Exact solutions of a mixed KdV + mKdV and Benjamin-Ono equation

Cârstea, A. S.; Grecu, D.; Vişinescu, Anca

doi:10.1016/s0375-9601(98)00486-1

Cited by 4 publications

(9 citation statements)

References 39 publications

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“…The long-range potential is of the inverse power-type or Kac-Baker form. In the case of inverse power type of the fourth-order, the continuum limit yields an integro-differential equation involving a Hilbert transform [15]. Using a perturbative technique the authors derive a mixed modified Korteweg-de-Vries with Benjamin-Ono equations [15], which is close to the result presented in the following.…”

Section: Introductionsupporting

confidence: 68%

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Kink dynamics in a lattice model with long-range interactions

Ioannidou¹,

Pouget²,

Aifantis³

2001

J. Phys. A: Math. Gen.

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This paper proposes a one-dimensional lattice model with long-range interactions which, in the continuum, keeps its nonlocal behaviour. In fact, the longtime evolution of the localized waves is governed by an asymptotic equation of the Benjamin-Ono type and allows the explicit construction of moving kinks on the lattice. The long-range particle interaction coefficients on the lattice are determined by the Benjamin-Ono equation.

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Section: Introductionsupporting

confidence: 68%

“…It is worthwhile mentioning some interesting studies of anharmonic atomic chain including long-range interactions due to large Coulomb coupling between particles [12,13,14,15]. The long-range potential is of the inverse power-type or Kac-Baker form.…”

Section: Introductionmentioning

confidence: 99%

Kink dynamics in a lattice model with long-range interactions

Ioannidou¹,

Pouget²,

Aifantis³

2001

J. Phys. A: Math. Gen.

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“…The analytical result for the position variance (26) agrees rather well with the simulation results (Fig. 2).…”

Section: Simulationssupporting

confidence: 84%

“…The time scale which describes the broadening of the exponential soliton and the diffusion is t r = 1.6ν hy γ 2 o where γ(t) is the inverse width of the exponential soliton We are interested in the soliton diffusion for large times t r . Substituting ( 27) into (26) and looking for the leading order of t r yields:…”

Section: Small-noise Expansionmentioning

confidence: 99%

“…The first studies in that direction date back to Ishimori [28] who studied anharmonic chains with Lennard-Jones (2n,n) intermolecular potential and showed that the dynamics is governed by the Benjamin-Ono equation in the case n = 2 or by the Korteweg-de Vries equation for n ≥ 4. In the case of cubic and quartic nearest neighbour interactions, a reductive perturbation method yields a KdV+mKdV and Benjamin-Ono equation wich was shown to possess exact nonsingular rational solutions [26].…”

Section: Introductionmentioning

confidence: 99%

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Long-range effects on superdiffusive algebraic solitons in anharmonic chains

2007

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Studies on thermal diffusion of lattice solitons in Fermi-Pasta-Ulam (FPU)-like lattices were recently generalized to the case of dispersive long-range interactions (LRI) of the Kac-Baker form. The position variance of the soliton shows a stronger than linear time-dependence (superdiffusion) as found earlier for lattice solitons on FPU chains with nearest neighbour interactions (NNI). In contrast to the NNI case where the position variance at moderate soliton velocities has a considerable linear time-dependence (normal diffusion), the solitons with LRI are dominated by a superdiffusive mechanism where the position variance mainly depends quadratic and cubic on time. Since the superdiffusion seems to be generic for nontopological solitons, we want to illuminate the role of the soliton shape on the superdiffusive mechanism. Therefore, we concentrate on a FPU-like lattice with a certain class of power-law long-range interactions where the solitons have algebraic tails instead of exponential tails in the case of FPU-type interactions (with or without Kac-Baker LRI). A collective variable (CV) approach in the continuum approximation of the system leads to stochastic integro-differential equations which can be reduced to Langevin-type equations for the CV position and width. We are able to derive an analytical result for the soliton diffusion which agrees well with the simulations of the discrete system. Despite of structurally similar Langevin systems for the two soliton types, the algebraic solitons reach the superdiffusive long-time limit with a characteristic t 1.5 time-dependence much faster than exponential solitons. The soliton shape determines the diffusion constant in the long-time limit that is approximately a factor of π smaller for algebraic solitons.

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