breathers.**

**Alan
Champneys** (Bristol, UK):
**Multistability of localised stationary and travelling modes
of lattice equations - discrete snaking [pdf]**

**Leonor
Cruzeiro** (Faro, Portugal):
**Nonlinear waves in biomolecules and HTSC**

**Vladimir Dubinko** (NSC Kharkov Inst Phys & Tec, Ukraine):
**The breather statistics in crystals far away from equilibrium
[pdf]**

**Dirk
Hennig** (Portsmouth, UK):
**When it helps to be purely Hamiltonian: acceleration of rare events
and enhanced escape dynamics**

**Guillaume
James** (Toulouse, France):
**Wave propagation in chains of beads with Hertzian contacts and
the discrete p-Schrodinger equation**

**Magnus
Johansson** (Linkoping, Sweden):
**Peierls-Nabarro energy surfaces and directional mobility of
discrete solitons in two-dimensional saturable nonlinear
Schrodinger lattices [pdf]**

**Nikos
Karachalios** (Samos, Greece): **Energy thresholds
for breathers and bifurcation of nonlinear states in lattices
[pdf]**

**Yuriy Kosevich** (Russian Academy of Sciences, Moscow):
**Tunnelling and self-trapping of breathers in lattices
and Bose-Einstein condensates**

**Faustino
Palmero** (Sevilla, Spain):
**Discrete breathers in electrical lattices**

**Mike Russell** (Heriot-Watt, UK):
**Breathers and nonlinear aspects of radiation damage in crystals.
Talk: [odt] [txt]
Slides: [pdf]**

** Conference Photo**:

Monday | Tuesday | ||
---|---|---|---|

9.00 | Registration | 9.00 | James |

10.00 | Archilla | 10.00 | Karachlios |

10.45 | Dubinko | 11.00 | Coffee / Tea |

11.15 | Coffee / Tea | 11.45 | Posters |

11.45 | Cruzeiro | 12.00 | Posters |

12.45 | Lunch | 12.30 | Lunch |

2.00 | Hennig | 2.00 | Johannson |

3.00 | Kosevich | 3.00 | Tea / Coffee |

3.30 | Tea / Coffee | 3.30 | Champneys |

4.00 | Palmero | 4.30 | End |

4.45 | Russell |

**Juan
Archilla** (Madrid, Spain)
**Kramer's reaction rate theory in the presence of breathers.**

The fact that the population of breathers becomes larger at higher energies that the phonon's one has been proposed as a mechanism that increases the reaction rate of reconstructive transformation. But the existence of breathers brings about a modulation of the potential barrier in Kramer's theory. This phenomenon in itself has a profound influence on the reaction rate, contributing to explain why some layers silicates experiences reconstructive transformations at low temperatures.

**Alan
Champneys** (Bristol, UK):
**Multistability of localised stationary and travelling modes
of lattice equations - discrete snaking**

A one-dimensional lattice equation is studied that models the light field in an optical system comprised of a periodic array of optical cavities pumped by a coherent light source. The model includes effects of linear detuning, linear and nonlinear dissipation and saturable nonlinearity. A wide variety of different parameter regions are studied using spatial dynamics methods. The presence of Maxwell points, where heteroclinic connections exist between different homogeneous states is found to lead to "snaking" bifurcation diagrams where there are infinitely many distinct stable solitons, both bright and grey. Mechanisms are revealed by which the snakes can be created and destroyed as a second parameter is varied. In particular, the bright solitons reach the boundary of the bistability region where the homogeneous state in the soliton's tail undergoes a fold, whereupon the snake splits into many separate loops.

The additional effects of a spatial gradient of the phase of the pump field are studied, which, unlike for continuum systems, is argued to not lead directly to moving solitons, but there remains a pinning region in which there are infinitely many distinct stable stationary solitons of arbitrarily large width. These solitons break up into discrete isolas. For large enough symmetry-breaking localised structures start to move. Careful numerical simulations reveal that branches of the moving solitons undergo unsual hysteresis with respect to the pump, for sufficiently large symmetry breaking.

