Applied Nonlinear Dynamical Systems in Mathematical Physics

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

Deadline for manuscript submissions: 31 December 2024 | Viewed by 5799

Special Issue Editor


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Guest Editor
Department of Economics, National and Kapodistrian University of Athens, Athens, Greece
Interests: mathematical physics; quantum gravity; quantum mechanics; representation theory; stochastic models in mathematical finance; functional analysis; linear algebra; dynamical systems; differential galois theory

Special Issue Information

Dear Colleagues,

Nonlinear phenomena that do not obey the superposition principle appear everywhere in nature. Their description and understanding is therefore of great interest from both theoretical and applicative points of view. Maxwell equations are linear, whereas Einstein equations are highly nonlinear. The nonlinear stability of black hole space-times is a fundamental problem in general relativity whose solution depends on the nonlinear structure of Einstein equations. However, even in the case of Maxwell equations, the behavior of light in nonlinear media gives rise to a host of new nonlinear phenomena which comprise the field of nonlinear optics. Finding solutions to nonlinear equations, be it the nonlinear Schrodinger equation, the equations appearing in nonlinear optics, Einstein equations, Yang–Mills equations, nonlinear wave equations or the nonlinear equations describing the behavior of complex fluids, is difficult. The employment of group theoretic methods facilitates the discovery of solutions to these equations and in generating new solutions from the old. Numerical techniques used to cope with the nonlinear character of the fundamental equations of mathematical physics have provided magnificent results and simulations that offer new insights into the understanding of phenomena. Nonlinear phenomena in gauge and gravity theories are not amenable to perturbation analysis and are essential for their understanding. A class of solutions which explicate such nonlinear phenomena includes solitons, kinks, instantons, and breathers. Such solutions have applications well beyond gauge and gravity theories. They have entered and changed such diverse fields as nonlinear optics, plasma physics, and ocean, atmospheric, and planetary sciences.  If a nonlinear phenomenon can be represented by an integrable system, the existence of sufficiently conserved quantities or first integrals restricts its evolution to a submanifold within its phase space. An interesting example of this kind is the Liouville–Arnold integrability. Integrability structures which emerge in gauge theories and string theory are of fundamental importance for their understanding.

Research areas may include (but are not limited to) the following:

  • Nonlinear equations of mathematical physics;
  • Nonlinear Schrodinger equation;
  • Difference equations and functional equations in mathematical physics;
  • (Lie) group theoretic methods for the solution of differential and difference equations;
  • Integrability and non-integrability, scattering/inverse scattering, and Painlevé analysis;
  • Lie-algebraic characterizations of integrability;
  • Tensor Poisson structures;
  • Classical and quantum many-body problems;
  • Geometry and algebra of nonlinear equations;
  • Integrability in gauge and string theory;
  • Monopoles, solitons, kinks, breathers, and instantons in gauge theory and in gravity theory;
  • Symmetries and invariants in mathematical physics;
  • Applications of Lie groups, Lie algebras, and higher structures;
  • Quasi-exactly solvable models in quantum mechanics;
  • Nonlinear stability of black hole space-times;
  • Complex fluids;
  • Nonlinear waves;
  • Nonlinear optics;
  • Numerical simulations in mathematical physics. 

Dr. Evangelos Melas
Guest Editor

Manuscript Submission Information

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Keywords

  • nonlinear equations of mathematical physics
  • nonlinear schrodinger equation
  • difference equations and functional equations in mathematical physics
  • (lie) group theoretic methods for the solution of differential and difference equations
  • lie-algebraic characterizations of integrability
  • tensor poisson structures
  • classical and quantum many-body problems
  • geometry and algebra of nonlinear equations
  • integrability in gauge and string theory
  • symmetries and invariants in mathematical physics
  • quasi-exactly solvable models in quantum mechanics
  • nonlinear stability of black hole space-times
  • complex fluids
  • nonlinear waves
  • nonlinear optics
  • numerical simulations in mathematical physics
 

Published Papers (6 papers)

