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12 pages, 1101 KB

Open AccessArticle

Resonant Solutions and Rogue Wave Solutions to the (2+1)-Dimensional Caudrey–Dodd–Gibbon Equation

by Yanmei Sun, Linlin Gui and Yufeng Zhang

Symmetry 2026, 18(2), 332; https://doi.org/10.3390/sym18020332 - 11 Feb 2026

Viewed by 265

The (2+1)-dimensional Caudrey–Dodd–Gibbon (CDG) equation, which can frequently be used to be describe the propagations of shallow-water waves and plasma physics, is solved by various methods in this paper, thereby revealing its nonlinear dynamical behavior. First, through the linear superposition principle in conjunction [...] Read more.

The (2+1)-dimensional Caudrey–Dodd–Gibbon (CDG) equation, which can frequently be used to be describe the propagations of shallow-water waves and plasma physics, is solved by various methods in this paper, thereby revealing its nonlinear dynamical behavior. First, through the linear superposition principle in conjunction with symmetric Hirota bilinear method, we obtain a bilinear form of the CDG equation, which possesses several symmetry properties, and construct the resonant solutions to exponential waves. Then, the one-rogue wave solutions to the CDG equation are constructed via the ansatz method. Finally, we show three-dimensional diagrams and density graphs of the yielded solutions to better show the dynamic characteristics. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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25 pages, 506 KB

Open AccessArticle

Solution Dynamics of the (1 + 1)-Dimensional Fisher’s Equation Using Lie Symmetry Analysis

by Phillipos Masindi and Lazarus Rundora

Symmetry 2026, 18(2), 279; https://doi.org/10.3390/sym18020279 - 3 Feb 2026

Viewed by 546

Abstract

Reaction–diffusion equations provide a fundamental framework for modelling spatial population dynamics and invasion processes in mathematical biology. Among these, Fisher’s equation combines diffusion with logistic growth to describe the spread of an advantageous gene and the formation of travelling population fronts. In this [...] Read more.

Reaction–diffusion equations provide a fundamental framework for modelling spatial population dynamics and invasion processes in mathematical biology. Among these, Fisher’s equation combines diffusion with logistic growth to describe the spread of an advantageous gene and the formation of travelling population fronts. In this work, we investigate the one-dimensional Fisher’s equation using Lie symmetry analysis to obtain a deeper analytical understanding of its wave propagation behaviour. The Lie point symmetries of the partial differential equation are derived and used to construct similarity variables that reduce Fisher’s equation to ordinary differential equations. These reduced equations are then solved by a combination of direct integration and the tanh method, yielding explicit invariant and travelling-wave solutions. Symbolic computations in MAPLE are employed to compute the symmetries, verify the reductions, and generate illustrative plots of the resulting wave profiles. The computed solutions capture sigmoidal fronts connecting stable and unstable steady states, providing clear information about wave speed and shape. Overall, this study demonstrates that Lie group methods, combined with hyperbolic-function techniques, offer a powerful and systematic approach for analysing Fisher-type reaction–diffusion models and interpreting their biologically relevant invasion dynamics. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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23 pages, 361 KB

Open AccessArticle

BiHom–Lie Brackets and the Toda Equation

by Botong Gai, Chuanzhong Li, Jiacheng Sun, Shuanhong Wang and Haoran Zhu

Symmetry 2025, 17(12), 2176; https://doi.org/10.3390/sym17122176 - 17 Dec 2025

Viewed by 550

Abstract

We introduce a BiHom-type skew-symmetric bracket on general linear Lie algebra

G L (V)

built from two commuting inner automorphisms

α = A d_{ψ}

and

β = A d_{ϕ}

, with [...] Read more.

We introduce a BiHom-type skew-symmetric bracket on general linear Lie algebra

G L (V)

built from two commuting inner automorphisms

α = A d_{ψ}

and

β = A d_{ϕ}

, with

ψ, ϕ \in G L (V)

and integers

i, j

. We prove that

(G L (V), {[\cdot, \cdot]}_{(ψ, ϕ)}^{(i, j)}, α, β)

is a BiHom–Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice

ϕ = ψ

with

(i, j) = (0, 0)

, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via

\tilde{L} = ψ^{- 1} L ψ

, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded

2 \times 2

blocks, we derive explicit trigonometric and hyperbolic formulae that make symmetry constraints (e.g., tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit

2 \times 2

solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

25 pages, 1934 KB

Open AccessArticle

A Tripartite Analytical Framework for Nonlinear (1+1)-Dimensional Field Equations: Painlevé Analysis, Classical Symmetry Reduction, and Exact Soliton Solutions

by Muhammad Uzair, Aljethi Reem Abdullah and Irfan Mahmood

Symmetry 2025, 17(12), 2049; https://doi.org/10.3390/sym17122049 - 1 Dec 2025

Viewed by 498

Abstract

This study presents a tripartite analytical framework for the (1+1)-dimensional nonlinear Klein–Fock–Gordon equation, a key model for spinless particles in relativistic quantum mechanics. The investigation begins with a Painlevé analysis showing that the equation is completely integrable via the Painlevé test by using [...] Read more.

