Article
Mathematics, Applied
Hanze Liu, Cheng-Lin Bai, Xiangpeng Xin
Summary: In this paper, the combination of Painleve analysis and symmetry classification was used to investigate reaction-diffusion equations, obtaining Painleve properties and Backlund transformations under certain conditions. The point symmetries of the equations were determined using Lie group classification method, and the complete generalized symmetry classifications of the general R-D equation were provided based on predetermined order characteristics. Additionally, exact solutions to the equations derived from Painleve expansions and symmetry reductions were examined.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Applied
Alexander G. Rasin, Jeremy Schiff
Summary: In this paper, we revisit the symmetry structure of integrable partial differential equations (PDEs) by studying the specific example of the KdV equation. We identify four nonlocal symmetries, known as generating symmetries, depending on a parameter in KdV. We explain that the commutator algebra of these nonlocal symmetries is not uniquely determined, and we present three possibilities for the algebra. We demonstrate that three of the four symmetries can be regarded as infinitesimal double Backlund transformations. These generating symmetries encompass all known symmetries of the KdV equation and exhibit novel structures due to their nonlocality. We propose that this structure is also present in other integrable PDEs.
JOURNAL OF NONLINEAR SCIENCE
(2022)
Article
Engineering, Mechanical
Sandeep Malik, Sachin Kumar, Amiya Das
Summary: In this study, the (2+1)-dimensional combined Korteweg-de Vries and modified Korteweg-de Vries equation is considered for the first time. Lie symmetries are generated and corresponding transformations are used to reduce the equation to ordinary differential equations. Various soliton solutions are constructed using different techniques, and the stability of the corresponding dynamical system is investigated using phase plane theory.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Applied
Philip Rosenau, Alexander Oron
Summary: We study a class of sublinear Gardner equations that exhibit compactons with finite span and richer morphology compared to unidirectional KdV-like equations. The multi-dimensional extension of the equations reveals a spectrum of multi-nodal compactons.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
Amiya Das, Uttam Kumar Mandal
Summary: This study focuses on the mathematical modeling of the KdV equation, investigating integrability through Painleve analysis and constructing Backlund transformations and conservation laws. Soliton solutions are obtained using various methods, demonstrating their propagation and interaction.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Physics, Multidisciplinary
Ozlem Yesiltas, Ikram Imane Kouachi
Summary: In this paper, the Dirac equation in the presence of PT / non-PT-symmetric potential interactions on a two dimensional gravitational static background with position-dependent mass has been studied. The exponential metric component allows the reduction of the Dirac operator to a general supersymmetric model with mass changing along coordinates. By using Lie algebras and supersymmetric quantum mechanical approaches, the eigenvalues of the Dirac operator for complex Morse and trigonometric complex Scarf-II potentials SL(2, C) are obtained. Additionally, a general Sturm-Liouville type equation is derived through a convenient mapping, which enables the study of the system within ?-pseudo-Hermiticity. The ? operator is determined for the examples of complex trigonometric Rosen-Morse potential and complex Morse potentials with real and complex parameters, and the solutions are obtained with energy values and probability densities.
Article
Physics, Applied
Ijaz Ali, Aly R. Seadawy, Syed Tahir Raza Rizvi, Muhammad Younis
Summary: By utilizing the Painleve test, this paper aims to analyze integrability of three famous nonlinear models: unstable nonlinear Schrodinger equation (UNLSE), modified UNLSE (MUNLSE), and (2+1)-dimensional cubic NLSE (CNLSE). The non-appearance of certain singularities such as movable branch points suggests a high probability of complete integrability. If an NLSE passes the P-test, the model can be solved using the inverse scattering transformation (IST).
