Search

We prove the existence of exponentially and superexponentially localized breather solutions for discrete nonlinear Klein–Gordon systems. Our approach considers $d$-dimensional infinite lattice models with general on-site potentials and interaction potentials being bounded by an arbitrary power law, as well as, systems with purely anharmonic forces, cases which are much less studied particularly in a higher-dimensional set-up. The existence problem is formulated in terms of a fixed-point equation considered in weighted sequence spaces, which is solved by means of Schauder's Fixed-Point Theorem. The proofs provide energy bounds for the solutions depending on the lattice parameters and its dimension under physically relevant non-resonance conditions.

We prove the existence of periodic travelling wave solutions for general discrete nonlinear Klein-Gordon systems, considering both cases of hard and soft on-site potentials. In the case of hard on-site potentials we implement a fixed point theory approach, combining Schauder’s fixed point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for non-trivial solutions to exist, energy (norm) thresholds for their existence and upper bounds on their velocity. In the case of soft on-site potentials, the proof of existence of periodic travelling wave solutions is facilitated by a variational approach based on the Mountain Pass Theorem. The proof of the existence of travelling wave solutions satisfying Dirichlet boundary conditions establishes rigorously the presence of compactons in discrete nonlinear Klein-Gordon chains. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.

While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear Schrödinger equation, which is more significant for physical applications, is not. We prove closeness of the solutions of both systems in the sense of a “continuous dependence” on their initial data in the and metrics. The most striking relevance of the analytical results is that small amplitude solutions of the Ablowitz-Ladik system persist in the Discrete Nonlinear Schrödinger one. It is shown that the closeness results are also valid in higher dimensional lattices, as well as, for generalised nonlinearities. For illustration of the applicability of the approach, a brief numerical study is included, showing that when the 1-soliton solution of the Ablowitz-Ladik system is initiated in the Discrete Nonlinear Schrödinger system with cubic or saturable nonlinearity, it persists for long-times. Thereby, excellent agreement of the numerical findings with the theoretical predictions is obtained. Regional Government of Andalusia and EU (FEDER program) project P18-RT-3480 Regional Government of Andalusia and EU (FEDER program) project US-1380977 MICINN, AEI and EU (FEDER program) project PID2019-110430GB-C21 MICINN, AEI and EU (FEDER program) project PID2020-112620GB-I00

We study two damped and forced discrete nonlinear Schr\"odinger equations on the one-dimensional infinite lattice. Without damping and forcing they are represented by the integrable Ablowitz-Ladik equation (AL) featuring non-local cubic nonlinear terms, and its standard (nonintegrable) counterpart with local cubic nonlinear terms (DNLS). The global existence of a unique solution to the initial value problem for both, the damped and forced AL and DNLS, is proven. It is further shown that for sufficiently close initial data, their corresponding solutions stay close for all times. Concerning the asymptotic behaviour of the solutions to the damped and forced AL and DNLS, for the former a sufficient condition for the existence of a restricted global attractor is established while it is shown that the latter possesses a global attractor. Finally, we prove the congruence of the restricted global AL attractor and the DNLS attractor for dynamics ensuing from initial data contained in an appropriate bounded subset in a Banach space.

The Ablowitz-Ladik system, being one of the few integrable nonlinear lattices, admits a wide class of analytical solutions, ranging from exact spatially localised solitons to rational solutions in the form of the spatiotemporally localised discrete Peregrine soliton. Proving a closeness result between the solutions of the Ablowitz-Ladik and a wide class of Discrete Nonlinear Schr\"odinger systems in a sense of a continuous dependence on their initial data, we establish that such small amplitude waveforms may be supported in the nonintegrable lattices, for significant large times. The nonintegrable systems exhibiting such behavior include a generalisation of the Ablowitz-Ladik system with a power-law nonlinearity and the Discrete Nonlinear Schr\"odinger with power-law and saturable nonlinearities. The outcome of numerical simulations illustrates in an excellent agreement with the analytical results the persistence of small amplitude Ablowitz-Ladik analytical solutions in all the nonintegrable systems considered in this work, with the most striking example being that of the Peregine soliton., Comment: arXiv admin note: text overlap with arXiv:2102.05332

The existence of nonzero localised periodic solutions for general one-dimensional discrete nonlinear Klein–Gordon systems with convex on-site potentials is proved. The existence problem of localised solutions is expressed in terms of a fixed point equation for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem.

