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21 pages, 1620 KB

Open AccessArticle

New Interaction Patterns for the Truncated M-Fractional Kadomtsev–Petviashvili Equation in High-Dimensional Space

by Lihua Zhang, Shuqi Chong, Hongbing Jiao, Bo Shen and Gangwei Wang

Fractal Fract. 2025, 9(9), 572; https://doi.org/10.3390/fractalfract9090572 - 30 Aug 2025

Viewed by 171

A truncated M-fractional Kadomtsev–Petviashvili (KP) equation in high-dimensional space has been proposed. The model provides the oretical support for studying the interaction patterns among waves. According to the attributes of the truncated M-fractional derivative, the truncated M-fractional KP equation can be reduced to [...] Read more.

A truncated M-fractional Kadomtsev–Petviashvili (KP) equation in high-dimensional space has been proposed. The model provides the oretical support for studying the interaction patterns among waves. According to the attributes of the truncated M-fractional derivative, the truncated M-fractional KP equation can be reduced to the new extended KP equation. New interaction patterns which are compositions of cnoidal functions and soliton or trigonometric functions have been derived by the consistent Riccati expansion method. Applying the simple direct method, the finite symmetry transformation group and Bäcklund transformation have been constructed. Based on the known dark soliton solution and lump solution, new interaction patterns have been derived, including compositions of a dark soliton and an exponential function, compositions of a dark soliton and a trigonometric sine function, and compositions of a lump and a trigonometric sine function. The innovative aspect lies in the way that we find two effective ways to construct new interplay patterns of fractional differential equations. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)

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26 pages, 306 KB

Open AccessArticle

Osculating Mate of a Curve in Minkowski 3-Space

by İskender Öztürk, Hasan Çakır and Mustafa Özdemir

Axioms 2025, 14(6), 468; https://doi.org/10.3390/axioms14060468 - 16 Jun 2025

Viewed by 262

Abstract

In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We [...] Read more.

In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We then derive the transformation matrix between these frames and investigate the curvature and torsion relations under varying causal characterizations of the curves—timelike and spacelike. Furthermore, we determine the conditions under which these generalized osculating pairs reduce to well-known curve pairs such as Bertrand, Mannheim, and Bäcklund pairs. Our results extend existing theories by unifying several known curve pair classifications under a single geometric framework in Lorentzian space. Full article

(This article belongs to the Section Geometry and Topology)

28 pages, 11557 KB

Open AccessReview

Physics-Informed Neural Networks for Higher-Order Nonlinear Schrödinger Equations: Soliton Dynamics in External Potentials

by Leonid Serkin and Tatyana L. Belyaeva

Mathematics 2025, 13(11), 1882; https://doi.org/10.3390/math13111882 - 4 Jun 2025

Viewed by 2132

Abstract

This review summarizes the application of physics-informed neural networks (PINNs) for solving higher-order nonlinear partial differential equations belonging to the nonlinear Schrödinger equation (NLSE) hierarchy, including models with external potentials. We analyze recent studies in which PINNs have been employed to solve NLSE-type [...] Read more.

This review summarizes the application of physics-informed neural networks (PINNs) for solving higher-order nonlinear partial differential equations belonging to the nonlinear Schrödinger equation (NLSE) hierarchy, including models with external potentials. We analyze recent studies in which PINNs have been employed to solve NLSE-type evolution equations up to the fifth order, demonstrating their ability to obtain one- and two-soliton solutions, as well as other solitary waves with high accuracy. To provide benchmark solutions for training PINNs, we employ analytical methods such as the nonisospectral generalization of the AKNS scheme of the inverse scattering transform and the auto-Bäcklund transformation. Finally, we discuss recent advancements in PINN methodology, including improvements in network architecture and optimization techniques. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Dynamics and Nonautonomous Solitons)

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15 pages, 2360 KB

Open AccessFeature PaperArticle

Analytic Investigation of a Generalized Variable-Coefficient KdV Equation with External-Force Term

by Gongxun Li, Zhiyan Wang, Ke Wang, Nianqin Jiang and Guangmei Wei

Mathematics 2025, 13(10), 1642; https://doi.org/10.3390/math13101642 - 17 May 2025

Viewed by 385

Abstract

This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair [...] Read more.

