On solutions of a Boussinesq-type equation with displacement-dependent nonlinearity: A soliton doublet

Peets, Tanel; Tamm, Kert; Simson, Päivo; Engelbrecht, Jüri

doi:10.1016/j.wavemoti.2018.11.001

Cited by 6 publications

(4 citation statements)

References 42 publications

(87 reference statements)

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“…( 14)) lead to the bounded solution and permit regulation of the width of the soliton needed to match the experimental results. In addition, it has been shown that depending on the coefficients, several types of solutions may exist, including trains of solitons, oscillatory solutions, and the soliton doublet [60,61]. It has also been demonstrated that as Eq.…”

Section: Remarks On Mathematics Needed For Modellingmentioning

confidence: 99%

Continuum mechanics and signals in nerves

Engelbrecht¹,

Peets²,

Tamm³

2021

PEAS

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Section: Remarks On Mathematics Needed For Modellingmentioning

confidence: 99%

Continuum mechanics and signals in nerves

Engelbrecht¹,

Peets²,

Tamm³

2021

PEAS

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“…The propagation of the AP is accompanied also by the dynamical deformation of the biomembrane. A mathematical model for the longitudinal deformation is proposed by Heimburg and Jackson [31] and later improved by Engelbrecht et al [18,20,52,53].…”

Section: Mathematical Models and Modellingmentioning

confidence: 99%

Mechanical waves in myelinated axon wall

Tamm,

Peets,

Engelbrecht

2021

Preprint

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The propagation of an action potential is accompanied by mechanical and thermal effects. Several mathematical models explain the deformation of the unmyelinated axon wall. In this paper, the deformation of the myelinated axon wall is studied. The mathematical model is inspired by the mechanics of microstructured materials. The model involves the improved Heimburg-Jackson equation together with another equation of wave motion that describes the process in the myelin sheath. The dispersion analysis of such a model explains the behaviour of group and phase velocities. In addition, it is shown how dissipative effects may influence the process. Numerical calculations demonstrate the changes in velocities and wave profiles in the myelinated axon wall.

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“…Two forced Boussinesq-type models are derived for a circular rod with axially symmetric loading and longitudinal pre-stretch from the full problem formulation using the presence of small parameters. Basic dynamical properties of the models are compared, and the equations are suggested as models for the related experimental research.Soliton solutions of a Boussinesq-type equation with displacement-type nonlinearities linked to the Heimburg-Jackson model describing longitudinal waves (density charge) in biomembranes are investigated in the paper by T. Peets et al [7]. The possibility of the existence of a soliton doublet, an ensemble of two co-existing…”

mentioning

confidence: 99%

“…Soliton solutions of a Boussinesq-type equation with displacement-type nonlinearities linked to the Heimburg-Jackson model describing longitudinal waves (density charge) in biomembranes are investigated in the paper by T. Peets et al [7]. The possibility of the existence of a soliton doublet, an ensemble of two co-existing…”

mentioning

confidence: 99%

Special issue of Wave Motion - “Nonlinear waves in solids: In memory of Alexander M. Samsonov”

Хуснутдинова

Grimshaw

et al. 2020

Wave Motion

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Throughout his career Alexander's main research was in the area of nonlinear bulk strain solitary waves in solid waveguides. Alexander has initiated and developed, together with his students and collaborators, a new research direction which, eventually, formed the basis for the book [1]. His work included the development of mathematical models of Boussinesq type describing long nonlinear longitudinal bulk strain waves in various elastic waveguides (e.g., rods, plates, shells, etc.) within the scope of the Murnaghan model of nonlinear dynamic elasticity, and application of the derived model equations to the description of solitary waves and their evolution in various non-homogeneous waveguides (e.g. rods of variable cross-section, layered waveguides, etc.) using analytical approaches and direct numerical simulations. Alexander has inspired and guided some invaluable and detailed experimental work carried out in the Ioffe Institute. The experimental group originally included Yury Ostrovsky, Galina Dreiden and Irina Semenova. Together, they worked on the generation and observation of longitudinal bulk strain solitary waves in various solid waveguides.Alexander had very broad research interests. In particular, he worked on application of methods of nonlinear mathematical physics to computational and systems biology, including, but not limited to, modelling of gene networks controlling early Drosophila embryogenesis [2], mechanistic aspect of molecular transport [3], as well as of the microRNA regulatory role in cells [4].Alexander was a passionate and inspiring teacher for his students and younger colleagues, dedicating much of his time to their training, and challenging them to think creatively and independently. Many of these young scientists now also work on various waves-related problems. Alexander's ideas continue to influence researchers working in the area of nonlinear waves in solids.Alexander was a very optimistic person loving travel, literature, photography, fine arts and music, of which German and Italian operas were his favourites. We will always remember him as a wonderful colleague and friend. We hope that this special issue will serve both as a reference for those who are already actively working on nonlinear waves in solids, and will help to engage a new generation of researchers.The special issue includes 11 papers devoted to various aspects of nonlinear waves in solids, including analytical, numerical and experimental developments.The paper by A. Pau and F. Vestroni [5] belongs to the classical area of acousto-elasticity, and is devoted to a detailed study of the individual effects of material and geometric nonlinearities on the changes in shear and longitudinal speeds of bulk waves, and their polarization as a function of the initial pre-stress. Three types of pre-stress are considered: hydrostatic, biaxial and uniaxial pre-stress. It is shown that material nonlinearity has a significant effect on the amount of speed change, and that pre-stress can lead to anisotropic behaviour and coupling betwe...

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On solutions of a Boussinesq-type equation with displacement-dependent nonlinearity: A soliton doublet

Cited by 6 publications

References 42 publications

Continuum mechanics and signals in nerves

Continuum mechanics and signals in nerves

Mechanical waves in myelinated axon wall

Special issue of Wave Motion - “Nonlinear waves in solids: In memory of Alexander M. Samsonov”

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