Soliton and antisoliton interactions in the ‘‘good’’ Boussinesq equation

Manoranjan, V. S.; Ortega, Tomás; Sanz‐Serna, J. M.

doi:10.1063/1.527850

Cited by 71 publications

(46 citation statements)

References 5 publications

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“…where the data u°, v° are 1-periodic functions, which are assumed to be smooth enough for (2.1)-(2.4) to have a unique solution, classical or generalized (see [10]). While, for simplicity, we have chosen the period in (2.2) to be 1, it is clear that what follows can be readily extended to cover the case of an arbitrary period.…”

Section: A Numerical Methodsmentioning

confidence: 99%

“…Since the amplitudes are small, the solitons emerge from the interaction without changes in shape or velocity. The theoretical solution is given by a rather complicated expression that can be seen in [9] or [10]. Again, T = 40, but now xL = 60, xR = 60, and U1 was taken from the theoretical solutions.…”

Section: _2mentioning

confidence: 99%

“…Two recent papers are [9,10]. The first of these articles provides an exact formula for the interaction of solitons.…”

Section: Introductionmentioning

confidence: 99%

“…However, the GB equation possesses some remarkable peculiarities. For instance, solitary waves (i) only exist for a finite range of velocities, (ii) can merge into a single solitary wave, (iii) can interact to give rise to so-called anti-solitons [10].…”

Section: Introductionmentioning

confidence: 99%

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Pseudospectral method for the ‘‘good” Boussinesq equation

Frutos¹,

Ortega²,

Sanz‐Serna³

1991

Math. Comp.

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Section: A Numerical Methodsmentioning

confidence: 99%

Section: _2mentioning

confidence: 99%

“…Two recent papers are [9,10]. The first of these articles provides an exact formula for the interaction of solitons.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pseudospectral method for the ‘‘good” Boussinesq equation

Frutos¹,

Ortega²,

Sanz‐Serna³

1991

Math. Comp.

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“…The total area change in the transition of DPPC is 24.6% (2), which means that, at the peak maximum, the soliton forces the membrane Ϸ85% through the lipid melting transition. The energy density of a soliton has both potential and kinetic energy contributions and can be calculated by using a Lagrangian formalism (24). The energy density is given by…”

Section: [9]mentioning

confidence: 99%

On soliton propagation in biomembranes and nerves

Heimburg

Jackson

2005

Proc. Natl. Acad. Sci. U.S.A.

499

636

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The lipids of biological membranes and intact biomembranes display chain melting transitions close to temperatures of physiological interest. During this transition the heat capacity, volume and area compressibilities, and relaxation times all reach maxima. Compressibilities are thus nonlinear functions of temperature and pressure in the vicinity of the melting transition, and we show that this feature leads to the possibility of soliton propagation in such membranes. In particular, if the membrane state is above the melting transition solitons will involve changes in lipid state. We discuss solitons in the context of several striking properties of nerve membranes under the influence of the action potential, including mechanical dislocations and temperature changes.sound ͉ action potential ͉ compressibility ͉ Hodgkin-Huxley theory T he lipid membrane is the major building block of biological membranes, which consist mainly of large numbers of different lipids and proteins with a composition specific to the particular membrane under consideration. The isolated lipids of biomembranes display order-disorder transitions in the temperature regime of about Ϫ20°C to ϩ60°C in which membranes absorb heat (25-40 kJ͞mol), and both the lateral order and chain order of the lipid molecules are lost. This transition is accompanied by an increase in volume of Ϸ4% and an increase in area of Ϸ25%. The low and high temperature phases are called solid-ordered and liquid-disordered, respectively, indicating the simultaneous change in lateral crystalline arrangement and chain order. They are also known as gel and fluid phase, respectively. Mixed systems display a wealth of different phase diagrams. The melting profiles of lipid mixtures are therefore generally more complex than those of single lipids and cover a wider temperature range. Both peripheral and integral proteins change lipid melting caused by molecular interactions that influence the cooperative nature of the membrane fluctuations as a whole (1). Fluctuations in volume and area, and the related fluctuations in curvature, give rise to pronounced changes in elastic constants, e.g. compressibilities, bending elasticity, and relaxation times, all of which have maxima in the region of the chain melting transition. It has been suggested on theoretical and experimental grounds that these response functions are all simple functions of the heat capacity (2-4). The sound velocities of lipid dispersions obtained with ultrasonic measurements and the bending elasticities of giant vesicles are practically identical to the profiles calculated from the heat capacity (5-8). Within certain limits, it is thus possible to calculate the response functions from the heat capacity without detailed knowledge of the composition of the lipid mixture. In the transition region, membranes thus become more compressible and easier to bend. Relaxation times grow and are found to be in the range of 10 Ϫ3 s Ϫ1 ⅐min (4, 9). In unilamellar vesicles of single lipids, these changes can be pronounced. In compar...

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Efficient structure‐preserving schemes for good Boussinesq equation

Chen

Kong

Hong

2018

Math Methods in App Sciences

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The good Boussinesq equation is endowed with symplectic conservation law and energy conservation law. In this paper, some new highly efficient structure‐preserving methods for the good Boussinesq equation are proposed by improving the standard finite difference method (FDM). The new methods only use and calculate values at the odd (or even) nodes to reduce the computational cost. We call this kind of methods odd‐even method (OEM). Numerical results show that the OEM and the standard FDM have nearly the same numerical errors under the same mesh partition. However, the OEM is much more efficient than the standard FDM, such as the consumed CPU time and occupied memory.

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Soliton and antisoliton interactions in the ‘‘good’’ Boussinesq equation

Cited by 71 publications

References 5 publications

Pseudospectral method for the ‘‘good” Boussinesq equation

Pseudospectral method for the ‘‘good” Boussinesq equation

On soliton propagation in biomembranes and nerves

Efficient structure‐preserving schemes for good Boussinesq equation

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