On solitary waves in case of amplitude-dependent nonlinearity

Tamm, Kert; Peets, Tanel

doi:10.1016/j.chaos.2015.01.013

Cited by 8 publications

(8 citation statements)

References 17 publications

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“…It is clear that while the magnitude of the amplitude of a solitary wave depends on the ratio of the parameters P and Q together with the velocity c, the sign of the amplitude is determined only by the parameter P : in case of P < 0 positive solitary wave emerges and in case of P > 0 the amplitude will be negative. It can also be shown that higher values of c result in lower amplitudes meaning that the lower amplitude solitary waves travel faster as it has been shown earlier Tamm and Peets, 2015).…”

Section: Steady Solutionssupporting

confidence: 64%

“…(8) it is evident that we have a mixed partial derivative term U XXT T . We use a change of variables for transforming the governing equation (8) for allowing the application of the PSM Tamm and Peets, 2015). The basic idea of the PSM is to find the spatial derivatives by making use of the properties of the Fourier transform and then solve the resulting ODE with respect to time derivate by making use of the commonly available schemes for numerical solving of the ODE's.…”

Section: Solutions Emerging From Arbitrary Initial Conditionsmentioning

confidence: 99%

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On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Engelbrecht

Tamm

Peets

2017

Philosophical Magazine

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Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with amplitude-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson (2005) for describing longitudinal waves in biomembranes and later improved by Engelbrecht et al. (2015) taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.

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Section: Steady Solutionssupporting

confidence: 64%

Section: Solutions Emerging From Arbitrary Initial Conditionsmentioning

confidence: 99%

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Engelbrecht

Tamm

Peets

2017

Philosophical Magazine

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“…The effect of the additional dispersive term h 2 u xxtt on the numerical and analytical solutions has been analysed in previous studies [12,13,30,31]. The mixed derivative term does not only limit the speed of the higher harmonics [10] but also controls the width of the solitary wave solution [13,30,31] which is an effect related to the properties of the structure (inertia) of the lipid bilayer.…”

Section: Mathematical Modelmentioning

confidence: 99%

“…The mixed derivative term does not only limit the speed of the higher harmonics [10] but also controls the width of the solitary wave solution [13,30,31] which is an effect related to the properties of the structure (inertia) of the lipid bilayer. It has been shown by numerical and analytical analysis [12,13,30] that solitary and oscillatory wave solutions may emerge with different sets of parameters demonstrating a rich spectrum of possible solutions emerging from Eq. (1).…”

Section: Mathematical Modelmentioning

confidence: 99%

“…This is the case of biomembranes which have an important role in biophysics. The governing equation is of a Boussinesq-type, which like in the case of wave-guides [8] is of the second order with two dispersive terms [10] and special nonlinearity [11,12]. Such a structure reflects the elastic properties and the microstructure of biomembranes.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2019

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In this paper the permanent profile waves governed by a Boussinesq-type wave equation are analysed. The model involves displacement-type nonlinearities and dispersion terms. Physically such a model equation describes longitudinal waves (density change) in biomembranes which have an internal structure composed by lipid molecules. The possible solutions are constructed and analysed. The phase plane analysis and numerical simulation reveal a novel phenomenon: the possible existence of a soliton doublet.

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Mathematics of Nerve Signals

Peets

Tamm

2019

Applied Wave Mathematics II

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On solitary waves in case of amplitude-dependent nonlinearity

Cited by 8 publications

References 17 publications

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

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

Mathematics of Nerve Signals

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