Special analytical solutions of the damped-anharmonic-oscillator equation

Dixon, J.M.; Tuszyński, Jack A.; Otwinowski, M.

doi:10.1103/physreva.44.3484

Cited by 18 publications

(12 citation statements)

References 22 publications

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“…Numerous studies have discovered classes of closed-form analytical solutions for polynomial nonlinearities F (U ; ω) (Dixon et al [1991], Powell and Tabor [1992], Goriely [1995], Benguria and Depassier [1994]). Powell and Tabor [1992] describe a set of exact solutions for F (U ; ω) of the form (1.2) when f (U ) has two terms with a special relation,…”

Section: Exact Solutionsmentioning

confidence: 99%

“…This model appears in many classical studies of the propagation of phase transitions in time-dependent Ginzburg-Landau problems (Dixon et al [1991]). For different forms of the nonlinear function F (u; ω), (1.1) has been used in population genetics to study the spread of a favored trait, originally given by Fisher's equation (Murray [1990]).…”

mentioning

confidence: 99%

“…In all of these problems, solutions of the partial differential equation exist whose spatial structure is given by the solution of (1.1) and the speed of propagation c is determined by a balance of nonlinear and dissipative effects in the ordinary differential equation. Analytical determination of the speed of propagation in closed form has been studied extensively for classes of polynomial nonlinearities F (U ; ω) (Benguria and Depassier [1994], Dixon et al [1991], Powell and Tabor [1992]). However, some open questions still remain regarding the calculation of c for more general problems (Goriely [1995], Cisternas and Depassier [1996]).…”

mentioning

confidence: 99%

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On Axisymmetric Traveling Waves and Radial Solutions of Semi‐linear Elliptic Equations

Witelski¹,

Ono²,

Kaper³

2000

Natural Resource Modeling

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ABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementary approach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowly varying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisymmetric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensively discuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Many of the results contained in this article are known, and in presenting known results, it is intended that this article be expository in nature, providing elementary demonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radially symmetric solutions of semilinear elliptic problems, especially to those articles in which more advanced mathematical theory is developed.

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Section: Exact Solutionsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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On Axisymmetric Traveling Waves and Radial Solutions of Semi‐linear Elliptic Equations

Witelski¹,

Ono²,

Kaper³

2000

Natural Resource Modeling

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“…1), v 0 = 300 m/s is the sound velocity in the medium, and v the velocity of kink excitations which amazingly emerges as special exact analytical solutions of this nonlinear ordinary differential equation with damping. This solution has been found earlier in a different context [60] and the reader is referred to it for a thorough mathematical analysis. The most important consequence of Eq.…”

Section: The Model Of Tubulin Tail Kink Excitationsmentioning

confidence: 92%

Modelling the Role of Intrinsic Electric Fields in Microtubules as an Additional Control Mechanism of Bi-directional Intracellular Transport

Satarić

Budinski-Petković

Lončarević

et al. 2008

Cell Biochem Biophys

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Active transport is essential for cellular function, while impaired transport has been linked to diseases such as neuronal degeneration. Much long distance transport in cells uses opposite polarity molecular motors of the kinesin and dynein families to move cargos along microtubules. It is clear that many types of cargo are moved by both sets of motors, and frequently in a reverse direction. The general question of how the direction of transport is regulated is still open. The mechanism of the cell's differential control of diverse cargos within the same cytoplasmic background is still unclear as is the answer to the question how endosomes and mitochondria move to different locations within the same cell. To answer these questions we postulate the existence of a local signaling mechanism used by the cell to specifically control different cargos. In particular, we propose an additional physical mechanism that works through the use of constant and alternating intrinsic (endogenous) electric fields as a means of controlling the speed and direction of microtubule-based transport. A specific model is proposed and analyzed in this paper. The model involves the rotational degrees of freedom of the C-termini of tubulin, their interactions and the coupling between elastic and dielectric degrees of freedom. Viscosity of the solution is also included and the resultant equation of motion is found as a nonlinear elliptic equation with dissipation. A particular analytical solution of this equation is obtained in the form of a kink whose properties are analyzed. It is concluded that this solution can be modulated by the presence of electric fields and hence may correspond to the observed behavior of motor protein transport along microtubules.

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“…In the context of damped anharmonic oscillators, Dixon et al [8] studied equations of the type (in this section, we use…”

Section: Equations Of the Dixon-tuszyński-otwinowski Typementioning

confidence: 99%

Supersymmetric pairing of kinks for polynomial nonlinearities

Rosu

Cornejo-Pérez

2005

Phys. Rev. E

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We show how one can obtain kink solutions of ordinary differential equations with polynomial nonlinearities by an efficient factorization procedure directly related to the factorization of their nonlinear polynomial part. We focus on reaction-diffusion equations in the travelling frame and damped-anharmonic-oscillator equations. We also report an interesting pairing of the kink solutions, a result obtained by reversing the factorization brackets in the supersymmetric quantum mechanical style. In this way, one gets ordinary differential equations with a different polynomial nonlinearity possessing kink solutions of different width but propagating at the same velocity as the kinks of the original equation. This pairing of kinks could have many applications. We illustrate the mathematical procedure with several important cases, among which the generalized Fisher equation, the FitzHugh-Nagumo equation, and the polymerization fronts of microtubules.

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Special analytical solutions of the damped-anharmonic-oscillator equation

Cited by 18 publications

References 22 publications

On Axisymmetric Traveling Waves and Radial Solutions of Semi‐linear Elliptic Equations

On Axisymmetric Traveling Waves and Radial Solutions of Semi‐linear Elliptic Equations

Modelling the Role of Intrinsic Electric Fields in Microtubules as an Additional Control Mechanism of Bi-directional Intracellular Transport

Supersymmetric pairing of kinks for polynomial nonlinearities

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