Mobility of high-power solitons in saturable nonlinear photonic lattices

Naether, Uta; Vicencio, Rodrigo A.; Stepić, Milutin

doi:10.1364/ol.36.001467

Cited by 22 publications

(17 citation statements)

References 15 publications

(23 reference statements)

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“…Therefore, the effective size of this solution will drastically decrease being, for some value of the norm, smaller than the ring solution. We may thus expect an exchange of fundamental properties between the ring and the one-peak solutions if compared at a given norm (cf, e.g., similar features for saturable systems, where multiple changes on the effective size of fundamental solutions result in stability exchanges [18][19][20]). We construct the family of one-peak solutions by implementing a multi-dimensional Newton-Raphson iterative method [1], demanding a (norm-square) accuracy of at least 10 −15 .…”

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confidence: 98%

Discrete flat-band solitons in the kagome lattice

Vicencio

Johansson

2013

Phys. Rev. A

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We consider a model for a two-dimensional kagome lattice with defocusing nonlinearity, and show that families of localized discrete solitons may bifurcate from localized linear modes of the flat band with zero power threshold. Each family of such fundamental nonlinear modes corresponds to a unique configuration in the strong-nonlinearity limit. By choosing well-tuned dynamical perturbations, small-amplitude, strongly localized solutions from different families may be switched into each other, as well as moved between different lattice positions. In a window of small power, the lowest-energy state is a symmetry-broken localized state, which may appear spontaneously

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confidence: 98%

Discrete flat-band solitons in the kagome lattice

Vicencio

Johansson

2013

Phys. Rev. A

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“…Discrete nonlinear dispersive systems arise in nonlinear optics, e.g. [2,1,10,11,35,55,53], the dynamics of biological molecules, e.g. [7,9], and condensed matter physics.…”

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confidence: 99%

Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls–Nabarro Barrier

Jenkinson

Weinstein

2017

Commun. Math. Phys.

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We study a class of discrete focusing nonlinear Schrödinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive waves. We also prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls-Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states. These results extend those of the authors for the case of nearest neighbor interactions to a large class of nonlocal short-range and long-range interactions. The appearance of distinct onsite and offsite states is a consequence of the breaking of continuous spatial translation invariance. The PNB plays a role in the dynamics of energy transport in such nonlinear Hamiltonian lattice systems.Our class of nonlocal interactions is defined in terms of coupling coefficients, Jm, where m P Z is the lattice site index, with Jm » m´1´2 s , s P r1, 8q and Jm " e´γ |m| , s " 8, γ ą 0, (Kac-Baker). For s ě 1, the bifurcation is seeded by solutions of the (effective / homogenized) cubic focusing nonlinear Schrödinger equation (NLS). However, for 1{4 ă s ă 1, the bifurcation is controlled by the fractional nonlinear Schrödinger equation, FNLS, with p´∆q s replacing´∆. The proof is based on a Lyapunov-Schmidt reduction strategy applied to a momentum space formulation. The PN barrier bounds require appropriate uniform decay estimates for the discrete Fourier transform of DNLS discrete solitary waves. A key role is also played by non-degeneracy of the ground state of FNLS, recently proved by Frank, Lenzmann & Silvestre.Key words. Discrete nonlocal nonlinear Schrödinger equation, onsite and offsite solitary waves, bifurcation from continuous spectrum, Peierls-Nabarro energy barrier, intrinsic localized modes Here, L is a linear and nonlocal interaction operator:(1.2)The range of the nonlocal interaction is characterized by the rate of decay of the coupling sequence, tJ m u. Nonlocal interactions arising in applications typically have coupling sequences, J m " J s m with polynomial decay J s m » m´1´2 s , 0 ă s ă 8, or exponential (Kac-Baker) decay: J 8 m » e´γ m , γ ą 0. We refer to the case s ą 1 as the short-range interaction case, and the case 0 ă s ă 1 as long-range interaction case. The case s " 1 is the critical or marginal range interaction. The case J 0 " 0, J˘1 " 1 and J m " 0, |m| ě 1, the operator L corresponds to the nearest-neighbor discrete Laplacian. 1 arXiv:1601.04598v2 [nlin.PS] 9 Jan 2017 J pQq for λ s pαq´1 ď |Q| ď π{α, (λ s pαq ą α and λ s pαq Ó 0 as α Ñ 0 ) in terms of those for 0 ď |Q| ď λ s pαq´1 (low frequency components of z E α,σ J). This yields a closed system for the low-frequency components, which we study perturbatively about the continuum FNLS limit using the implicit function theorem.

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“…Many works have followed discussing various properties of these modes, of which we here just mention a few. Khare et al [42] obtained analytical solutions for a complete family of intermediate solutions, Cuevas and Eilbeck [13] studied discrete soliton interactions, Melvin et al [49] found, as mentioned above, radiationless travelling waves at "special" velocities, and Naether et al [57] analyzed the PN potential landscape in the stability exchange regimes using a constraint method to be described in the next section. We will also return to discuss the 2D version of the saturable DNLS and its mobility properties in the next section.…”

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confidence: 99%

Breather Mobility and the Peierls-Nabarro Potential: Brief Review and Recent Progress

Johansson

Jason

2015

Quodons in Mica

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The question whether a nonlinear localized mode (discrete soliton/breather) can be mobile in a lattice has a standard interpretation in terms of the PeierlsNabarro (PN) potential barrier. For the most commonly studied cases, the PN barrier for strongly localized solutions becomes large, rendering these essentially immobile. Several ways to improve the mobility by reducing the PN-barrier have been proposed during the last decade, and the first part gives a brief review of such scenarios in 1D and 2D. We then proceed to discuss two recently discovered novel mobility scenarios. The first example is the 2D Kagome lattice, where the existence of a highly degenerate, flat linear band allows for a very small PN-barrier and mobility of highly localized modes in a small-power regime. The second example is a 1D waveguide array in an active medium with intrinsic (saturable) gain and damping, where exponentially localized, travelling discrete dissipative solitons may exist as stable attractors. Finally, using the framework of an extended Bose-Hubbard model, we show that while quantum fluctuations destroy the mobility of slowly moving, strongly localized classical modes, coherent mobility of rapidly moving states survives even in a strongly quantum regime.

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Mobility of high-power solitons in saturable nonlinear photonic lattices

Cited by 22 publications

References 15 publications

Discrete flat-band solitons in the kagome lattice

Discrete flat-band solitons in the kagome lattice

Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls–Nabarro Barrier

Breather Mobility and the Peierls-Nabarro Potential: Brief Review and Recent Progress

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