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Symmetry
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Nonlinear Science and Numerical Simulation with Symmetry
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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: 30 November 2025 | Viewed by 8139

Special Issue Editors

Prof. Dr. Boris Malomed

E-Mail Website
Guest Editor
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
Special Issues, Collections and Topics in MDPI journals
Dr. Hidetsugu Sakaguchi

E-Mail Website
Guest Editor
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)
Prof. Dr. David Laroze

E-Mail Website
Guest Editor
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

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (6 papers)

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Research

11 pages, 262 KiB  
Article
The Exact Traveling Wave Solutions of a KPP Equation
by Eugene Kogan
Symmetry 2024, 16(12), 1621; https://doi.org/10.3390/sym16121621 - 6 Dec 2024
Viewed by 483
Abstract
We obtain the exact analytical traveling wave solutions of the Kolmogorov–Petrovskii–Piskunov equation, with the reaction term belonging to the class of functions, which includes that of the (generalized) Fisher equation, for the particular values of the wave’s speed. Additionally we obtain the exact [...] Read more.
We obtain the exact analytical traveling wave solutions of the Kolmogorov–Petrovskii–Piskunov equation, with the reaction term belonging to the class of functions, which includes that of the (generalized) Fisher equation, for the particular values of the wave’s speed. Additionally we obtain the exact analytical traveling wave solutions of the generalized Burgers–Huxley equation. Full article
(This article belongs to the Special Issue Nonlinear Science and Numerical Simulation with Symmetry)
18 pages, 37584 KiB  
Article
Breather Bound States in a Parametrically Driven Magnetic Wire
by Camilo José Castro, Ignacio Ortega-Piwonka, Boris A. Malomed, Deterlino Urzagasti, Liliana Pedraja-Rejas, Pablo Díaz and David Laroze
Symmetry 2024, 16(12), 1565; https://doi.org/10.3390/sym16121565 - 22 Nov 2024
Viewed by 555
Abstract
We report the results of a systematic investigation of localized dynamical states in the model of a one-dimensional magnetic wire, which is based on the Landau–Lifshitz–Gilbert (LLG) equation. The dissipative term in the LLG equation is compensated by the parametric drive imposed by [...] Read more.
We report the results of a systematic investigation of localized dynamical states in the model of a one-dimensional magnetic wire, which is based on the Landau–Lifshitz–Gilbert (LLG) equation. The dissipative term in the LLG equation is compensated by the parametric drive imposed by the external AC magnetic field, which is uniformly applied perpendicular to the rectilinear wire. The existence and stability of the localized states is studied in the plane of the relevant control parameters, namely, the amplitude of the driving term and the detuning of its frequency from the parametric resonance. With the help of systematically performed simulations of the LLG equation, the existence and stability areas are identified in the parameter plane for several species of the localized states: stationary single- and two-soliton modes, single and double breathers, drifting double breathers with spontaneously broken inner symmetry, and multisoliton complexes. Multistability occurs in this system. The breathers emit radiation waves (which explains their drift caused by the spontaneous symmetry breaking, as it breaks the balance between the recoil from the waves emitted to left and right), while the multisoliton complexes exhibit cycles of periodic transitions between three-, five-, and seven-soliton configurations. Dynamical characteristics of the localized states are systematically calculated too. These include, in particular, the average velocity of the asymmetric drifting modes, and the largest Lyapunov exponent, whose negative and positive values imply that the intrinsic dynamics of the respective modes is regular or chaotic, respectively. Full article
(This article belongs to the Special Issue Nonlinear Science and Numerical Simulation with Symmetry)
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9 pages, 614 KiB  
Article
Manifestation of Superfluidity in Atom-Number-Imbalanced Two-Component Bose–Einstein Condensates
by Saeed Majed Al-Marzoug, Bakhtiyor Baizakov, Usama Al Khawaja and Hocine Bahlouli
Symmetry 2024, 16(7), 910; https://doi.org/10.3390/sym16070910 - 17 Jul 2024
Viewed by 728
Abstract
Superfluid and dissipative regimes in the dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate (BEC) with unequal atom numbers in the two components have been explored. The system supports localized waves of the symbiotic type owing to the same-species repulsion and cross-species attraction. The [...] Read more.
Superfluid and dissipative regimes in the dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate (BEC) with unequal atom numbers in the two components have been explored. The system supports localized waves of the symbiotic type owing to the same-species repulsion and cross-species attraction. The minority BEC component moves through the majority component and creates excitations. To quantify the emerging excitations, we introduce a time-dependent function called disturbance. Through numerical simulations of the coupled Gross–Pitaevskii equations with periodic boundary conditions, we have identified a critical velocity of the localized wave, above which a transition from the superfluid to dissipative regime occurs, as evidenced by a sharp increase in the disturbance function. The factors responsible for the discrepancy between the actual critical velocity and the speed of sound, expected from theoretical arguments, have been discussed. Full article
(This article belongs to the Special Issue Nonlinear Science and Numerical Simulation with Symmetry)
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17 pages, 767 KiB  
Article
Approximation Conjugate Gradient Method for Low-Rank Matrix Recovery
by Zhilong Chen, Peng Wang and Detong Zhu
Symmetry 2024, 16(5), 547; https://doi.org/10.3390/sym16050547 - 2 May 2024
Viewed by 1118
Abstract
Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. [...] Read more.
Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. In this paper, we propose an approximation conjugate gradient method for solving the low-rank matrix recovery problem, i.e., the low-rank matrix is obtained to replace the original symmetric and asymmetric matrices such that the approximation error is the smallest. The conjugate gradient search direction is given through matrix addition and matrix multiplication. The new conjugate gradient update parameter is given by the F-norm of matrix and the trace inner product of matrices. The conjugate gradient generated by the algorithm avoids SVD decomposition. The backtracking linear search is used so that the approximation conjugate gradient direction is computed only once, which ensures that the objective function decreases monotonically. The global convergence and local superlinear convergence of the algorithm are given. The numerical results are reported and show the effectiveness of the algorithm. Full article
(This article belongs to the Special Issue Nonlinear Science and Numerical Simulation with Symmetry)
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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 2 | Viewed by 2662
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 - 5 May 2023
Cited by 2 | Viewed by 1586
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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