Discrete Ginzburg-Landau (DGL) equations with non-local nonlinearities have been established as significant inherently discrete models in numerous physical contexts, similar to their counterparts with local nonlinear terms. We study two prototypical examples of non-local and local DGLs on the one-dimensional infinite lattice. For the non-local DGL, we identify distinct scenarios for the asymptotic behavior of the globally existing in time solutions depending on certain parametric regimes. One of these scenarios is associated with a restricted compact attractor according to J. K. Hale's definition. We also prove the closeness of the solutions of the two models in the sense of a "continuous dependence on their initial data" in the $l^2$ metric under general conditions on the intrinsic linear gain or loss incorporated in the model. As a consequence of the closeness results, in the dissipative regime we establish the congruence of the attractors possessed by the semiflows of the non-local and of the local model respectively, for initial conditions in a suitable domain of attraction defined by the non-local system., Comment: 18 pages, 1 figure. To appear in Nonlinear Analysis (2022)

The problem of showing the existence of localised modes in nonlinear lattices has attracted considerable efforts from the physical but also from the mathematical viewpoint where a rich variety of methods has been employed. In this paper we prove that a fixed point theory approach based on the celebrated Schauder's Fixed Point Theorem may provide a general method to establish concisely not only the existence of localised structures but also a required rate of spatial localisation. As a case study we consider lattices of coupled particles with nonlinear nearest neighbour interaction and prove the existence of exponentially spatially localised breathers exhibiting either even-parity or odd-parity symmetry under necessary non-resonant conditions accompanied with the proof of energy bounds of the solutions., Comment: 12 pages. To appear in Journal of Mathematical Physics

Contains fulltext : 215333.pdf (Publisher’s version ) (Closed access) BACKGROUND: Patients with stable coronary artery disease and diabetes with previous percutaneous coronary intervention (PCI), particularly those with previous stenting, are at high risk of ischaemic events. These patients are generally treated with aspirin. In this trial, we aimed to investigate if these patients would benefit from treatment with aspirin plus ticagrelor. METHODS: The Effect of Ticagrelor on Health Outcomes in diabEtes Mellitus patients Intervention Study (THEMIS) was a phase 3 randomised, double-blinded, placebo-controlled trial, done in 1315 sites in 42 countries. Patients were eligible if 50 years or older, with type 2 diabetes, receiving anti-hyperglycaemic drugs for at least 6 months, with stable coronary artery disease, and one of three other mutually non-exclusive criteria: a history of previous PCI or of coronary artery bypass grafting, or documentation of angiographic stenosis of 50% or more in at least one coronary artery. Eligible patients were randomly assigned (1:1) to either ticagrelor or placebo, by use of an interactive voice-response or web-response system. The THEMIS-PCI trial comprised a prespecified subgroup of patients with previous PCI. The primary efficacy outcome was a composite of cardiovascular death, myocardial infarction, or stroke (measured in the intention-to-treat population). FINDINGS: Between Feb 17, 2014, and May 24, 2016, 11 154 patients (58% of the overall THEMIS trial) with a history of previous PCI were enrolled in the THEMIS-PCI trial. Median follow-up was 3.3 years (IQR 2.8-3.8). In the previous PCI group, fewer patients receiving ticagrelor had a primary efficacy outcome event than in the placebo group (404 [7.3%] of 5558 vs 480 [8.6%] of 5596; HR 0.85 [95% CI 0.74-0.97], p=0.013). The same effect was not observed in patients without PCI (p=0.76, pinteraction=0.16). The proportion of patients with cardiovascular death was similar in both treatment groups (174 [3.1%] with ticagrelor vs 183 (3.3%) with placebo; HR 0.96 [95% CI 0.78-1.18], p=0.68), as well as all-cause death (282 [5.1%] vs 323 [5.8%]; 0.88 [0.75-1.03], p=0.11). TIMI major bleeding occurred in 111 (2.0%) of 5536 patients receiving ticagrelor and 62 (1.1%) of 5564 patients receiving placebo (HR 2.03 [95% CI 1.48-2.76], p

We prove the existence of nonlinear normal modes for general systems of two coupled nonlinear oscillators. Facilitating the comparison principle for ordinary differential equations, it is shown that there exist exact solutions representing a vibration in unison of the system. The associated spatially localised time-periodic solutions feature out-of-phase and in-phase motion of the oscillators.

The conservative dynamics of a 1D chain of units coupled with (FPU type) nonlinear interactions is considered. Stationary patterns in such chains emerge due to a balance of coupling energy between neighbouring units. Particularly interesting are the nontrivial stationary states which contain segments of positive and negative slope. This results in a zig-zag pattern, in the case of periodic boundary conditions, and in kink (or anti-kink) solutions in the case of the free boundary conditions. Imposing constraints on the chain, by way of two confining infinitely high walls, has repercussions for the stability of these stationary states. Here, such stationary states, commensurable with the available space between the two walls, are examined in detail, and their respective stability properties are determined analytically by invoking the transfer matrix method. Strikingly, stationary anti-kink solutions and periodic zig-zag states, being unstable in the absence of confining walls, become stable when confining walls are introduced. Furthermore, simulations reveal that chains with randomly generated initial conditions can seek out these patterns, thus localising energy, and persist for considerable time.

