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Entropy
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Recent Advances in the Theory of Nonlinear Lattices
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Recent Advances in the Theory of Nonlinear Lattices

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: 30 April 2025 | Viewed by 3976

Special Issue Editors

Prof. Dr. Lars English

E-Mail Website
Guest Editor
Department of Physics and Astronomy, Dickinson College, Carlisle, PA 17013, USA
Interests: nonlinear-lattice dynamics; intrinsic localized modes; synchronization phenomena; flat bands; left-handed electrical lines; Ising machines for analog comp
Prof. Dr. Faustino Palmero

E-Mail Website
Guest Editor
Grupo de Física No Lineal. Escuela Técnica Superior de Ingeniería Informática. Departamento de Física Aplicada I, Universidad de Sevilla, 41012 Sevilla, Spain
Interests: discrete breathers; solitons; control of chaos; nonlinear discrete lattices

Special Issue Information

Dear Colleagues,

The combination of nonlinearity and discreteness has long been an area of intense research, giving birth to the fundamental excitation variously called intrinsic localized modes, discrete breathers, or discrete solitons, as well as more generally providing a setting for wave instability, chaos, and pattern formation. More recently, the focus pertaining to nonlinear-lattice dynamics has broadened to encompass other fully nonlinear states, such as kinks, rogue waves, nanopterons, and pterobreathers, as well as topologically nontrivial excitations such as vortices, skyrmions, and chiral edge states that enjoy robustness due to topological protection. Similarly, lattices with carefully designed geometries can feature flat bands where compact localized states arise. In such topologically nontrivial systems, the role of nonlinearity and disorder is beginning to be explored, as is the potential utility of nonlinear and/or topological excitations to energy transport or information storage applications. Another active research arena examines parity–time symmetry (PT) and symmetry breaking in nonlinear lattices with damping and amplification featuring non-Hermitian physics.

Such recent paradigms are being investigated in a wide variety of physical systems, including (but not limited to) photonic lattices and waveguide arrays, electrical or neuronal lattices, optical lattices and matter–wave/BEC systems, MEMS structures, Josephson-junction arrays, or atomic lattices (vibrational, electronic, or spin). Review or research articles (theoretical, numerical, or experimental) exploring such nonlinear-lattice themes as outlined above in any of these systems (or beyond) would be highly welcome for this Special Issue.

Prof. Dr. Lars English
Prof. Dr. Faustino Palmero
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • nonlinear lattices
  • solitons
  • rogue waves
  • topological excitations
  • chiral edge states
  • PT symmetry
  • non-Hermitian physics
  • flat band physics
  • energy localization/transport
  • fractional lattice models

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Published Papers (3 papers)

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Research

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17 pages, 4909 KiB  
Article
Stability of Breathers for a Periodic Klein–Gordon Equation
by Martina Chirilus-Bruckner, Jesús Cuevas-Maraver and Panayotis G. Kevrekidis
Entropy 2024, 26(9), 756; https://doi.org/10.3390/e26090756 - 4 Sep 2024
Viewed by 470
Abstract
The existence of breather-type solutions, i.e., solutions that are periodic in time and exponentially localized in space, is a very unusual feature for continuum, nonlinear wave-type equations. Following an earlier work establishing a theorem for the existence of such structures, we bring to [...] Read more.
The existence of breather-type solutions, i.e., solutions that are periodic in time and exponentially localized in space, is a very unusual feature for continuum, nonlinear wave-type equations. Following an earlier work establishing a theorem for the existence of such structures, we bring to bear a combination of analysis-inspired numerical tools that permit the construction of such waveforms to a desired numerical accuracy. In addition, this enables us to explore their numerical stability. Our computations show that for the spatially heterogeneous form of the ϕ4 model considered herein, the breather solutions are generically unstable. Their instability seems to generically favor the motion of the relevant structures. We expect that these results may inspire further studies towards the identification of stable continuous breathers in spatially heterogeneous, continuum nonlinear wave equation models. Full article
(This article belongs to the Special Issue Recent Advances in the Theory of Nonlinear Lattices)
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19 pages, 1568 KiB  
Article
Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks
by Siyuan Xing and Efstathios G. Charalampidis
Entropy 2024, 26(5), 396; https://doi.org/10.3390/e26050396 - 30 Apr 2024
Viewed by 1162
Abstract
In this paper, we apply a machine-learning approach to learn traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) [...] Read more.
In this paper, we apply a machine-learning approach to learn traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs that treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into the so-called co-traveling wave frame. This reformulation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions (known as “leftons”) within the (1+1)-dimensional, b-family of PDEs. In addition, we expand our investigations, and explore not only peakon solutions in the ab-family but also compacton solutions in (2+1)-dimensional, Rosenau-Hyman family of PDEs. A comparative analysis with multi-layer perceptron (MLP) reveals that SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader application in solving complex nonlinear PDEs. Full article
(This article belongs to the Special Issue Recent Advances in the Theory of Nonlinear Lattices)
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Review

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37 pages, 3477 KiB  
Review
Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings
by Boris A. Malomed
Entropy 2024, 26(2), 137; https://doi.org/10.3390/e26020137 - 2 Feb 2024
Cited by 1 | Viewed by 1328
Abstract
This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete [...] Read more.
This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross–Pitaevskii (GP) equations with the Lee–Huang–Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole–dipole and quadrupole–quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin–orbit-coupled (SOC) binary Bose–Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others. Full article
(This article belongs to the Special Issue Recent Advances in the Theory of Nonlinear Lattices)
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Sergej Flach, Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea
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