**Leonor
Cruzeiro** (Faro, Portugal):
**Nonlinear waves in biomolecules and HTSC**

Proteins are the mega molecules that mediate most of the processes that go on in living cells. In spite of many decades of study, the way they fold and function is still very obscure. I will summarize the state of the art in the protein folding field, and present my own unorthodox answers to some of the unsolved questions. Central to those proposed answers is the so-called VES hypothesis, that is, the possibility that protein conformational steps are driven by quantum vibrational excited states. The formal mathematical similarities between the nonlinear models for vibrational energy transfer in proteins and for electron transport in lattices will be emphasized and an application of these models to high temperature superconductivity (HTSC) will be presented.

**Vladimir Dubinko** (NSC Kharkov Inst Phys & Tec, Ukraine):
**The breather statistics in crystals far away from equilibrium**

The statistics of discrete breathes (DBs) in solids under irradiation with swift particles is considered, and the corresponding reaction rate amplification factors are derived. Radiation-induced formation of DBs is shown to change mechanical properties of materials under irradiation, which is confirmed by experimental data. The present model shows that the effects due to the crystal anharmonicity are of both fundamental significance and of considerable technological importance in the fields of nuclear engineering and radiation effects.

**Dirk
Hennig** (Portsmouth, UK):
**When it helps to be purely Hamiltonian: acceleration of rare events
and enhanced escape dynamics**

We consider the self-organized escape of a, linear chain of coupled units from a metastable state over a barrier in a microcanonical situation. Initially the units of the chain are situated near the bottom, of the potential well forming a flat state. In the underlying conservative and deterministic dynamics such a uniform and linear lattice state with comparatively little energy content; seems to be restrained to oscillations around the, potential bottom preventing escape from the well. It is demonstrated that even small deviations from the flat state entail internal energy redistribution leading to such strong localization that the lattice chain spontaneously adopts a localized pattern resembling a hairpin-like structure. The latter corresponds to a critical equilibrium configuration, that is a transition state, and, being dynamically unstable, constitutes the starting point for the escape process. The collective barrier crossing of the units takes place as kink-antikink motions along the chain. It turns out that this nonlinear barrier crossing in a microcanonical situation is more efficient compared with a thermally activated chain for small ratios between the total energy of the chain and the barrier energy.

**Guillaume
James** (Toulouse, France):
**Wave propagation in chains of beads with Hertzian contacts and
the discrete p-Schrodinger equation**

Perturbative methods like modulation equations and local continuation techniques have been used to describe important classes of waves in nonlinear lattices, like solitons, nonlinear normal modes and breathers. Because such approaches require sufficient smoothness and often concern weakly nonlinear waves, their application to uncompressed granular chains is delicate due to Hertz's contact forces between grains, which are fully nonlinear under compression and vanish when beads are not in contact, yielding a limited smoothness of the stress-strain relation at the contact point. We adapt some perturbative methods to granular chains including Newton's cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz's contact forces. A nonlinear Schrodinger equation with discrete p-Laplacian captures many dynamical features of this system, e.g. the existence of small amplitude periodic travelling waves, and modulational instabilities yielding to energy trapping in the form of discrete breathers. In granular chains without local oscillators, we prove the existence of periodic travelling waves close to binary oscillations. These waves can be numerically continued with respect to wavelength until reaching the long wave limit.

**Magnus
Johansson** (Linkoping, Sweden):
**Peierls-Nabarro energy surfaces and directional mobility of
discrete solitons in two-dimensional saturable nonlinear
Schrodinger lattices**

We address the problem of directional mobility of discrete solitons in two-dimensional rectangular lattices, in the framework of a discrete nonlinear Schrodinger model with saturable on-site nonlinearity. A numerical constrained Newton-Raphson method is used to calculate two-dimensional Peierls-Nabarro energy surfaces, which describe a pseudopotential landscape for the slow mobility of coherent localized excitations, corresponding to continuous phase-space trajectories passing close to stationary modes. Investigating the two-parameter space of the model through independent variations of the nonlinearity constant and the power, we show how parameter regimes and directions of good mobility are connected to existence of smooth surfaces connecting the stationary states. In particular, directions where solutions can move with minimum radiation can be predicted from flatter parts of the surfaces. For such mobile solutions, slight perturbations in the transverse direction yield additional transverse oscillations with frequencies determined by the curvature of the energy surfaces, and with amplitudes that for certain velocities may grow rapidly. We also describe how the mobility properties and surface topologies are affected by inclusion of weak lattice anisotropy. (U Naether, RA Vicencio and M Johansson, to appear in Phys. Rev. E)

