Nonlinear Sciences

New submissions

[ total of 13 entries: 1-13 ]
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New submissions for Thu, 23 Nov 17

[1]
Title: Stabilization of the Peregrine soliton and Kuznetsov-Ma breathers by means of nonlinearity and dispersion management
Subjects: Pattern Formation and Solitons (nlin.PS)

We demonstrate a possibility to make rogue waves (RWs) in the form of the Peregrine soliton (PS) and Kuznetsov-Ma breathers (KMBs) effectively stable objects, with the help of properly defined dispersion or nonlinearity management applied to the continuous-wave (CW) background supporting the RWs. In particular, it is found that either management scheme, if applied along the longitudinal coordinate, making the underlying nonlinear Schr\"odinger equation (NLSE) selfdefocusing in the course of disappearance of the PS, indeed stabilizes the global solution with respect to the modulational instability of the background. In the process, additional excitations are generated, namely, dispersive shock waves and, in some cases, also a pair of slowly separating dark solitons. Further, the nonlinearity-management format, which makes the NLSE defocusing outside of a finite domain in the transverse direction, enables the stabilization of the KMBs, in the form of confined oscillating states. On the other hand, a nonlinearity-management format applied periodically along the propagation direction, creates expanding patterns featuring multiplication of KMBs through their cascading fission.

[2]
Title: Effect of self-deflection on a totally asymmetric simple exclusion process with functions of site-assignments
Subjects: Cellular Automata and Lattice Gases (nlin.CG); Statistical Mechanics (cond-mat.stat-mech); Physics and Society (physics.soc-ph)

This study proposes a model of a totally asymmetric simple exclusion process on a single channel lane with functions of site-assignments along the pitlane. The system model attempts to insert a new particle to the leftmost site at a certain probability by randomly selecting one of the empty sites in the pitlane, and reserving it for the particle. Thereafter, the particle is directed to stop at the site only once during its travel. Recently, the system was determined to show a self-deflection effect, in which the site usage distribution biases spontaneously toward the leftmost site, and the throughput becomes maximum when the site usage distribution is slightly biased to the rightmost site, instead of being an exact uniform distribution. Our exact analysis describes this deflection effect and show a good agreement with simulations.

[3]
Title: Theory of Heat Equations for Sigma Functions
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Algebraic Geometry (math.AG); Number Theory (math.NT)

We consider the heat equations satisfied by the sigma function associated with a planar curve, extending and developing earlier pioneering work of Buchstaber and Leykin. These heat equations lead to useful {\em linear} recursive relations for the coefficients of power series expansion of the sigma function. In particular we exhibit explicit results for curves of genus 3, and give a new constructive proof of an explicit expression for the main matrix in the theory for {\em any} hyperelliptic curve. We also state and prove a new explicit formula for the eigenvalues of the linear operators associated with this matrix, as well as other practical formulae.

[4]
Title: Nonlocal reductions of the Ablowitz-Ladik equation
Comments: 20 pages, 1 (png) figure, LaTeX
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Pattern Formation and Solitons (nlin.PS)

The purpose of the present paper is to develop the inverse scattering transform for the nonlocal semi-discrete nonlinear Schrodinger equation (known as Ablowitz-Ladik equation) with PT-symmetry. This includes: the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data and the fundamental analytic solutions. In addition, the paper studies the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann-Hilbert problem, the 1- and 2-soliton solutions for the nonlocal Ablowitz-Ladik equation are derived. Finally, the completeness relation for the associated Jost solutions is proved. Based on this, the expansion formula over the complete set of Jost solutions is derived. This will allow one to interpret the inverse scattering transform as a generalised Fourier transform.

Cross-lists for Thu, 23 Nov 17

[5]  arXiv:1708.06323 (cross-list from math-ph) [pdf, ps, other]
Title: Quantum groups, Yang-Baxter maps and quasi-determinants
Authors: Zengo Tsuboi
Journal-ref: Nuclear Physics B 926 (2018) 200-238
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Quantum Algebra (math.QA); Exactly Solvable and Integrable Systems (nlin.SI)

For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators. Thereby obtained a quasi-determinant expression of the quantum Yang-Baxter map associated with the quantum algebra $U_{q}(gl(n))$. Moreover, the map is identified with products of quasi-Pl\"{u}cker coordinates over a matrix composed of the L-operators. We also consider the quasi-classical limit, where the underlying quantum algebra reduces to a Poisson algebra. The quasi-determinant expression of the quantum Yang-Baxter map reduces to ratios of determinants, which give a new expression of a classical Yang-Baxter map.