We investigate the interplay between chaotic and integrable Hamiltonian systems. In detail, a fully connected four-site lattice system associated with the discrete nonlinear Schrodinger equation is studied. On an embedded two-site segment (dimer) of the four-site system (tetramer) the coupling element between its two sites is time-periodically modified by an external driving term rendering the dimer dynamics chaotic, along with delocalisation of initially single-site excitations. Starting from an isolated dimer system the strength of the coupling to the remaining two sites of the tetramer is treated as a control parameter. It is striking that when the dimer interacts globally with the remaining two sites, thus constituting a fully connected tetramer, a non-trivial dependence of the degree of localisation on the strength of the coupling is found. There even exist ranges of optimal coupling strengths for which the driven tetramer dynamics becomes not only regular but also restores complete single-site localisation. We relate the re-establishment of complete localisation with transitions from permanent chaos via regular transients to permanent stable motion on a torus in the higher-dimensional phase space. In conclusion, increasing the dimension of a system can have profound effects on the character of the dynamics in higher-dimensional mixed phase spaces such that even full stabilisation of motion can be accomplished.

We construct a prototypical example of a spatially-open autonomous Hamiltonian system in which localised, but otherwise unbiased, ensembles of initial conditions break spatio-temporal symmetries in the subsequent ensemble dynamics, despite time reversal symmetry of the equations of motion. Together with transient chaos, this provides the mechanism for the occurrence of a current. Transporting trajectories pass through transient chaos and subsequently cross surfaces of no-return, after which they perform solely regular motion so that the current is of continual ballistic nature.

We study collective escape phenomena in nonlinear chain models. First we investigate the fragmentation of an overdamped polymer chain due to thermal fluctuations in the absence of an external force. We calculate the activation times of individual bonds in the coupled chain system and compare them with times obtained from Brownian dynamics simulations. We also consider a grafted chain exposed to an external force which monotonically grows as time goes on. In underdamped situations we show that collective localized excitations in a nonlinear force field with absorbing states can cause polymer fragmentation. In a similar fashion, localized modes assist a thermally activated escape of interacting particles in a metastable potential landscape which is additionally subjected to a periodic driving. The latter is necessary to obtain overcritical elongations which create localized modes even in case of stronger damping.

We study the dynamics of particles evolving in a two-dimensional periodic, spatially-symmetric potential landscape. The system is subjected to weak external time-periodic forces rocking the potential in either direction which, inter alia, breaks integrability. In particular, chaotic layers arising around separatrices which connect unstable equilibria constitute a network of interlinked paths along which trajectories can diffuse. Attention is paid to the emergence of directed motion for an ensemble of particles being trapped initially inside one well of the two-dimensional potential landscape. Interestingly, with properly chosen values of the strength, frequencies, and phases, of the external modulation forces, the particle motion can not only be initiated but also targeted in a desired direction. We demonstrate that, although unstable motion in the form of Arnold diffusion is possible, non-biased diffusion of an ensemble of particles is restricted to certain regions in phase space such that the resulting net motion proceeds directedly. This gives evidence that higher-dimensional tori play the role of partial barriers impeding unhindered motion in phase space and, in addition, serve to capture trajectories in transporting channels.

We consider the dynamics of a chain of coupled units evolving in a periodic substrate potential. The chain is initially in a flat state and situated in a potential well. A bias force, acting as a weak driving mechanism, is applied at a single unit of the chain. We study the instigation of directed transport in two types of system: (i) a microcanonical situation associated with deterministic and conservative dynamics and (ii) the Langevin dynamics when the system is in contact with a heat bath. Interestingly, for the deterministic and conservative dynamics the directed transport is drastically enhanced compared with its Langevin counterpart. In particular, in the deterministic and conservative regime a self-organised redistribution of energy triggers huge-sized avalanches yielding ultimately accelerated transport of the chain. In contrast, in the thermally-assisted process between avalanches the chain settles always into a pinned metastable state impeding continual accelerated chain motion.