**Nikos
Karachalios** (Samos, Greece): **Energy thresholds
for breathers and bifurcation of nonlinear states in lattices**

We present results on the existence of energy thresholds and the bifurcation of nonlinear states in lattices of DNLS type. The derivation of energy thresholds is considered for a DNLS model with nonlinear hopping terms, and the method is taking into account the localization of the true breather solutions. Bifurcation of nonlinear states is considered for a discretization of the Gross-Pitaevskii equation, describing an array of Bose-Einstein condensate droplets confined in the wells of an optical lattice. Stability issues for these states are also dicussed.

**Yuriy Kosevich** (Russian Academy of Sciences, Moscow):
**Tunnelling and self-trapping of breathers in lattices
and Bose-Einstein condensates**

We present analytical and numerical studies of the phase-coherent dynamics of intrinsically localized excitations (breathers) in a system of two weakly coupled nonlinear one-dimensional lattices. We show that there are two qualitatively different dynamical regimes of the coupled breathers, either immovable or slowly moving: the periodic transverse translation (tunneling) of the low-amplitude breather between the chains and the one-chain-localization (self-trapping) of the high-amplitude breather. We also show that these two regimes of coupled phase-coherent breathers are similar and are described by a similar pair of equations to the two regimes in the nonlinear tunneling dynamics of two weakly linked interacting (nonideal) Bose-Einstein condensates. On the basis of this profound analogy, we predict a tunneling mode of two weakly coupled Bose-Einstein condensates in which their relative phase oscillates around pi/2 mod pi.

**Faustino
Palmero** (Sevilla, Spain):
**Discrete breathers in electrical lattices**

We focus on the production of both stationary and travelling intrinsic localized modes (ILMs), also known as discrete breathers, in two closely related electrical lattices. We describe a novel mechanism that is responsible for the motion of driven ILMs in this system, and quantify this effect by modelling in some detail the electrical components comprising the lattice. Comparison between theory and experiments show a reasonable agreement.

**Mike Russell** (Heriot-Watt, UK):
**Breathers and nonlinear aspects of radiation damage in crystals.**

Transient displacements of atoms in crystals by scattering of swift particles generate a bewildering variety of effects, most of which involve nonlinear forces. Chris Eilbeck has played a pivotal role in helping to understand the discrete particle transport processes involved, especially breathers. Breathers help explain long standing problems of missing energy in molecular dynamic studies and the observation of enhanced migration of atoms far removed from radiation damage sites. Recent developments on the evolution and scattering of breathers will be described. Recent experiments in metals are discussed and possible applications mentioned.

The registration fee of £ 30 will be handled by ICMS,
and used to provide tea/coffee/lunches.

Accommodation can be found in locally,
further information is available
here.

The International Centre for Mathematical Sciences, (ICMS)

The London Mathematical Society (LMS)

The Edinburgh Mathematical Society (EMS)

The Glasgow Mathematics Journal Trust Fund (GMJTF)

We have a some funds to help postgraduate students attend
the workshop, and also some funds to support researchers from
Africa and former Soviet Union countries to attend, (at any stage
of their career). Please apply to one of the organisers for
details, or to apply for such help in attending.

Jonathan Wattis | Gabriel Lord | |

Jonathan.Wattis at nottingham.ac.uk | Gabriel at ma.hw.ac.uk | |

School of Mathematical Sciences, | Department of Mathematics, | |

University of Nottingham, | Heriot-Watt University, | |

University Park, | Riccarton, | |

Nottingham NG7 2RD, UK. | Edinburgh, EH14 4AS, UK. |

*Last updated 12 April 2011*