INTERNATIONAL JOURNAL OF MODERN PHYSICS B
(2021)
Article
Mathematics
Nikolay A. Kudryashov, Sofia F. Lavrova
Summary: The Chavy-Waddy-Kolokolnikov model for bacterial colonies is studied. The Painleve test is used to determine if the mathematical model is integrable, providing restrictions on the parameters. The inverse scattering transform method is found ineffective for solving the Cauchy problem due to the requirement of stationary solutions. The stability of stationary points and construction of periodic and solitary solutions are also explored.
Article
Mathematics
Nan Liu, Boling Guo
Summary: The study applies the method of nonlinear steepest descent to analyze the long-time asymptotics of an extended modified KdV equation with decaying initial data in two transition regions, resulting in different asymptotic expressions in each region.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Hanze Liu, Cheng-Lin Bai, Xiangpeng Xin
Summary: In this paper, the authors combined Painleve analysis and Lie group method to investigate the nonlinear phi(4) type equations, obtaining the Painleve properties and Backlund transformations under certain conditions. The point symmetries of the generalized phi(4) model were determined using Lie group classification method, and exact solutions to the equations were explored through symmetry reductions and analytic methods.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Dig Vijay Tanwar, Abdul-Majid Wazwaz
Summary: This article investigates the nonlinear behavior of ion acoustic waves in a plasma with superthermal electrons and isothermal positrons. It analyzes the Lie symmetries and exact solutions of the KdV-Burgers equation, and discusses the characteristics of the solutions through numerical simulations.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Interdisciplinary Applications
O. M. Kiselev
Summary: Solutions of the perturbed Painleve-2 equation are used to describe a dynamic bifurcation of soft loss of stability. The bifurcation boundary has a spiral structure, and the equations of modulation of this boundary are obtained. The boundary separates solutions of different types.
CHAOS SOLITONS & FRACTALS
(2021)
Article
Mathematics, Applied
Wen-Bo Bo, Ru-Ru Wang, Wei Liu, Yue-Yue Wang
Summary: The symmetry breaking of solitons in the nonlinear Schrodinger equation with cubic-quintic competing nonlinearity and parity-time symmetric potential is studied. It is found that symmetric fundamental solitons and symmetric tripole solitons tend to be stable, while asymmetric solitons are unstable in both high and low power regions. Increasing saturable nonlinearity widens the stability region of fundamental symmetric solitons and symmetric tripole solitons.
Article
Physics, Multidisciplinary
Geza Levai
Summary: In this study, exactly solvable potentials are derived from the formal solutions of the confluent Heun equation, with conditions for PT symmetry identified. The symmetrical canonical form of the equation is found to be more suitable for implementing PT symmetry. The potentials obtained depend on twelve parameters and include Natanzon-class potentials as special cases. Comparison with earlier research shows the importance of considering the symmetrical canonical form for PT symmetry.
Article
Mathematics
K. Krishnakumar, A. Durga Devi, V Srinivasan, P. G. L. Leach
Summary: In this study, the symmetry and integrability of a Generalized Modified Camassa-Holm Equation (GMCH) were investigated. It was observed that the nonlinearity of the equation rapidly increases as the parameter n increases. However, despite this, the equation family exhibits similar symmetries and other characteristics. It was also shown that the resultant second-order nonlinear ODE generated from the GMCH equation is linearizable and that the family passes the Painleve Test.
INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS
(2023)
Article
Physics, Mathematical
Andreas Fring, Samuel Whittington
LETTERS IN MATHEMATICAL PHYSICS
(2020)
Article
Physics, Multidisciplinary
Andreas Fring, Rebecca Tenney
EUROPEAN PHYSICAL JOURNAL PLUS
(2020)
Article
Physics, Multidisciplinary
Julia Cen, Francisco Correa, Andreas Fring
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2020)
Article
Astronomy & Astrophysics
Andreas Fring, Takano Taira
Article
Physics, Multidisciplinary
Andreas Fring, Takanobu Taira
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2020)
Article
Physics, Multidisciplinary
Andreas Fring, Rebecca Tenney
Summary: The study proposes a perturbative method to determine the time-dependent Dyson map and metric operator associated with time-dependent non-Hermitian Hamiltonians. By applying the method to different systems, it demonstrates a recursive constructive approach to solving coupled differential equations and reveals a set of inequivalent solutions for the Dyson maps and metric operators, leading to different physical behavior.
