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Axioms
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Geometry and Nonlinear Computations in Physics
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Geometry and Nonlinear Computations in Physics

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 April 2023) | Viewed by 5223

Special Issue Editor

Dr. Solomon Manukure

E-Mail Website
Guest Editor
Department of Mathematics, Florida A & M University, Tallahassee, FL 32307, USA
Interests: partial differential equations; nonlinear wave theory; integrable systems; soliton theory; Hamiltonian systems; algebro-geometric solutions; Riemann–Hilbert problem
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinearity is a very interesting concept in mathematical physics. Over the years, this phenomenon has attracted significant attention from many disciplines, including mathematics, physics, and engineering. This is because most real, physical systems are inherently nonliner in nature.

Space–time dynamics are usually modeled in terms of partial differential equations, which are often nonlinear. There are many advanced tools and techniques to study nonlinear PDEs that appear in physical and engineering sciences, some of which include the inverse scattering transform for Cauchy problems, symmetry methods, the Hamiltonian framework, and the Hirota bilinear method. There are also many geometric and numerical methods. Nevertheless, solving nonlinear partial differential equations poses substantial challenges.

The aim of this Special Issue is to highlight the important role played by geometric and computational techniques in solving and analyzing nonlinear systems arising in physics.  Submissions of original research and review articles from diverse areas are welcomed. The topics of this Special Issue include (but are not limited to):

  • Nonlinear equations of mathematical physics.
  • Riemann–Hilbert problems.
  • Quantum integrable systems.
  • Discrete integrable systems.
  • Applications of Lie group theory and Lie algebras to differential equations.
  • Integrability and nonintegrability, Painleve analysis.
  • Inverse scattering method.
  • Bifurcation techniques.
  • Symmetry and conservation laws.
  • Theory of solitons and solitary waves.
  • Long-time asymptotics and stability theory.
  • Differential geometry and mathematical physics.
  • Numerical methods in nonlinear wave theory.
  • Hamiltonian theory.
  • Inverse problems.

Dr. Solomon Manukure
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. Axioms 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

  • nonlinear wave equations
  • integrable systems
  • inverse scattering method
  • solitons
  • inverse problems

Published Papers (4 papers)

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Research

15 pages, 682 KiB  
Article
Kinematic Differential Geometry of a Line Trajectory in Spatial Movement
by Areej A. Almoneef and Rashad A. Abdel-Baky
Axioms 2023, 12(5), 472; https://doi.org/10.3390/axioms12050472 - 14 May 2023
Viewed by 927
Abstract
This paper investigates the kinematic differential geometry of a line trajectory in spatial movement. Specifically, we provide a theoretical expression of inflection line congruence, which is the spatial equivalent of the inflection circle of planar kinematics. Additionally, we introduce new proofs for the [...] Read more.
This paper investigates the kinematic differential geometry of a line trajectory in spatial movement. Specifically, we provide a theoretical expression of inflection line congruence, which is the spatial equivalent of the inflection circle of planar kinematics. Additionally, we introduce new proofs for the Euler–Savary and Disteli formulae and thoroughly analyze their spatial equivalence. Full article
(This article belongs to the Special Issue Geometry and Nonlinear Computations in Physics)
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16 pages, 1198 KiB  
Article
A Swarming Meyer Wavelet Computing Approach to Solve the Transport System of Goods
by Zulqurnain Sabir, Tareq Saeed, Juan L. G. Guirao, Juan M. Sánchez and Adrián Valverde
Axioms 2023, 12(5), 456; https://doi.org/10.3390/axioms12050456 - 08 May 2023
Viewed by 1172
Abstract
The motive of this work is to provide the numerical performances of the reactive transport model that carries trucks with goods on roads by exploiting the stochastic procedures based on the Meyer wavelet (MW) neural network. An objective function is constructed by using [...] Read more.
The motive of this work is to provide the numerical performances of the reactive transport model that carries trucks with goods on roads by exploiting the stochastic procedures based on the Meyer wavelet (MW) neural network. An objective function is constructed by using the differential model and its boundary conditions. The optimization of the objective function is performed through the hybridization of the global and local search procedures, i.e., swarming and interior point algorithms. Three different cases of the model have been obtained, and the exactness of the stochastic procedure is observed by using the comparison of the obtained and Adams solutions. The negligible absolute error enhances the exactness of the proposed MW neural networks along with the hybridization of the global and local search schemes. Moreover, statistical interpretations based on different operators, histograms, and boxplots are provided to validate the constancy of the designed stochastic structure. Full article
(This article belongs to the Special Issue Geometry and Nonlinear Computations in Physics)
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16 pages, 726 KiB  
Article
Monoparametric Families of Orbits Produced by Planar Potentials
by Thomas Kotoulas
Axioms 2023, 12(5), 423; https://doi.org/10.3390/axioms12050423 - 26 Apr 2023
Cited by 1 | Viewed by 795
Abstract
We study the motion of a test particle on the xy−plane. The particle trajectories are given by a one-parameter family of orbits f(x,y) = c, where c = const. By using the tools of [...] Read more.
We study the motion of a test particle on the xy−plane. The particle trajectories are given by a one-parameter family of orbits f(x,y) = c, where c = const. By using the tools of the 2D inverse problem of Newtonian dynamics, we find two-dimensional potentials that produce a pre-assigned monoparametric family of regular orbits f(x,y)=c that can be represented by the “slope function” γ=fyfx uniquely. We apply a new methodology in order to find potentials depending on specific arguments, i.e., potentials of the form V(x,y)=P(u) where u=x2+y2,xy,x3−y3,xy (x,y≠ 0). Then, we establish one differential condition for the family of orbits f(x,y) = c. If it is satisfied, it guarantees the existence of such a potential, generating the above family of planar orbits. Then, the potential function V=V(x,y) is found by quadratures. For known families of curves, e.g., ellipse, the logarithmic spiral, the lemniscate of Bernoulli, and circles, we find homogeneous and polynomial potentials that are compatible with this family of orbits. We offer pertinent examples that cover all of the cases, and we examine which of these potentials are integrable. We also study one-dimensional potentials. The families of straight lines in 2D space are also examined. Full article
(This article belongs to the Special Issue Geometry and Nonlinear Computations in Physics)
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13 pages, 6393 KiB  
Article
Various Solitons and Other Wave Solutions to the (2+1)-Dimensional Heisenberg Ferromagnetic Spin Chain Dynamical Model
by Feng Shi and Kang-Jia Wang
Axioms 2023, 12(4), 354; https://doi.org/10.3390/axioms12040354 - 03 Apr 2023
Cited by 4 | Viewed by 1163
Abstract
This paper outlines a study into the exact solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation that is used to illustrate the ferromagnetic materials of magnetic ordering by applying two recent techniques, namely, the Sardar-subequation method and extended rational sine–cosine and sinh–cosh [...] Read more.
This paper outlines a study into the exact solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation that is used to illustrate the ferromagnetic materials of magnetic ordering by applying two recent techniques, namely, the Sardar-subequation method and extended rational sine–cosine and sinh–cosh methods. Abundant exact solutions such as the bright soliton, dark soliton, combined bright–dark soliton, singular soliton and other periodic wave solutions expressed by the generalized trigonometric, generalized hyperbolic, trigonometric and hyperbolic functions are obtained. The numerical results are illustrated in the form of 3D plots, 2D contours and 2D curves by choosing proper parametric values to interpret the physical behavior of the model. The obtained results in this work are expected to provide a rich platform for constructing the soliton solutions of PDEs in physics. Full article
(This article belongs to the Special Issue Geometry and Nonlinear Computations in Physics)
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