Standard and embedded solitons in nematic optical fibers

Rodrı́guez, Rogelio; Reyes, J. A.; Espinosa-Cerón, A.; Fujioka, Jorge; Malomed, Boris A.

doi:10.1103/physreve.68.036606

Cited by 43 publications

(28 citation statements)

References 47 publications

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“…At first, ES were found in nonlinear-optical models including quadratic (χ (2) ) nonlinearities [1]- [6], and later they were discovered in hydrodynamic [7] and liquidcrystal dynamical models [8]. Quite recently, ES were found in discrete systems too, viz., in a finite-difference version of a higher-order NLS (nonlinear Schrödinger) equation [9], and in a model of an array of linearlycoupled waveguides with the χ (2) and χ (3) (cubic) nonlinearities [10].…”

Section: Introductionmentioning

confidence: 99%

“…It is worth observing that the coefficient in front of the nonlinear term in the cmKdV equation may be positive or negative, depending on the physical application. For example, in the description of light propagation in liquidcrystal waveguides, this coefficient is positive [8]. On the other hand, the derivation of the cmKdV equation for strongly dispersive waves in a weakly nonlinear medium by means of a multiple-scale expansion, presented in Ref.…”

Section: The Model and Its Family Of Moving Lattice-soliton Solutmentioning

confidence: 99%

“…If ρ − 2δ = 6 Eq. (2) is precisely the cmKdV equation studied in [8]. It is worth observing that the coefficient in front of the nonlinear term in the cmKdV equation may be positive or negative, depending on the physical application.…”

Section: The Model and Its Family Of Moving Lattice-soliton Solutmentioning

confidence: 99%

“…The ES may be stable against small perturbations in the linear approximation, but general arguments and direct simulations demonstrate that they are usually unstable at the second order, which is why they are often called "semi-stable" solutions. Nevertheless, examples of completely stable ES have been also found [8,13,15], and their stability has been demonstrated in Refs. [13,15].…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the recent discoveries of continuous families of ES in the complex modified Korteweg-de Vries (cmKdV) equation [8] and in a generalized third-order nonlinear Schrödinger equation [13,15], in the present paper we will consider a differential-difference equation of the following form:…”

Section: The Model and Its Family Of Moving Lattice-soliton Solutmentioning

confidence: 99%

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Moving embedded lattice solitons

Malomed

Fujioka

Espinosa-Cerón

et al. 2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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It was recently proved that solitons embedded in the spectrum of linear waves may exist in discrete systems, and explicit solutions for isolated unstable embedded lattice solitons (ELS) of a differentialdifference version of a higher-order NLS equation were found [Physica D 197 (2004) 86]. The discovery of these ELS gives rise to relevant questions such as the following: are there continuous families of ELS?, can ELS be stable?, is it possible for ELS to move along the lattice?, how do ELS interact?. The present work addresses these questions by showing that a novel equation (a discrete version of a complex modified KdV equation which includes next-nearest-neighbor couplings) has a two-parameter continuous family of exact ELS. These solitons can move with arbitrary velocities across the lattice, and the numerical simulations demonstrate that these ELS are completely stable. Moreover, the numerical tests show that these ELS are robust enough to withstand collisions, and the result of a collision is only a shift in the positions of the solitons. The model may apply to the description of a Bose-Einstein condensate with dipole-dipole interactions between the atoms, trapped in a deep optical-lattice potential. In nonlinear systems where solitons can exist, the propagation of small-amplitude linear waves, which obey the linearized version of the nonlinear equations, is possible too. However, for a soliton to exist, it is absolutely necessary that no resonances occur between the soliton and these linear waves. Otherwise, the soliton would decay due to an energy transfer towards the linear waves. Based on this no-resonance argument, it was frequently assumed that the solitons' internal frequencies could not be contained in the linear spectrum of the system, i.e., they could not lie within the band of frequencies permitted to linear waves. However, at the end of the nineties exceptions to this rule were found, and a special type of solitons were discovered, which do not resonate with linear waves, in spite of having frequencies immersed in the spectrum of these waves. In 1999 these peculiar solitary waves were given the name of embedded solitons (ES), and in the following years a number of models supporting ES were identified. Most of these models describe continuous systems. However, some examples of discrete ES were recently found too. These embedded lattice solitons (ELS) are isolated solutions which are stable against small perturbations in the linear approximation, but are nonlinearly unstable. The discovery of these isolated unstable ELS triggered the search for models admitting continuous families of stable ELS. In this article we present a novel differential-difference equation which has a two-parameter continuous family of exact ELS. These solitons can move with arbitrary velocities across the lattice, and the numerical simulations show that they are completely stable solutions.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: The Model and Its Family Of Moving Lattice-soliton Solutmentioning

confidence: 99%

Section: The Model and Its Family Of Moving Lattice-soliton Solutmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: The Model and Its Family Of Moving Lattice-soliton Solutmentioning

confidence: 99%

See 3 more Smart Citations

Moving embedded lattice solitons

Malomed

Fujioka

Espinosa-Cerón

et al. 2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Solutions of Local and Nonlocal Discrete Complex Modified Korteweg–De Vries Equations and Continuum Limits

Hu,

Shen,

Zhao

2024

Math Methods in App Sciences

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Cauchy matrix approach for the discrete Ablowitz–Kaup–Newell–Segur equations is reconsidered, where two “proper” discrete Ablowitz–Kaup–Newell–Segur equations and two “unproper” discrete Ablowitz–Kaup–Newell–Segur equations are derived. The “proper” equations admit local reduction, while the “unproper” equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz–Kaup–Newell–Segur equations, two local and nonlocal discrete complex modified Korteweg–de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg–de Vries equations, soliton solutions and Jordan‐block solutions are presented by solving the determining equation set. The dynamical behaviors of 1‐soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg–de Vries equations are discussed.

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A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier–Lebesgue Spaces

Chapouto

2021

J Dyn Diff Equat

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We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2021), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside $$H^\frac{1}{2}({\mathbb {T}})$$ H 1 2 ( T ) . Here, we employ the method introduced by Deng et al. (Commun Math Phys 384(1):1061–1107, 2021) to prove local well-posedness of the second renormalized mKdV equation in the Fourier–Lebesgue spaces $${\mathcal {F}}L^{s,p}({\mathbb {T}})$$ F L s , p ( T ) for $$s\ge \frac{1}{2}$$ s ≥ 1 2 and $$1\le p <\infty $$ 1 ≤ p < ∞ . As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier–Lebesgue spaces with infinite momentum.

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Standard and embedded solitons in nematic optical fibers

Cited by 43 publications

References 47 publications

Moving embedded lattice solitons

Moving embedded lattice solitons

Solutions of Local and Nonlocal Discrete Complex Modified Korteweg–De Vries Equations and Continuum Limits

A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier–Lebesgue Spaces

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