We study the conservative and deterministic dynamics of two nonlinearly interacting particles evolving in a one-dimensional spatially periodic washboard potential. A weak tilt of the washboard potential is applied biasing one direction for particle transport. However, the tilt vanishes asymptotically in the direction of bias. Moreover, the total energy content is not enough for both particles to be able to escape simultaneously from an initial potential well; to achieve transport the coupled particles need to interact cooperatively. For low coupling strength the two particles remain trapped inside the starting potential well permanently. For increased coupling strength there exists a regime in which one of the particles transfers the majority of its energy to the other one, as a consequence of which the latter escapes from the potential well and the bond between them breaks. Finally, for suitably large couplings, coordinated energy exchange between the particles allows them to achieve escapes -- one particle followed by the other -- from consecutive potential wells resulting in directed collective motion. The key mechanism of transport rectification is based on the asymptotically vanishing tilt causing a symmetry breaking of the non-chaotic fraction of the dynamics in the mixed phase space. That is, after a chaotic transient, only at one of the boundaries of the chaotic layer do resonance islands appear. The settling of trajectories in the ballistic channels associated with transporting islands provides long-range directed transport dynamics of the escaping dimer.

Based upon the structural properties of DNA duplexes and their counterion-water surrounding in solution, we have introduced here a screw model which may describe translocation of DNA duplexes through artificial nanopores of the proper diameter (where the DNA counterion–hydration shell can be intact) in a qualitatively correct way. This model represents DNA as a kind of "screw," whereas the counterion-hydration shell is a kind of "nut." Mathematical conditions for stable dynamics of the DNA screw model are investigated in detail. When an electrical potential is applied across an artificial membrane with a nanopore, the "screw" and "nut" begin to move with respect to each other, so that their mutual rotation is coupled with their mutual translation. As a result, there are peaks of electrical current connected with the mutual translocation of DNA and its counterion–hydration shell, if DNA is possessed of some non-regular base-pair sequence. The calculated peaks of current strongly resemble those observed in the pertinent experiments. An analogous model could in principle be applied to DNA translocation in natural DNA–protein complexes of biological interest, where the role of "nut" would be played by protein-tailored "channels." In such cases, the DNA screw model is capable of qualitatively explaining chemical-to-mechanical energy conversion in DNA–protein molecular machines via symmetry breaking in DNA–protein friction.

In the present paper we consider the deterministic escape dynamics of a dimer from a metastable state over an anharmonic potential barrier. The underlying dynamics is conservative and noiseless and thus, the allocated energy has to suffice for barrier crossing. The two particles comprising the dimer are coupled through a spring. Their motion takes place in a two-dimensional plane. Each of the two constituents for itself is unable to escape, but as the outcome of strongly chaotic coupled dynamics the two particles exchange energy in such a way that eventually exit from the domain of attraction may be promoted. We calculate the corresponding critical dimer configuration as the transition state and its associated activation energy vital for barrier crossing. It is found that there exists a bounded region in the parameter space where a fast escape entailed by chaotic dynamics is observed. Interestingly, outside this region the system can show Fermi resonance which, however turns out to impede fast escape.

We study the Hamiltonian dynamics of a one-dimensional chain of linearly coupled particles in a spatially periodic potential which is subjected to a time-periodic mono-frequency external field. The average over time and space of the related force vanishes and hence, the system is effectively without bias which excludes any ratchet effect. We pay special attention to the escape of the entire chain when initially all of its units are distributed in a potential well. Moreover for an escaping chain we explore the possibility of the successive generation of a directed flow based on large accelerations. We find that for adiabatic slope-modulations due to the ac-field transient long-range transport dynamics arises whose direction is governed by the initial phase of the modulation. Most strikingly, that for the driven many particle Hamiltonian system directed collective motion is observed provides evidence for the existence of families of transporting invariant tori confining orbits in ballistic channels in the high dimensional phase spaces.

We scrutinize the available (seemingly disparate) theories of protein folding and propose a new concept which brings them under one roof. First, we single out dipole–dipole coupling within protein backbone as the main reason for intrinsic double-well nature of the protein potential. Then, protein folding as a whole ought to be (at least) a two-stage process, namely: (a) both amino-acid side chains and solvent enslave the dynamics of the backbone to reach the folding transition state with the help of stochastic resonance, and (b) the backbone funnels the whole protein into the global potential energy minimum by enslaving the dynamics of the amino-acid side chains plus solvent, and simultaneously arresting the stochastic resonance prerequisites to lock the protein in its folded state. The latter is accomplished owing to the concerted action of the protein compactization (enthalpic contribution) and thermal motion intensification (entropic contribution), which is, in fact, a physical hallmark of enthalpy–entropy compensation.

Assuming either classical electrodynamics or the quantum mechanical tight-binding of an electron to a nonlinear lattice with exponentially repulsive potential interactions we show how in both cases electron trapping can be mediated by solitons thus forming similar robust bound states (solectrons).

