Nonlinearity and Discreteness: Solitons in Lattices

Malomed, Boris A.

doi:10.1007/978-3-030-44992-6_4

Cited by 19 publications

(17 citation statements)

References 126 publications

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“…The hydrolysis of a single ATP molecule is sufficient to initiate three amide I quanta, which could then propagate along the protein α-helices and be utilized for the execution of various physiological activities. The quantum system of equations of motion ( 14) and ( 15) for amide I energy quanta coupled to lattice phonons in proteins could be analyzed within the continuum approximation in order to derive self-trapped propagating quasiparticles referred to as solitons [11,12,37,[56][57][58][59]. The lifetime of these solitons is potentially infinite at zero temperature, T = 0 K, which allows for analytic derivation of soliton energy, effective mass, and speed of energy transportation [13,19,38,60,61].…”

Section: Discussionmentioning

confidence: 99%

Thermal stability of solitons in protein $α$-helices

Georgiev,

Glazebrook

2022

Preprint

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Protein α-helices provide an ordered biological environment that is conducive to soliton-assisted energy transport. The nonlinear interaction between amide I excitons and phonon deformations induced in the hydrogen-bonded lattice of peptide groups leads to self-trapping of the amide I energy, thereby creating a localized quasiparticle (soliton) that persists at zero temperature. The presence of thermal noise, however, could destabilize the protein soliton and dissipate its energy within a finite lifetime. In this work, we have computationally solved the system of stochastic differential equations that govern the quantum dynamics of protein solitons at physiological temperature, T = 310 K, for either a single isolated α-helix spine of hydrogen bonded peptide groups or the full protein α-helix comprised of three parallel α-helix spines. The simulated stochastic dynamics revealed that although the thermal noise is detrimental for the single isolated α-helix spine, the cooperative action of three amide I exciton quanta in the full protein α-helix ensures soliton lifetime of over 30 ps, during which the amide I energy could be transported along the entire extent of an 18-nm-long α-helix. Thus, macromolecular protein complexes, which are built up of protein α-helices could harness soliton-assisted energy transport at physiological temperature. Because the hydrolysis of a single adenosine triphosphate molecule is able to initiate three amide I exciton quanta, it is feasible that multiquantal protein solitons subserve a variety of specialized physiological functions in living systems.

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Section: Discussionmentioning

confidence: 99%

Thermal stability of solitons in protein $α$-helices

Georgiev,

Glazebrook

2022

Preprint

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“…parameterised by the initial phase q 1 (0) and ε, with T the period of the periodic orbit; the map Υ is basically the variation over the period T of the Hamiltonian flow (a part from the coordinate p 1 ). The main result (proved in [29]) used in the examples is the following Theorem 2.1 Consider the map Υ defined in (17) in a neighbourhood of the lower dimensional torus p = 0, ξ = η = 0 and let x * (ε) = (q * (ε), 0, 0, 0), with q * (ε) satisfying (16). Assume that…”

Section: Normal Form Algorithm (1mentioning

confidence: 99%

“…The discrete nonlinear Schrödinger (dNLS) equation is a paradigmatic physical model in many different areas of Physics, such as condensed matter, photonic crystals and waveguides. Indeed, it is a widely investigated nonlinear lattice model (see, e.g., [10,6,13,32,17]) thanks to the possibility to mathematically combine rigorous different analytic approaches (e.g., perturbative, variational, spectral, etc.) and to accurately explore its dynamical features with reliable numerical methods.…”

Section: Introductionmentioning

confidence: 99%

Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms

Danesi,

Sansottera,

Paleari

et al. 2021

Preprint

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We consider the problem of the continuation with respect to a small parameter ε of spatially localised and time periodic solutions in 1-dimensional dNLS lattices, where ε represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal form algorithm in order to investigate the continuation and the linear stability of degenerate localised periodic orbits on lower and full dimensional invariant resonant tori. We recover results already existing in the literature and provide new insightful ones, both for discrete solitons and for invariant subtori.

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“…[25][26][27] for recent studies. On the other hand, the SM has provided an excellent platform for exploring the interplay between nonlinear localized structures and near-linear extended ones, between nonlinearity and dispersion, on the path between integrability and non-integrability; see, e.g., [28] for a recent review.…”

Section: Introductionmentioning

confidence: 99%

Thermalization in the one-dimensional Salerno model lattice

et al. 2021

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The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (non-integrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS towards AL) is varied, the region in the space of initial energy and norm-densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.

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Nonlinearity and Discreteness: Solitons in Lattices

Cited by 19 publications

References 126 publications

Thermal stability of solitons in protein $α$-helices

Thermal stability of solitons in protein $α$-helices

Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms

Thermalization in the one-dimensional Salerno model lattice

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