Review
Physics, Mathematical
Andreas Fring, Samuel Whittington
Summary: This study proposes construction principles for new classes of Toda field theories based on root systems defined on Lorentzian lattices. The Painleve integrability test reveals that most of these Lorentzian Toda field theories are not integrable, with only one integer valued resonance found. The classical mass spectra of several massive variants are analyzed in detail, showing that Lorentzian Toda field theories can be viewed as perturbed versions of integrable theories with an algebraic framework.
REVIEWS IN MATHEMATICAL PHYSICS
(2021)
Article
Physics, Multidisciplinary
Andreas Fring, Rebecca Tenney
Summary: Complex point transformations can be used to construct non-Hermitian first integrals, time-dependent Dyson maps and metric operators for non-Hermitian quantum systems. By applying the point transformation to the system, solutions to the corresponding equations can be obtained.
Article
Physics, Multidisciplinary
Andreas Fring, Rebecca Tenney
Summary: The study suggests a scheme that leads to an infinite series of time-dependent Dyson maps, which associate different Hermitian Hamiltonians with a uniquely specified time-dependent non-Hermitian Hamiltonian. Through identifying the underlying symmetries and utilizing Lewis-Riesenfeld invariants, the explicit construction of the Dyson maps and metric operators is facilitated. A specific example of a two-dimensional system of oscillators coupled in a non-Hermitian PT-symmetrical fashion is presented to illustrate the scheme.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2021)
Article
Physics, Multidisciplinary
Julia Cen, Francisco Correa, Andreas Fring, Takano Taira
Summary: This article investigates the stability of different types of soliton solutions in nonlocal integrable systems, and discovers various scenarios and phenomena.
Article
Physics, Multidisciplinary
Andreas Fring, Takanobu Taira
Summary: We investigate the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism in non-Hermitian field theories with local non-Abelian gauge symmetry in different regions of their parameter spaces. We find that the mechanism remains synchronized in all regimes characterized by a modified CPT symmetry, giving mass to gauge vector bosons while preventing the existence of massless Goldstone bosons. However, at the zero exceptional points, where the eigenvalues of the mass squared matrix vanish, the mechanism breaks down as the Goldstone bosons cannot be identified and the gauge vector bosons remain massless.
EUROPEAN PHYSICAL JOURNAL PLUS
(2022)
Article
Physics, Multidisciplinary
Andreas Fring, Takanobu Taira, Rebecca Tenney
Summary: This paper introduces time-dependent C(t)-operators and investigates their properties in relation to Lewis-Riesenfeld invariants. The application of these concepts in a non-Hermitian two-level matrix Hamiltonian demonstrates their practical working nature.
Article
Physics, Mathematical
Andreas Fring, Rebecca Tenney
Summary: In this study, a new method is proposed to construct Lewis-Riesenfeld invariants for coupled oscillator systems using two-dimensional point transformations. The invariants are obtained by solving a set of coupled differential equations, and it is demonstrated that point transformations can also be used to construct time-dependent Dyson maps.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Physics, Multidisciplinary
Andreas Fring, Takanobu Taira, Rebecca Tenney
Summary: We demonstrate the reality of the instantaneous energies by showing the existence of a Hermitian time-dependent intertwining operator that maps the non-Hermitian time-dependent energy operator to its Hermitian conjugate. This property holds in all three PT-regimes, including the time-independent PT-symmetric regime, the exceptional point, and the spontaneously broken PT-regime. We also propose a modified adiabatic approximation using instantaneous eigenstates of the energy operator, which always leads to real Berry phases.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2023)
Article
Astronomy & Astrophysics
Andreas Fring, Takanobu Taira