This study presents a tripartite analytical framework for the (1+1)-dimensional nonlinear Klein–Fock–Gordon equation, a key model for spinless particles in relativistic quantum mechanics. The investigation begins with a Painlevé analysis showing that the equation is completely integrable via the Painlevé test by using Maple. Subsequently, classical Lie symmetry analysis is employed to derive the infinitesimal generators of the equation. A Lagrangian formulation is constructed for these generators, from which similarity variables are systematically obtained. This framework enables a complete similarity reduction, transforming the complex nonlinear partial differential equation into a more tractable ordinary differential equation. To solve this reduced ordinary differential equation and to obtain a spectrum of soliton solutions, we implement the new generalized exponential differential rational function method. This advanced technique utilizes a rational trial function based on the

i t h

derivatives of exponentials, generating a diverse spectrum of closed-form soliton solutions through strategic choices of arbitrary constants. The novelty of this approach provides a unified framework for handling higher-order nonlinearities, yielding solutions such as multi-peakons and lump solitons, which are vividly characterized using Mathematica-generated 3D, 2D, and contour plots. These findings provide significant insights into nonlinear wave dynamics with potential applications in quantum field theory, nonlinear optics, plasma physics, etc. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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27 pages, 392 KB

Open AccessArticle

Non-Autonomous Soliton Hierarchies

by Maciej Błaszak, Krzysztof Marciniak and Błażej M. Szablikowski

Symmetry 2025, 17(7), 1103; https://doi.org/10.3390/sym17071103 - 9 Jul 2025

Viewed by 531

Abstract

A formalism for the systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is shown that this IVP has a [...] Read more.

A formalism for the systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is shown that this IVP has a formal solution, and within the framework of two particular subalgebras of the hereditary Lie algebra, the explicit forms of this formal solution are derived. Finally, this formalism is applied to the Korteveg-de Vries, dispersive water waves and Ablowitz–Kaup–Newell–Segur soliton hierarchies. The zero-curvature representations and Hamiltonian structures of the considered non-autonomous soliton hierarchies are also provided. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

19 pages, 4454 KB

Open AccessArticle

Continuous Maximum Coverage Location Problem with Arbitrary Shape of Service Areas and Regional Demand

by Sergiy Yakovlev, Sergiy Shekhovtsov, Lyudmyla Kirichenko, Olha Matsyi, Dmytro Podzeha and Dmytro Chumachenko

Symmetry 2025, 17(5), 676; https://doi.org/10.3390/sym17050676 - 29 Apr 2025

Cited by 2 | Viewed by 2661

Abstract

This paper addresses the maximum coverage location problem in a generalized setting, where both facilities (service areas) and regional demand are modeled as continuous entities. Unlike traditional formulations, our approach allows for arbitrary shapes for both service areas and demand regions, with additional [...] Read more.

This paper addresses the maximum coverage location problem in a generalized setting, where both facilities (service areas) and regional demand are modeled as continuous entities. Unlike traditional formulations, our approach allows for arbitrary shapes for both service areas and demand regions, with additional constraints on facility placement. The key novelty of this work is its ability to handle complex, irregularly shaped service areas, including approximating them as unions of centrally symmetric shapes. This enables the use of an analytical approach based on spatial symmetry, which allows for efficient estimation of the covered area. The problem is formulated as a nonlinear optimization task. We analyze the properties of the objective function and leverage the Shapely library in Python 3.13.3 for efficient geometric computations. To improve computational efficiency, we develop an extended elastic model that significantly reduces processing time. This model generalizes the well-known quasi-physical, quasi-human algorithm for circle packing, extending its applicability to more complex spatial configurations. The effectiveness of the proposed approach is validated through test cases in which service areas take the form of circles, ellipses, and irregular polygons. Our method provides a robust and adaptable solution for various settings of practically interesting continuous maximum coverage location problems involving irregular regional demand and service areas. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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18 pages, 422 KB

Open AccessArticle

On Higher-Order Generalized Fibonacci Hybrid Numbers with q-Integer Components: New Properties, Recurrence Relations, and Matrix Representations

by Can Kızılateş, Emrah Polatlı, Nazlıhan Terzioğlu and Wei-Shih Du

Symmetry 2025, 17(4), 584; https://doi.org/10.3390/sym17040584 - 11 Apr 2025

Cited by 3 | Viewed by 982

Abstract

Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q [...] Read more.

Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q-integer components are defined through the utilization of q-integers and higher-order generalized Fibonacci numbers. Several special cases of these newly established hybrid numbers are presented. The article explores the integration of q-calculus and hybrid numbers, resulting in the derivation of a Binet-like formula, novel identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of hybrid numbers with quantum integer coefficients. Furthermore, new identities for these types of hybrids are obtained using two novel special matrices. To substantiate the findings, numerical examples are provided, generated with the assistance of Maple. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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24 pages, 347 KB

Open AccessArticle

The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line

by Jiawei Hu and Ning Zhang

Symmetry 2024, 16(10), 1258; https://doi.org/10.3390/sym16101258 - 25 Sep 2024

Viewed by 1610

Abstract

The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and [...] Read more.

The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and this study addresses this issue by extending the application of the Fokas method to the higher-order Gerdjikov-Ivanov equation on the half line

(- \infty, 0]

. We have demonstrated the exclusive representation of the potential function

u (z, t)

in the higher-order Gerdjikov–Ivanov equation through the solution of a Riemann–Hilbert problem. The characteristic function is partitioned on the complex plane, and we obtain the jump matrix between each partition based on the positive and negative values of the partition as well as the spectral matrix determined by the initial data and boundary value data. The findings suggest that the spectral functions are not mutually independent; instead, they conform to a global relationship. The novel aspect of this study is the application of the Fokas method to a previously unexplored case, contributing to the theoretical and practical understanding of complex partial differential equations and filling a gap in the treatment of boundary conditions. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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19 pages, 310 KB

Open AccessArticle

Onthe Reducibility of a Class Nonlinear Almost Periodic Hamiltonian Systems

by Nina Xue and Yanmei Sun

Symmetry 2024, 16(6), 656; https://doi.org/10.3390/sym16060656 - 26 May 2024

Viewed by 1458

Abstract

Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic [...] Read more.

Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic Hamiltonian system with zero equilibrium points for most small enough parameters. As an example, we discuss the reducibility and stability of an almost periodic Hill’s equation. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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