Travelling solitary waves in the discrete Schrödinger equation with saturable nonlinearity: Existence, stability and dynamics

Melvin, Thomas R.O.; Champneys, Alan R; Kevrekidis, P. G.; Cuevas, J.

doi:10.1016/j.physd.2007.09.026

Cited by 53 publications

(53 citation statements)

References 41 publications

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“…As the system is symplectic and Hamiltonian, the linear stability of the solutions requires that the monodromy eigenvalues, λ (also called Floquet multipliers) must lie on the unit circle (see, e.g., [35,36] Figure 4 shows the dependence of the frequency of the internal mode corresponding to the oscillation of the trap with respect to ω 0 for fixed µ = 5 and compares it with the frequency predicted by Eq. (22).…”

Section: Dark-dark Solitons As Periodic Orbits In the Manakov Modelmentioning

confidence: 99%

Beating dark–dark solitons in Bose–Einstein condensates

Yan¹,

Chang²,

Hamner³

et al. 2012

J. Phys. B: At. Mol. Opt. Phys.

131

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Motivated by recent experimental results, we study beating dark-dark solitons as a prototypical coherent structure that emerges in two-component Bose-Einstein condensates. We showcase their connection to darkbright solitons via SO(2) rotation, and infer from it both their intrinsic beating frequency and their frequency of oscillation inside a parabolic trap. We identify them as exact periodic orbits in the Manakov limit of equal interand intra-species nonlinearity strengths with and without the trap and showcase the persistence of such states upon weak deviations from this limit. We also consider large deviations from the Manakov limit illustrating that this breathing state may be broken apart into dark-antidark soliton states. Finally, we consider the dynamics and interactions of two beating dark-dark solitons in the absence and in the presence of the trap, inferring their typically repulsive interaction.

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Section: Dark-dark Solitons As Periodic Orbits In the Manakov Modelmentioning

confidence: 99%

Beating dark–dark solitons in Bose–Einstein condensates

Yan¹,

Chang²,

Hamner³

et al. 2012

J. Phys. B: At. Mol. Opt. Phys.

131

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“…Thus, there can be values of parameters at which exact travelling solitons that do not shed radiation can be shown to occur [28,25,26]. Albeit, theses solitons cannot occur for arbitrarily small wave speeds, due to additional resonances with phonons [24,23].…”

mentioning

confidence: 99%

“…In truth it could be that these jumps are caused by fold bifurcations, in order to find which we would need to be able to continue unstable versions of these travelling localised periodic solutions. Perhaps dedicated continuation algorithms for solutions of this type, such as those used in [23], can be used to probe more details.…”

mentioning

confidence: 99%

Discrete Snaking: Multiple Cavity Solitons in Saturable Media

Yulin¹,

Champneys²

2010

SIAM J. Appl. Dyn. Syst.

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Abstract. This paper continues an investigation into a one-dimensional lattice equation that models the light field in a system comprised of a periodic array of pumped optical cavities with saturable nonlinearity. The additional effects of a spatial gradient of the phase of the pump field are studied, which in the presence of loss terms is shown to break the spatial reversibility of the steady problem. Unlike for continuum systems, small symmetry-breaking is argued to not lead directly to moving solitons, but there remains a pinning region in which there are infinitely many distinct stable stationary solitons of arbitrarily large width. These solitons are no-longer arranged in a homoclinic snaking bifurcation diagrams, but instead break up into discrete isolas. For large enough symmetry-breaking, the fold bifurcations of the lowest intensity solitons no longer overlap, which is argued to be the trigger point of moving localised structures. Due to the dissipative nature of the problem, any radiation shed by these structures is damped and so they appear to be true attractors. Careful direct numerical simulations reveal that branches of the moving solitons undergo unsual hysteresis with respect to the pump, for sufficiently large symmetry breaking. Introduction.Recently, the present authors [33,34] studied the rich variety of stable localised structures that can occur in a spatially discrete model for an optical cavity with an imposed periodic structure and saturable nonlinearity. In particular, it was shown how infinitely many distinct stable localised structures can occur for a wide range of parameter values due to the so-called homoclinic snaking mechanism that has received a lot of attention in continuum models of pattern formation [1,4,5,6,8,14,20,22,35]. This paper explores the consequences of introducing a spatial gradient to the optical pumping field (the forcing term in the model), which breaks the spatial reversibility of the steady system.The model can be written in dimensionless form as the one-dimensional lattice equation for a complex field A n ∈ C, n ∈ Z:

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“…In physics literature, the most popular nonlinearity is the so-called Kerr, or cubic, nonlinearity which is a representative of the wide class of superlinear at infinity nonlinearities. However, in recent years one can see a growing interest to saturable, i.e., asymptotically linear at infinity, nonlinearities (see [1][2][3]5,[8][9][10][11]17]). Such nonlinearities serve certain models of photorefractive media.…”

Section: Introductionmentioning

confidence: 99%

“…(1.6) In physics literature the favorite saturable nonlinearity is that given by (1.5) with p = 2. In [5,10,17] the corresponding translation invariant DNLS is studied numerically, while in [8] exact standing waves has been found, still in translation invariant case.…”

Section: Introductionmentioning

confidence: 99%

Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

Панков

2010

Journal of Mathematical Analysis and Applications

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The main result of the paper concerns the existence of nontrivial exponentially decaying solutions to periodic stationary discrete nonlinear Schrödinger equations with saturable nonlinearities, provided that zero belongs to a spectral gap of the linear part. The proof is based on the critical point theory in combination with periodic approximations of solutions. As a preliminary step, we prove also the existence of nontrivial periodic solutions with arbitrarily large periods.

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Travelling solitary waves in the discrete Schrödinger equation with saturable nonlinearity: Existence, stability and dynamics

Cited by 53 publications

References 41 publications

Beating dark–dark solitons in Bose–Einstein condensates

Beating dark–dark solitons in Bose–Einstein condensates

Discrete Snaking: Multiple Cavity Solitons in Saturable Media

Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

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