Interactions of nonlocal dark solitons under competing cubic–quintic nonlinearities

Chen, Wei; Shen, Ming; Kong, Qian; Shi, Jianhui; Wang, Qi; Królikowski, Wiesław

doi:10.1364/ol.39.001764

Cited by 56 publications

(27 citation statements)

References 36 publications

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“…This is the case in non-neutral plasmas [33], where there is a net Coulomb 1/r interaction, in the Calogero-Sutherland * plestird@mcmaster.ca † dodell@mcmaster.ca (CS) model [34][35][36], where particles interact via a 1/r 2 potential, and in dipolar BECs [37][38][39][40][41][42][43][44], where dipoledipole interactions lead to 1/r 3 interactions. Nonlocal interactions also occur in optical systems, such as those mediated by thermal conduction [45][46][47][48][49], and their consequences have been observed experimentally [50,51]. The CS model is integrable and hence supports true solitons [52][53][54][55], whereas non-neutral plasmas are not integrable systems and only support solitary waves.…”

Section: Introductionmentioning

confidence: 94%

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

Plestid

O’Dell

2019

Phys. Rev. E

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The Hamiltonian Mean Field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in 1D and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE) which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wavefunction and the depth of the meanfield potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the meanfield potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.

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Section: Introductionmentioning

confidence: 94%

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

Plestid

O’Dell

2019

Phys. Rev. E

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“…In the first instance, we approximate the two-soliton state as a linear superposition of two individual solitons, with the interaction treated as the spatial overlap of the soliton envelopes. This technique has been previously applied to interaction problems in the NLS [53,54] and Gross-Pitaevskii equations [55,56], in addition to several others [57][58][59][60][61]. An advantage of this method is the ability to derive a set of variational equations which describe the motion of the solitons.…”

Section: The Variational Analysismentioning

confidence: 99%

Non-integrable dynamics of matter-wave solitons in a density-dependent gauge theory

et al. 2018

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We study interactions between bright matter-wave solitons which acquire chiral transport dynamics due to an optically-induced density-dependent gauge potential. Through numerical simulations, we find that the collision dynamics feature several non-integrable phenomena, from inelastic collisions including population transfer and radiation losses to the formation of short-lived bound states and soliton fission. An effective quasi-particle model for the interaction between the solitons is derived by means of a variational approximation, which demonstrates that the inelastic nature of the collision arises from a coupling of the gauge field to velocities of the solitons. In addition, we derive a set of interaction potentials which show that the influence of the gauge field appears as a short-range potential, that can give rise to both attractive and repulsive interactions.

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“…Using the definition of M (see (13) and 17), the fact that H s pR d q, is an algebra since s ą d{2, and the fact that the exact and numerical solutions stay in a bounded set of H s pR d q, it is easy to prove the following lemma.…”

Section: Let Us Define the Operatorsmentioning

confidence: 99%

Energy preserving relaxation method for space-fractional nonlinear Schrödinger equation

Cheng

Wang

Yang

2020

Applied Numerical Mathematics

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This paper is concerned with the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analog of it. The Crank-Nicolson method is a well known method of order 2 but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in [10] for the cubic nonlinear Schrödinger equation. This method is also an energy preserving method and numerical simulations have shown that its order is 2. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.

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Interactions of nonlocal dark solitons under competing cubic–quintic nonlinearities

Cited by 56 publications

References 36 publications

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

Non-integrable dynamics of matter-wave solitons in a density-dependent gauge theory

Energy preserving relaxation method for space-fractional nonlinear Schrödinger equation

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