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Research

15 pages, 658 KiB  
Article
Real and Complex Potentials as Solutions to Planar Inverse Problem of Newtonian Dynamics
by Thomas Kotoulas
Axioms 2024, 13(2), 88; https://doi.org/10.3390/axioms13020088 - 29 Jan 2024
Viewed by 707
Abstract
We study the motion of a test particle in a conservative force field. In the framework of the 2D inverse problem of Newtonian dynamics, we find 2D potentials that produce a preassigned monoparametric family of regular orbits [...] Read more.
We study the motion of a test particle in a conservative force field. In the framework of the 2D inverse problem of Newtonian dynamics, we find 2D potentials that produce a preassigned monoparametric family of regular orbits f(x,y)=c on the xy-plane (where c is the parameter of the family of orbits). This family of orbits can be represented by the “slope functionγ=fyfx uniquely. A new methodology is applied to the basic equation of the planar inverse problem in order to find potentials of a special form, i.e., V=F(x+y)+G(xy), V=F(x+iy)+G(xiy) and V=P(x)+Q(y), and polynomial ones. According to this methodology, we impose differential conditions on the family of orbits f(x,y) = c. If they are satisfied, such a potential exists and it is found analytically. For known families of curves, e.g., circles, parabolas, hyperbolas, etc., we find potentials that are compatible with them. We offer pertinent examples that cover all the cases. The case of families of straight lines is referred to. Full article
(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)
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10 pages, 230 KiB  
Article
Solution of High-Order Nonlinear Integrable Systems Using Darboux Transformation
by Xinhui Wu, Jiawei Hu and Ning Zhang
Axioms 2023, 12(11), 1032; https://doi.org/10.3390/axioms12111032 - 3 Nov 2023
Viewed by 835
Abstract
The 4×4 trace-free complex matrix set is introduced in this study. By using it, we are able to create a novel soliton hierarchy that is reduced to demonstrate its bi-Hamiltonian structure. Additionally, we give the Darboux matrix T, whose elements are [...] Read more.
The 4×4 trace-free complex matrix set is introduced in this study. By using it, we are able to create a novel soliton hierarchy that is reduced to demonstrate its bi-Hamiltonian structure. Additionally, we give the Darboux matrix T, whose elements are connected to the spectral parameter in accordance with the various positions and numbers of the spectral parameter λ. The Darboux transformation approach has also been successfully applicated to superintegrable systems. Full article
(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)
18 pages, 1941 KiB  
Article
On the Propagation Model of Two-Component Nonlinear Optical Waves
by Aleksandr O. Smirnov and Eugeni A. Frolov
Axioms 2023, 12(10), 983; https://doi.org/10.3390/axioms12100983 - 18 Oct 2023
Viewed by 850
Abstract
Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference [...] Read more.
Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. To construct the hierarchical equations, we use the monodromy matrix method, as first proposed by B.A. Dubrovin. The method we use consists of solving the following sequence of problems. First, using the Lax operator, we find the monodromy matrix, which is a polynomial in the spectral parameter. More precisely, we find a sequence of monodromy matrices dependent on the degree of this polynomial. Each Lax operator has its own sequence of monodromy matrices. Then, using the terms from the decomposition of the monodromy matrix, we construct a sequence of second operators from a Lax pair. A hierarchy of evolutionary integrable nonlinear equations follows from the conditions of compatibility of the sequence of Lax pairs. Also, knowledge of the monodromy matrix allows us to find stationary equations that are analogs of the Novikov equations for the Korteweg–de Vries equation. In addition, the characteristic equation of the monodromy matrix corresponds to the spectral curve equation of the relevant multiphase solution for the integrable nonlinear equation. Since the coefficients of the spectral curve equation are integrals of the hierarchical equations, they can be utilized to find the simplest solutions of the constructed integrable nonlinear equations. In this paper, we demonstrate the operation of this method, starting with the assignment of the Lax operator and ending with the construction of the simplest solutions. Full article
(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)
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7 pages, 235 KiB  
Article
Causality in Scalar-Einstein Waves
by Mark D. Roberts
Axioms 2023, 12(9), 843; https://doi.org/10.3390/axioms12090843 - 30 Aug 2023
Viewed by 520
Abstract
A wavelike scalar-Einstein solution is found and indicating vectors constructed from the Bel-Robinson tensor are used to study which objects co-move with the wave and whether gravitational energy transfer is null. It is found that this Bel-Robinson energy criteria gives null energy transfer [...] Read more.
A wavelike scalar-Einstein solution is found and indicating vectors constructed from the Bel-Robinson tensor are used to study which objects co-move with the wave and whether gravitational energy transfer is null. It is found that this Bel-Robinson energy criteria gives null energy transfer for both the vacuum plane wave and the scalar plane wave. Full article
(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)
21 pages, 2003 KiB  
Article
The Zakharov–Shabat Spectral Problem for Complexification and Perturbation of the Korteweg–de Vries Equation
by Tatyana V. Redkina, Arthur R. Zakinyan and Robert G. Zakinyan
Axioms 2023, 12(7), 703; https://doi.org/10.3390/axioms12070703 - 19 Jul 2023
Viewed by 822
Abstract
In this paper we consider examples of complex expansion (cKdV) and perturbation (pKdV) of the Korteweg–de Vries equation (KdV) and show that these equations have a representation in the form of the zero-curvature equation. In this case, we use the Lie algebra of [...] Read more.
In this paper we consider examples of complex expansion (cKdV) and perturbation (pKdV) of the Korteweg–de Vries equation (KdV) and show that these equations have a representation in the form of the zero-curvature equation. In this case, we use the Lie algebra of 4-dimensional quadratic nilpotent matrices. Moreover, it is shown that the simplest possible matrix representation of this algebra leads to the possibility of constructing a countable number of conservation laws for these equations. Full article
(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)
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11 pages, 4838 KiB  
Article
On the Wave Structures to the (3+1)-Dimensional Boiti–Leon–Manna–Pempinelli Equation in Incompressible Fluid
by Yan-Nan Chen and Kang-Jia Wang
Axioms 2023, 12(6), 519; https://doi.org/10.3390/axioms12060519 - 25 May 2023
Cited by 1 | Viewed by 845
Abstract
In the present study, two effective methods, the Exp-function method and He’s frequency formulation, are employed to investigate the dynamic behaviors of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, which is used widely to describe the incompressible fluid. A variety of the wave structures, including the [...] Read more.
In the present study, two effective methods, the Exp-function method and He’s frequency formulation, are employed to investigate the dynamic behaviors of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, which is used widely to describe the incompressible fluid. A variety of the wave structures, including the dark wave, bright-dark wave and periodic wave solutions, are successfully constructed. Compared with the results attained by the methods, the obtained solutions are all new and have not been presented in the other literature. The diverse wave structures of the solutions are presented through numerical results in the form of three-dimensional plots and two-dimensional curves. It reveals that the proposed methods are powerful and straightforward, which are expected to be helpful for the study of travelling-wave theory in fluid. Full article
(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)
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