Nonlinear Science and Numerical Simulation with Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 July 2024 | Viewed by 3112

Special Issue Editors

Department of Interdisciplinary Studies, the Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel
Interests: optical solitons and optical communications; dynamics of long josephson junctions; nonlinear dynamical lattices; pattern formation in one- and two-dimensional homogeneous and inhomogeneous nonlinear dissipative media perturbation theory and variational methods; ginzburg-landau equations
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Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
Interests: topological soliton; spin-orbit coupling (SOC); Bose-Einstein condensates (BECs)
Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile
Interests: nonlinear phenomena; magnetism; thermal and electronic transport in quantum systems; hydrodynamic instabilities; radiation problems

Special Issue Information

Dear Colleagues,

Nonlinear science is a huge research area, which comprises a wide variety of topics in modern physics, mathematics, engineering, biology, etc. In most cases, the theoretical models of nonlinear systems with local interactions are based on ordinary or partial differential equations (ODEs or PDEs), or coupled systems of such equations. Nonlocal nonlinear models are typically represented by integral equations. A fundamental problem in this area is that, with the exception of a few celebrated integrable models, the underlying nonlinear ODEs, PDEs, and integral equations do not admit analytical solutions. Therefore, a majority of studies in nonlinear science rely on numerical simulations (which are often combined with the use of approximate analytical methods once exact solutions are no longer available).  Many algorithms are used for the numerical solution of diverse nonlinear problems, with well-known examples including the split-step and Newton’s methods for the dynamical and static settings, respectively. In all cases, the type of nonlinearity (quadratic, cubic, quartic, quintic, etc., as well as mixed and nonpolynomial ones), spatial dimensions, number of components in the system, and symmetry play important roles. It should also be stressed that nonlinear static and dynamical states take different forms in conservative and dissipative systems. Basic problems addressed by the numerical simulations of nonlinear systems are the stability of stationary states (particularly the stability of states with specific symmetry, such as multidimensional axisymmetric modes carrying vorticity), and the evolution of dynamical states (in particular, spontaneous symmetry breaking). This Special Issue is designed as a collection of original and review articles (including those focused on the methodological aspects of numerical simulations), which aim to present the state of the art in numerical studies of nonlinear phenomenology in diverse disciplines.

Prof. Dr. Boris Malomed
Dr. Hidetsugu Sakaguchi
Prof. Dr. David Laroze
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. Symmetry 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 2400 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

  • breathers
  • collapse
  • dynamical chaos
  • integrability
  • pattern formation
  • reaction–diffusion systems
  • rogue waves
  • shock waves
  • solitons
  • stability
  • topological dynamics
  • vortices
  • skyrmions
  • cardiac dynamics
  • biological applications

Published Papers (2 papers)

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Research

31 pages, 445 KiB  
Article
Borel Transform and Scale-Invariant Fractional Derivatives United
by Simon Gluzman
Symmetry 2023, 15(6), 1266; https://doi.org/10.3390/sym15061266 - 15 Jun 2023
Cited by 1 | Viewed by 1697
Abstract
The method of Borel transformation for the summation of asymptotic expansions with the power-law asymptotic behavior at infinity is combined with elements of scale-invariant fractional analysis with the goal of calculating the critical amplitudes. The fractional order of specially designed scale-invariant fractional derivatives [...] Read more.
The method of Borel transformation for the summation of asymptotic expansions with the power-law asymptotic behavior at infinity is combined with elements of scale-invariant fractional analysis with the goal of calculating the critical amplitudes. The fractional order of specially designed scale-invariant fractional derivatives u is used as a control parameter to be defined uniquely from u-optimization. For resummation of the transformed expansions, we employed the self-similar iterated roots. We also consider a complementary optimization, called b-optimization with the number of iterations b as an alternative fractional control parameter. The method of scale-invariant Fractional Borel Summation consists of three constructive steps. The first step corresponds to u-optimization of the amplitudes with fixed parameter b. When the first step fails, the second step corresponds to b-optimization of the amplitudes with fixed parameter u. However, when the two steps fail, the third step corresponds to the simplified, Borel-light technique. The marginal amplitude should be found by means of the self-similar iterated roots constructed for the transformed series, optimized with either of the two above approaches and corrected with a diagonal Padé approximants. The examples are given when the complementary optimizations,“horses-for-courses” approach outperforms other analytical methods in calculation of critical amplitudes. Full article
(This article belongs to the Special Issue Nonlinear Science and Numerical Simulation with Symmetry)
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17 pages, 814 KiB  
Article
Spot–Ladder Selection of Dislocation Patterns in Metal Fatigue
by Hiroyuki Shima, Yoshitaka Umeno and Takashi Sumigawa
Symmetry 2023, 15(5), 1028; https://doi.org/10.3390/sym15051028 - 05 May 2023
Cited by 2 | Viewed by 974
Abstract
Spontaneous pattern formation by a large number of dislocations is commonly observed during the initial stages of metal fatigue under cyclic straining. It was experimentally found that the geometry of the dislocation pattern undergoes a crossover from a 2D spot-scattered pattern to a [...] Read more.
Spontaneous pattern formation by a large number of dislocations is commonly observed during the initial stages of metal fatigue under cyclic straining. It was experimentally found that the geometry of the dislocation pattern undergoes a crossover from a 2D spot-scattered pattern to a 1D ladder-shaped pattern as the amplitude of external shear strain increases. However, the physical mechanism that causes the crossover between different dislocation patterns remains unclear. In this study, we theorized a bifurcation diagram that explains the crossover between the two dislocation patterns. The proposed theory is based on a weakly nonlinear stability analysis that considers the mutual interaction of dislocations as a nonlinearity. It was found that the selection rule among the two dislocation patterns, “spotted” and “ladder-shaped”, can be described by inequalities with respect to nonlinearity parameters contained in the governing equations. Full article
(This article belongs to the Special Issue Nonlinear Science and Numerical Simulation with Symmetry)
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