[6]  arXiv:1711.07788 (cross-list from hep-th) [pdf, other]
Title: Scattering of kinks in a non-polynomial model
Comments: 4 pages, 3 figures; Proceedings of the 3rd International Conference on Particle Physics and Astrophysics, Moscow, 2-5 October 2017
Subjects: High Energy Physics - Theory (hep-th); Pattern Formation and Solitons (nlin.PS)

We study a model described by a single real scalar field in the two-dimensional space-time. The model is specified by a potential which is non-polynomial and supports analytical kink-like solutions that are similar to the standard kink-like solutions that appear in the $\varphi^4$ model when it develops spontaneous symmetry breaking. We investigate the kink-antikink scattering problem in the non-polynomial model numerically and highlight some specific features, which are not present in the standard case.

[7]  arXiv:1711.08196 (cross-list from quant-ph) [pdf, other]
Title: Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata
Subjects: Quantum Physics (quant-ph); Cellular Automata and Lattice Gases (nlin.CG)

Active quantum error correction on topological codes is one of the most promising routes to long-term qubit storage. In view of future applications, the scalability of the used decoding algorithms in physical implementations is crucial. In this work, we focus on the one-dimensional Majorana chain and construct a strictly local decoder based on a self-dual cellular automaton. We study numerically and analytically its performance and exploit these results to contrive a scalable decoder with exponentially growing decoherence times in the presence of noise. Our results pave the way for scalable and modular designs of actively corrected one-dimensional topological quantum memories.

[8]  arXiv:1711.08318 (cross-list from quant-ph) [pdf, ps, other]
Title: Local box-counting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral statistics
Subjects: Quantum Physics (quant-ph); Chaotic Dynamics (nlin.CD)

Two decades ago, Wang and Ong [Phys. Rev. A 55, 1522 (1997)] hypothesized that the local box-counting dimension of a discrete quantum spectrum should depend exclusively on the nearest-neighbor spacing distribution (NNSD) of the spectrum. In this paper, we validate their hypothesis by deriving an explicit formula for the local box-counting dimension of a countably-infinite discrete quantum spectrum. This formula expresses the local box-counting dimension of a spectrum in terms of single and double integrals of the NNSD of the spectrum. As applications, we derive an analytical formula for Poisson spectra and closed-form approximations to the local box-counting dimension for spectra having Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) spacing statistics. In the Poisson and GOE cases, we compare our theoretical formulas with the published numerical data of Wang and Ong and observe excellent agreement between their data and our theory. We also study numerically the local box-counting dimensions of the Riemann zeta function zeros and the alternate levels of GOE spectra, which are often used as numerical models of spectra possessing GUE and GSE spacing statistics, respectively. In each case, the corresponding theoretical formula is found to accurately describe the numerically-computed local box-counting dimension.

[9]  arXiv:1711.08440 (cross-list from physics.optics) [pdf, ps, other]
Title: Self-adjustment of a nonlinear lasing mode to a pumped area in a two-dimensional microcavity
Journal-ref: Photonics Research, Vol. 5, No. 6, pp. B47-B53 (2017)
Subjects: Optics (physics.optics); Chaotic Dynamics (nlin.CD)

We numerically performed wave dynamical simulations based on the Maxwell-Bloch (MB) model for a quadrupole-deformed microcavity laser with spatially selective pumping. We demonstrate the appearance of an asymmetric lasing mode whose spatial pattern violates both the x- and y-axes mirror symmetries of the cavity. Dynamical simulations revealed that a lasing mode consisting of a clockwise or counterclockwise rotating-wave component is a stable stationary solution of the MB model. From the results of a passive-cavity mode analysis, we interpret these asymmetric rotating-wave lasing modes by the locking of four nearly degenerate passive-cavity modes. For comparison, we carried out simulations for a uniform pumping case and found a different locking rule for the nearly degenerate modes. Our results demonstrate a nonlinear dynamical mechanism for the formation of a lasing mode that adjusts its pattern to a pumped area.

Replacements for Thu, 23 Nov 17

[10]  arXiv:1212.1710 (replaced) [pdf]
Title: The information and its observer: external and internal information processes, information cooperation, and the origin of the observer intellect
Subjects: Adaptation and Self-Organizing Systems (nlin.AO); Information Theory (cs.IT)
[11]  arXiv:1710.10140 (replaced) [pdf, other]
Title: Efficient manifold tracing for planar maps
Subjects: Chaotic Dynamics (nlin.CD); Plasma Physics (physics.plasm-ph)
[12]  arXiv:1711.02063 (replaced) [pdf, other]
Title: Cluster integrable systems, q-Painleve equations and their quantization