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Mathematics
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Mathematical Methods in Nonlinear Waves and Dynamical Systems
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Mathematical Methods in Nonlinear Waves and Dynamical Systems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (29 February 2020) | Viewed by 12392

Special Issue Editor

Prof. Dr. Hadi Susanto

E-Mail Website
Guest Editor
Department of Mathematical Sciences, University of Essex, Colchester CO4 3SQ, UK
Interests: dynamical systems; applied analysis; ordinary and partial differential equations; nonlinear waves; stability theory; pattern formation

Special Issue Information

Dear Colleagues,

This Special Issue, "Mathematical Methods in Nonlinear Waves and Dynamical Systems" is open for submissions and welcomes research and review papers on the development of mathematical methods in the study of nonlinear waves and the broad interdisciplinary area of dynamical systems, including coherent structures, chaos, and nonlinear dynamics.
Recent progress in the natural sciences, such as the study of non-Hermitian physics, nonlocal differential equations, differential equations on metric graphs, dynamics of complex networks, etc., require new approaches and novel applications of established mathematical tools. This Special Issue aims at hosting a representative sample of works on these advances

Prof. Dr. Hadi Susanto
Guest Editor

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. Mathematics is an international peer-reviewed open access semimonthly 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 waves
  • Solitary waves
  • Coherent structures
  • Pattern formation
  • Oscillators
  • Chaos
  • Asymptotic methods
  • Variational formulations
  • Stability theory

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Published Papers (5 papers)

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Research

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12 pages, 265 KiB  
Article
Justification of the Lugiato-Lefever Model from a Damped Driven ϕ4 Equation
by Fiki Taufik Akbar, Bobby Eka Gunara and Hadi Susanto
Mathematics 2020, 8(5), 727; https://doi.org/10.3390/math8050727 - 6 May 2020
Viewed by 2301
Abstract
The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrödinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained much attention as it has [...] Read more.
The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrödinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained much attention as it has become the basic model describing microresonator (Kerr) frequency combs. Recent works derive the Lugiato-Lefever equation from a class of damped driven ϕ 4 equations closed to resonance. In this paper, we provide a justification of the envelope approximation. From the analysis point of view, the result is novel and non-trivial as the drive yields a perturbation term that is not square integrable. The main approach proposed in this work is to decompose the solutions into a combination of the background and the integrable component. This paper is the first part of a two-manuscript series. Full article
(This article belongs to the Special Issue Mathematical Methods in Nonlinear Waves and Dynamical Systems)
16 pages, 904 KiB  
Article
Dynamical Behavior of a Modified Leslie–Gower One Prey–Two Predators with Competition
by Dian Savitri, Agus Suryanto, Wuryansari Muharini Kusumawinahyu and Abadi
Mathematics 2020, 8(5), 669; https://doi.org/10.3390/math8050669 - 28 Apr 2020
Cited by 10 | Viewed by 2783
Abstract
We study the dynamics of a modified Leslie–Gower one prey–two predators model with competition between predator populations. The model describes complex dynamics in the permanence, global stability and bifurcation. It is shown that there are eight possible equilibrium states. Two equilibrium states, i.e., [...] Read more.
We study the dynamics of a modified Leslie–Gower one prey–two predators model with competition between predator populations. The model describes complex dynamics in the permanence, global stability and bifurcation. It is shown that there are eight possible equilibrium states. Two equilibrium states, i.e., the extinction of all of the species state and the extinction of both predators state are always unstable, while the other equilibrium states are conditionally locally and globally asymptotically stable. We also analyzed numerically the effect of competition between predators. Our numerical simulations showed that the competition rate of the second-predator may induce the transcritical bifurcation, the saddle-node bifurcation as well as the bi-stability phenomenon. Such numerical results are consistent with the analytical results. Full article
(This article belongs to the Special Issue Mathematical Methods in Nonlinear Waves and Dynamical Systems)
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12 pages, 2564 KiB  
Article
Synchronization of Butterfly Fractional Order Chaotic System
by Michal Fečkan, T. Sathiyaraj and JinRong Wang
Mathematics 2020, 8(3), 446; https://doi.org/10.3390/math8030446 - 19 Mar 2020
Cited by 15 | Viewed by 2542
Abstract
In this paper, we study the synchronization of a nonlinear fractional system, and analyze its time response and chaotic behaviors. We represent a solution for considered system by employing the Mittag-Leffler matrix function and Jacobian matrix. Thereafter, we prove synchronization of error system [...] Read more.
In this paper, we study the synchronization of a nonlinear fractional system, and analyze its time response and chaotic behaviors. We represent a solution for considered system by employing the Mittag-Leffler matrix function and Jacobian matrix. Thereafter, we prove synchronization of error system between drive-response systems using stability theory and linear feedback control methods. Finally, numerical simulations are presented to show the effectiveness of the theoretical results. Full article
(This article belongs to the Special Issue Mathematical Methods in Nonlinear Waves and Dynamical Systems)
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12 pages, 775 KiB  
Article
A Novel Decay Rate for a Coupled System of Nonlinear Viscoelastic Wave Equations
by Khaled Zennir and Sultan S. Alodhaibi
Mathematics 2020, 8(2), 203; https://doi.org/10.3390/math8020203 - 6 Feb 2020
Cited by 8 | Viewed by 2100
Abstract
The main goal of the present paper is to study the existence, uniqueness and behavior of a solution for a coupled system of nonlinear viscoelastic wave equations with the presence of weak and strong damping terms. Owing to the Faedo-Galerkin method combined with [...] Read more.
The main goal of the present paper is to study the existence, uniqueness and behavior of a solution for a coupled system of nonlinear viscoelastic wave equations with the presence of weak and strong damping terms. Owing to the Faedo-Galerkin method combined with the contraction mapping theorem, we established a local existence in [ 0 , T ] . The local solution was made global in time by using appropriate a priori energy estimates. The key to obtaining a novel decay rate is the convexity of the function χ , under the special condition of the initial energy E ( 0 ) . The condition of the weights of weak and strong damping has a fundamental role in the proof. The existence of both three different damping mechanisms and strong nonlinear sources make the paper very interesting from a mathematics point of view, especially when it comes to unbounded spaces such as R n . Full article
(This article belongs to the Special Issue Mathematical Methods in Nonlinear Waves and Dynamical Systems)

Review

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14 pages, 261 KiB  
Review
Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs
by Riccardo Adami, Filippo Boni and Alice Ruighi
Mathematics 2020, 8(4), 617; https://doi.org/10.3390/math8040617 - 17 Apr 2020
Cited by 9 | Viewed by 2031
Abstract
We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrödinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff’s), is non-trivially interacting. This [...] Read more.
We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrödinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff’s), is non-trivially interacting. This category includes Dirac’s delta conditions, delta prime, Fülöp-Tsutsui, and others. Full article
(This article belongs to the Special Issue Mathematical Methods in Nonlinear Waves and Dynamical Systems)
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