We study the combined implications of connectivity and heterogeneous inputs on the synchronization features of a one-dimensional chain of diffusively coupled FitzHugh Nagumo (FHN) systems. The uncoupled systems are triggered into a regime of chaotic firing by periodic parametric forces modeling external stimuli. Due to the parameter dispersion involved in randomly distributed amplitudes and/or phases of the forces the units are nonidentical and the firing events on the chain of uncoupled units will be asynchronous leading to a distribution of the spiking times. Interest is focused on mutually synchronized spikings arising through the coupling where the connectivity of the network may range from nearest-neighbor interaction to global interactions. From our studies we conclude that increasing the interaction radius does not necessarily entail better spike synchrony and the coupling strength plays a more important role than connectivity. It is found that for driving with random amplitudes together with random phases a critical interaction radius exists beyond which firing becomes suppressed if the coupling between the units is too strong. In such cases of ‘firing death’ the units perform only small-amplitude oscillations which are mutually synchronous. The optimal coupling for spike synchrony is of intermediate strength and altering the connectivity does not really matter for the degree of spike synchrony. Distinct to that, when all the phases are equal and only the amplitudes of the forces are randomly distributed enhanced spike synchrony is achieved for sufficiently strong coupling regardless of the interaction radius.

Assuming the quantum mechanical "tight binding" of an electron to a nonlinear lattice with Morse potential interactions we show how electric conduction can be mediated by solitons. For relatively high values of an applied electric field the current follows Ohm's law. As the field strength is lowered the current takes a finite, constant, field-independent value.

The formation of bubbles in nucleic acids (NAs) are fundamental in many biological processes such as DNA replication, recombination, telomeres formation, nucleotide excision repair, as well as RNA transcription and splicing. These precesses are carried out by assembled complexes with enzymes that separate selected regions of NAs. Within the frame of a nonlinear dynamics approach we model the structure of the DNA duplex by a nonlinear network of coupled oscillators. We show that in fact from certain local structural distortions there originate oscillating localized patterns, that is radial and torsional breathers, which are associated with localized H-bond deformations, being reminiscent of the replication bubble. We further study the temperature dependence of these oscillating bubbles. To this aim the underlying nonlinear oscillator network of the DNA duplex is brought in contact with a heat bath using the Nos$\rm{\acute{e}}$-Hoover-method. Special attention is paid to the stability of the oscillating bubbles under the imposed thermal perturbations. It is demonstrated that the radial and torsional breathers, sustain the impact of thermal perturbations even at temperatures as high as room temperature. Generally, for nonzero temperature the H-bond breathers move coherently along the double chain whereas at T=0 standing radial and torsional breathers result., Comment: 19 pages, 7 figures

We investigate the charge transport in synthetic DNA polymers built up from single types of base pairs. In the context of a polaron-like model, for which an electronic tight-binding system and bond vibrations of the double helix are coupled, we present estimates for the electron-vibration coupling strengths utilizing a quantum-chemical procedure. Subsequent studies concerning the mobility of polaron solutions, representing the state of a localized charge in unison with its associated helix deformation, show that the system for poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers, respectively possess quantitatively distinct transport properties. While the former supports unidirectionally moving electron breathers attributed to highly efficient long-range conductivity the breather mobility in the latter case is comparatively restrained inhibiting charge transport. Our results are in agreement with recent experimental results demonstrating that poly(dG)-poly(dC) DNA molecules acts as a semiconducting nanowire and exhibits better conductance than poly(dA)-poly(dT) ones., Comment: 11 pages, 5 figures

We present a model of the DNA double helix assigning three degrees of freedom to each pair of nucleotides. The model is an extension of the Barbi–Cocco–Peyrard (BCP) model in the sense that the current model allows for longitudinal motions of the nucleotides parallel to the helix axis. The molecular structure of the double helix is modelled by a system of coupled oscillators. The nucleotides are represented by point masses and coupled via point–point interaction potentials. The latter describe the covalent and hydrogen bonds responsible for the secondary structure of DNA. We obtain breather solutions using an established method for the construction of breathers on nonlinear lattices starting from the anti-coupling limit. In order to apply this method we analyse the phonon spectrum of the linearised system corresponding to our model. The obtained breathing motion consists of a local opening and re-closing of base pairs combined with a local untwist of the helix. The motions in longitudinal direction are of much lower amplitudes than the radial and angular elongations.

