Soliton dynamics in linearly coupled discrete nonlinear Schrödinger equations

Trombettoni, Andrea; Nistazakis, H. E.; Rapti, Zoi; Frantzeskakis, D. J.; Kevrekidis, P. G.

doi:10.1016/j.matcom.2009.08.033

Cited by 3 publications

(2 citation statements)

References 56 publications

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“…In particular, a (symmetric) 2 × 2 matrix NLS equation turned out to be of relevance for the description of a special Bose-Einstein condensate (with atoms in a spin 1 state) [102][103][104][105][106][107][108][109][110][111][112][113][114][115]. Semi-discrete matrix NLS equations appeared in [3,[116][117][118][119][120][121][122][123][124][125][126][127][128][129], a full discretisation has been elaborated in [130] and a dispersionless limit studied in [127].…”

Section: Introductionmentioning

confidence: 99%

Solutions of matrix NLS systems and their discretizations: a unified treatment

Dimakis¹,

Müller‐Hoissen²

2010

Inverse Problems

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Using a bidifferential graded algebra approach to "integrable" partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework, by specifying a representation of the bidifferential graded algebra and imposing a reduction. By application of a general result, corresponding families of exact solutions are obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The solutions are parametrised in terms of constant matrix data subject to a Sylvester equation (which previously appeared as a rank condition in the integrable systems literature). These data exhibit a certain redundancy, which we diminish to a large extent.More precisely, we first consider more general AKNS-type systems from which two different matrix NLS systems emerge via reductions. In the continuous case, the familiar Hermitian conjugation reduction leads to a continuous matrix (including vector) NLS equation, but it is well-known that this does not work as well in the discrete cases. On the other hand there is a complex conjugation reduction, which apparently has not been studied previously. It leads to square matrix NLS systems, but works in all three cases (continuous, semi-and fully-discrete). A large part of this work is devoted to an exploration of the corresponding solutions, in particular regularity and asymptotic behaviour of matrix soliton solutions.

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Section: Introductionmentioning

confidence: 99%

Solutions of matrix NLS systems and their discretizations: a unified treatment

Dimakis¹,

Müller‐Hoissen²

2010

Inverse Problems

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“…Seismic waves for example have well known regimes where the P and S wave energy density equilibrates in a unique way that is independent of the details of the scattering. Interaction of solitons in coupled nonlinear lattices (scalar models) have been considered for various classical configurations such as coupled Toda lattices [29], coupled nonlinear Schrodinger equations [30,31] or coupled Ablowitz-Ladik chains [32] for example. In the case of coupled Toda lattices, it was shown numerically by Kevrikidis et al [29] that solitonic excitations supplied to each one of the coupled chains may result in the two distinct dynamical regimes (attractors).…”

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confidence: 99%

Energy equipartition in two-dimensional granular systems with spherical intruders

et al. 2013

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We study the effects of a line of spherical interstitial particles (or intruders) placed between two adjacent uncompressed chains of larger particles in a square packing of spheres, using experiments and numerical simulations. We excite one of the chains of particles adjacent to the intruders with an impact and show how energy is transmitted across the system until equipartition is reached from the excited (or impacted) chain to the absorbing (or adjacent) chain. The coupling of the two chains, although a purely two-dimensional effect, is modeled by a simplified one-and-a-half-dimensional (1.5-D) system in which transverse motions of the particles are neglected.PACS numbers: 46.40.Cd Introduction. Granular crystals are unique nonlinear systems that exhibit interesting properties stemming from the nonlinear contact interactions (Hertzian [1]) between two individual particles. Uniform one-dimensional chains of spheres have been shown to support the propagation of a new type of solitary wave. The width of these waves is independent of its velocity [2][3][4][5]. The degree of nonlinearity can be tuned from highly nonlinear to linear by the addition of a precompression force [3,4,6]. Additionally, it was shown that homogeneous granular media can support families of strongly nonlinear traveling and standing waves [7], whereas heterogeneous media can exhibit resonance [8] and anti-resonance phenomena [9]. Another interesting property of these materials is the reflection of the solitary waves at an interface between two granular crystals [10][11][12], which could be used to develop new impulse trapping granular materials [13][14][15]. Several groups studied the interaction of a solitary wave with defect particles [16][17][18][19]. Twodimensional (2-D) granular crystals have been relatively unexplored, and prior works mainly consisted of numerical studies or experiments visualizing dynamic stress in photo-elastic disks [20][21][22][23]. Solitary waves have been observed in 2-D square packings of spherical particles [24]. Granular crystals have been proposed as new structured materials for the control and redirection of stress waves (see for example [10,13,14,25,28,33]). The experiments reported in this paper provide the first observation of energy equipartition between two adjacent and nonlinearly coupled chains of particles. In particular, we show that when one chain is excited by an impulse while the other is as rest, the energy is redistributed between the two chains within a short spatial distance. A similar equipartition phenomenon was studied numerically in an earlier work [26]. This phenomenon is of interest to create new acoustic wave guides, delay lines and stress mitigating materials. Energy transfer and equipartition phenomena in weakly coupled one-dimensional granular chains were studied [26,27], and in [28] through a macroscopic realization of the Landau-Zener tunneling quantum effect. The energy equipartition principle is well known for elastic waves. Seismic waves for example have well known regimes wh...

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The existence of discrete solitons for the discrete coupled nonlinear Schrödinger system

Huang

Zhou

2023

Bound Value Probl

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In this paper, we investigate the nonlinear coupled discrete Schrödinger equations with unbounded potentials. We find simple sufficient conditions for the existence of discrete soliton solution by using the Nehari manifold approach and the compact embedding theorem. Furthermore, by comparing the value of the action functional at the discrete soliton solution with those at nonzero solutions of one component zero, we demonstrate that both components of the discrete soliton solution are nontrivial.

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Soliton dynamics in linearly coupled discrete nonlinear Schrödinger equations

Cited by 3 publications

References 56 publications

Solutions of matrix NLS systems and their discretizations: a unified treatment

Solutions of matrix NLS systems and their discretizations: a unified treatment

Energy equipartition in two-dimensional granular systems with spherical intruders

The existence of discrete solitons for the discrete coupled nonlinear Schrödinger system

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