The initiation of the transcription process in DNA is linked with the dynamics originating from structural distortions of the double helix. It was proposed that such deformations might be caused by a ‘hit and run’ mechanism which is associated with the temporary attachment of some proteins, constituting activator factors, to regions of the DNA leaving it in deformed shape. In a nonlinear model approach we demonstrate that there exist such structural distortions of the double helix that appropriately serve to activate the formation of open regions in the form of oscillating bubble. The structure of the double helix form of DNA is modeled by a oscillator network model. We show that the underlying nonlinear dynamics supports localized solutions in the form of radial breathers and kink-shaped angular patterns. It is demonstrated that the radial breathers, which are attributed to localized H-bond deformations of the DNA molecule, move coherently along the double chain. We further illustrate that the breathers sustain the impact of heterogeneity due to the genetic code inscribed in DNA. Moreover, mobility of the breathers is also preserved when, the positions of the nucleotides are (randomly) modulated through fluctuational modes of the chemical environment, and energy dissipation due to non-elasticity damping of the motion of the nucleotides is incorporated. The amplitudes, oscillation periods and spatial extensions of the radial breathers resembles those found for the oscillating bubbles in real DNA molecules.

The physics of activated escape of objects out of a metastable state plays a key role in diverse scientific areas involving chemical kinetics, diffusion and dislocation motion in solids, nucleation, electrical transport, motion of flux lines superconductors, charge density waves, and transport processes of macromolecules and astrophysics, to name but a few. The underlying activated processes present the multidimensional extension of the Kramers problem of a single Brownian particle. In comparison to the latter case, however, the dynamics ensuing from the interactions of many coupled units can lead to intriguing novel phenomena that are not present when only a single degree of freedom is involved. In this review we report on a variety of such phenomena that are exhibited by systems consisting of chains of interacting units in the presence of potential barriers. In the first part we consider recent developments in the case of a deterministic dynamics driving cooperative escape processes of coupled nonlinear units out of metastable states. The ability of chains of coupled units to undergo spontaneous conformational transitions can lead to a self-organised escape. The mechanism at work is that the energies of the units become re-arranged, while keeping the total energy conserved, in forming localised energy modes that in turn trigger the cooperative escape. We present scenarios of significantly enhanced noise-free escape rates if compared to the noise-assisted case. The second part of the review deals with the collective directed transport of systems of interacting particles overcoming energetic barriers in periodic potential landscapes. Escape processes in both time-homogeneous and time-dependent driven systems are considered for the emergence of directed motion. It is shown that ballistic channels immersed in the associated mixed high-dimensional phase space are at the source for the directed long-range transport. Open problems and future directions are discussed in order to invigorate readers to engage in their own research.

We investigate the influence of energetic disorder, viscous damping and an external field on the electron transfer (ET) in DNA. The double helix structure of the λ-form of DNA is modeled by a steric oscillator network. In the context of the base-pair picture two different kinds of modes representing twist motions of the base pairs and H-bond distortions are coupled to the electron amplitude. Through the nonlinear interaction between the electronic and the vibrational degrees of freedom localized stationary states in the form of standing electron-vibron breathers are produced which we derive with a stationary map method. We show that in the presence of additional energetic disorder the degree of localization of such breathers is enhanced. It is demonstrated how an applied electric field initiates the long-range coherent motion of breathers along the bases of a DNA strand. These moving electron-vibron breathers, absorbing energy from the applied field, sustain energetic losses due to the viscous friction caused by the aqueous solvent as well as the impact of a moderate amount of energetic disorder. Moreover, it is illustrated that with the choice of the amplitude and frequency of the external field, the breather can be steered to a desired lattice position achieving control of the ET.

We consider the electron transfer (ET) in random helical protein models. The steric molecular structure of the protein matrix is modeled by a network of coupled oscillators. The latter, representing the peptide groups, are coupled via point–point interaction potentials describing the covalent and hydrogen bonds which stabilize the secondary structure of the helical protein scaffold. The electronic degree of freedom, expressed in terms of a tight-binding system, is coupled to intramolecular as well as bond vibrations. The effects of disorder and imperfections present in any real protein system are simulated by randomness in the system parameters and/or random equilibrium lengths of the bonds yielding a random protein cage. Interest is focused on the mobility of breather solutions accomplishing ET. We demonstrate that the coupling of the electron to the vibrational dynamics of the protein matrix is vital for the initiation of coherent ET. Furthermore, it is shown that the moving electron breathers of the ordered system may sustain the impact of randomness in the system parameters and persist as chaotic breathers establishing long-ranged ET along the transfer channels of the protein scaffold. The comparative analysis in dependence on the source of the randomness integrated into the breather dynamics is quantitatively performed with the help of transport coefficients. Interestingly, for relatively large degree of disorder in the system parameters the coupling of the polaron to the randomly distorted protein matrix leads even to enhanced ET in comparison with the case when randomness is only included in the system parameters and the bonds have equal equilibrium lengths. Particularly the last result concerning the amplifying modification of ET operations attributed to protein-inherent modes of the random protein matrix is exemplary for the constructive and yet paradoxical role played by disorder for the improvement of transport properties in biological systems.

An enhancement of localized nonlinear modes in coupled systems gives rise to a novel type of escape process. We study a spatially one dimensional set-up consisting of a linearly coupled oscillator chain of $N$ mass-points situated in a metastable nonlinear potential. The Hamilton-dynamics exhibits breather solutions as a result of modulational instability of the phonon states. These breathers localize energy by freezing other parts of the chain. Eventually this localised part of the chain grows in amplitude until it overcomes the critical elongation characterized by the transition state. Doing so, the breathers ignite an escape by pulling the remaining chain over the barrier. Even if the formation of singular breathers is insufficient for an escape, coalescence of moving breathers can result in the required concentration of energy. Compared to a chain system with linear damping and thermal fluctuations the breathers help the chain to overcome the barriers faster in the case of low damping. With larger damping, the decreasing life time of the breathers effectively inhibits the escape process.

We study coupled exciton vibration dynamics in a one-dimensional polypeptide chain interacting with the pulsating protein matrix. Special attention is paid to the initiation of polaron motion on the polypeptide chain through the coupling of the exciton transfer dynamics to a breathing mode of the protein matrix. We found that the dynamics of the protein matrix strongly influence the polaron mobility and the steady motion is activated by different types of breathing protein modes ranging from coherent driving with a single-frequency mode to noncoherent driving with modes whose frequencies and phases are site-dependent random quantities. Interestingly, up to a critical value of the width of the frequency distribution it holds that the broader the distribution of the random frequencies of the protein mode is the higher is the velocity of the mobile polaron.

We consider the exciton transfer along a one-dimensional molecular chain. The exciton motion is influenced by longitudinal vibrations evolving in a Toda lattice potential. It is shown how the soliton solutions of the vibron system coupled to the exciton system induce solitonic exciton transfer. To this aim the existence of a regime of suppressed energy exchange between the coupled excitonic and vibrational degrees of freedom is established in the case of which a nonlinear Schr\"odinger equation for the exciton variable is derived. The nonlinear Schr\"odinger equation possesses soliton solutions corresponding to coherent transfer of the localized exciton.

We study the solution behavior of a damped and parametrically driven nonlinear chain modeled by a discrete nonlinear Schr\"odinger equation. Special attention is paid to the impact of the damping and driving terms on the existence and stability of localized solutions. Dependent upon the strength of the driving force, we find rich lattice dynamics such as stationary solitonlike solutions and periodic and quasiperiodic breathers, respectively. The latter are characterized by regular motion on tori in phase space. For a critical driving amplitude the torus is destroyed in the course of time, leaving temporarily a chaotic breather on the lattice. We call this order-chaos transition a dynamical quasiperiodic route to chaos. Eventually the chaotic breather collapses to a stable localized multisite state. Finally, it is demonstrated that above a certain amplitude of the parametric driving force no localized states exist.

We study the energy exchange between excitonic and vibrational degrees of freedom, respectively, in the context of the Holstein model. For weak exciton-vibron coupling we derive bounds on the energy exchange based on Nekhoroshev-like arguments. In particular it is shown that for large differences of the excitonic and vibronic frequencies, respectively, the energy exchange is suppressed up to times growing exponentially with the ratio of the frequencies. Despite the partial energies being conserved there is equipartition of the excitonic energy among all lattice sites. However in the limit of large exciton-vibron coupling the excitonic actions are frozen separately which leads to exciton localization.

We study the escape of a chain of coupled units over the barrier of a metastable potential. It is demonstrated that a very weak external driving field with a suitably chosen frequency suffices to accomplish speedy escape. The latter requires passage through a transition state, the formation of which is triggered by permanent feeding of energy from a phonon background into humps of localized energy and elastic interaction of the arising breather solutions. In fact, cooperativity between the units of the chain entailing coordinated energy transfer is shown to be crucial for enhancing the rate of escape in an extremely effective and low-energy cost way where the effects of entropic localization and breather coalescence conspire.

We prove the existence of time-periodic solutions consisting of patterns built up from two states, one with small amplitude and the other one with large amplitude, in general nonlinear Hamiltonian finite-size lattices with global coupling. Utilising the comparison principle for differential equations it is demonstrated that for a two site segment of the nonlinear lattice one can construct solutions that are sandwiched between two oscillatory localised lattice states. Subsequently, it is proven that such a localised state can be embedded in the extended nonlinear lattice forming a breathing pattern with a single site of large amplitude against a background of uniform small-amplitude states. Furthermore, it is demonstrated that spatial patterns are possible that are built up from any combination of the small-amplitude state and the large-amplitude state. It is shown that for soft (hard) on-site potentials the range of allowed frequencies of the in-phase (out-of-phase) breathing patterns extends to values below (above) the lower (upper) value of the bivalued degenerate linear spectrum of phonon frequencies., Comment: arXiv admin note: text overlap with arXiv:1304.6370

We study the dynamics of excitation energy transfer along a lattice chain modeled by the discrete nonlinear Schr\"odinger (DNLS) equation. We prove that a segment carrying resonant motion can be decoupled from the remainder of the chain supporting quasiperiodic dynamics. The resonant segment from the extended chain is taken to be a four-site element, viz., a tetramer. First, we focus interest on the energy exchange dynamics along the tetramer viewed as two weakly coupled DNLS dimers. Hamiltonian methods are used to investigate the phase-space dynamics. We pay special attention to the role of the diffusion of the action variables inside resonance layers for the energy migration. When distributing the energy initially equally between the two dimers one observes a directed irreversible flow of energy from one dimer into the other assisted by action diffusion. Eventually on one dimer a stable self-trapped excitation of large amplitude forms at a single site while the other dimer exhibits equal energy partition over its two sites. Finally, we study the formation of localized structure on an extended DNLS lattice chain. In particular we explore the stability of the so-called even-parity and odd-parity localized modes, respectively, and explain their different stability properties by means of phase-space dynamics. The global instability of the even-parity mode is shown. For the excited even-parity mode a symmetry-breaking perturbation of the pattern leads to an intrinsic collapse of the even-parity mode to the odd-parity one. The latter remains stable with respect to symmetry-breaking perturbations. In this way we demonstrate that the favored stable localized states for the DNLS lattice chain correspond to one-site localized excitations.

The interaction of atomic hydrogen with the fcc(111) surfaces of Pd and Rh was investigated theoretically with an ab initio method, to find out the differences and similiarities between these neighboring metals. At the Rh surface the hcp site of the threefold-coordinated adsorption sites is preferred, while at Pd almost no difference between the hcp and fcc sites was found. For Pd, the occupation of subsurface positions was calculated to be more stable than bulklike positions. The energy gain caused by hydrogen absorption in subsurface positions is only about 100 meV lower than for hydrogen adsorption at the surface. In contrast, for Rh, significant differences between adsorption and absorption were calculated. The diffusion barrier for hydrogen diffusion from surface to subsurface positions was calculated and compared to the diffusion barrier in bulk. The hydrogen-induced work-function changes for the considered 4d transition-metal surfaces were positive for coverage \ensuremath{\theta}=1.

We consider a system of two coupled particles evolving in a periodic and spatially symmetric potential under the influence of external driving and damping. The particles are driven individually in such a way that in the uncoupled regime, one particle evolves on a chaotic attractor, while the other evolves on regular periodic attractors. Notably only the latter supports coherent particle transport. The influence of the coupling between the particles is explored, and in particular how it relates to the emergence of a directed current. We show that increasing the (weak) coupling strength subdues the current in a process, which in phase-space, is related to a merging crisis of attractors forming one large chaotic attractor in phase-space. Further, we demonstrate that complete current suppression coincides with a chaos-hyperchaos transition.

We investigate the solution properties oi. a generalized discrete nonlinear Schrodinger equation describing a nonlinear lattice chain. The generalized equation interpolates between the integrable discrete Ablowitz-Ladik equation and the nonintegrable discrete Schrodinger equation. Special interest is paid to the creation of stationary localized solutions called breathers. To tackle this problem we apply a map approach and illuminate the linkage of homoclinic and heteroclinic map orbits with localized lattice solutions. The homoclinic and heteroclinic orbits correspond to exact nonlinear solitonlike eigenstates of the lattice. Normal forms and the Melnikov method are used for analytical determinations of homoclinic orbits. Nonintegrability of the map leads to soliton pinning on the lattice. The soliton pinning energy is calculated and it is shown that it can be tuned by varying the ratio of the nonintegrability parameter versus the integrability parameter. The heteroclinic map orbit is derived on the basis of a variational principle. Finally, we use homoclinic and heteroclinic orbits as initial conditions to excite designed stationary localized solutions of desired width in the dynamics of the discrete nonlinear Schrodinger equation. In this way eve are able to construct coherent solitonlike structures of profile determined by the map parameters.