This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair for this equation is successfully constructed using the extended Ablowitz–Kaup–Newell–Segur system (AKNS system). Furthermore, the Riccati-type Bäcklund transformation (R-BT), Wahlquist–Estabrook-type Bäcklund transformation (WE-BT), and the nonlinear superposition formula are derived. In utilizing these transformations and the formula, explicit one-soliton-like and two-soliton-like solutions are constructed from a seed solution. Moreover, the infinite conservation laws of the equation are systematically derived. Finally, the influence of variable coefficients and the external-force term on the propagation characteristics of a solitory wave is discussed, and soliton interaction is illustrated graphically. Full article

(This article belongs to the Special Issue Research on Applied Partial Differential Equations)

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28 pages, 3560 KB

Open AccessArticle

Solitons, Cnoidal Waves and Nonlinear Effects in Oceanic Shallow Water Waves

by Huanhe Dong, Shengfang Yang, Yong Fang and Mingshuo Liu

Fractal Fract. 2025, 9(5), 305; https://doi.org/10.3390/fractalfract9050305 - 7 May 2025

Viewed by 457

Abstract

Gravity water waves in the shallow-ocean scenario described by generalized Boussinesq–Broer–Kaup–Whitham (gBBKW) equations are discussed. The residual symmetry and Bäcklund transformation associated with the gBBKW equations are systematically constructed. The time and space evolution of wave velocity and height are explored. Additionally, it [...] Read more.

Gravity water waves in the shallow-ocean scenario described by generalized Boussinesq–Broer–Kaup–Whitham (gBBKW) equations are discussed. The residual symmetry and Bäcklund transformation associated with the gBBKW equations are systematically constructed. The time and space evolution of wave velocity and height are explored. Additionally, it is demonstrated that the gBBKW equations are solvable through the consistent Riccati expansion method. Leveraging this property, a novel Bäcklund transformation, solitary wave solution, and soliton–cnoidal wave solution are derived. Furthermore, miscellaneous novel solutions of gBBKW equations are obtained using the modified Sardar sub-equation method. The impact of variations in the diffusion power parameter on wave velocity and height is quantitatively analyzed. The exact solutions of gBBKW equations provide precise description of propagation characteristics for a deeper understanding and the prediction of some ocean wave phenomena. Full article

(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)

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7 pages, 273 KB

Open AccessArticle

Bäcklund Transformation for Solving a (3+1)-Dimensional Integrable Equation

by Binlu Feng, Linlin Gui, Yufeng Zhang and Siqi Han

Axioms 2025, 14(3), 225; https://doi.org/10.3390/axioms14030225 - 18 Mar 2025

Viewed by 411

Abstract

A new generalized (3+1)-dimensional Kadomtsev–Petviashvil (3dKP) equation is derived from the inverse scattering transform method. This equation can be reduced to the standard KP equation and the well-know (3+1)-dimensional equation. In making use of the Lax pair transformation, a Bäcklund transformation of the [...] Read more.

A new generalized (3+1)-dimensional Kadomtsev–Petviashvil (3dKP) equation is derived from the inverse scattering transform method. This equation can be reduced to the standard KP equation and the well-know (3+1)-dimensional equation. In making use of the Lax pair transformation, a Bäcklund transformation of the generalized (3+1)-dimensional KP equation is constructed and some soliton solutions are produced. Finally, a superposition formula is singled out as well by making use of the Bäcklund transformation. As far as we know, the work presented in this paper has not been studied up to now. Full article

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15 pages, 238 KB

Open AccessArticle

Prolongation Structure of a Development Equation and Its Darboux Transformation Solution

by Lixiu Wang, Jihong Wang and Yangjie Jia

Mathematics 2025, 13(6), 921; https://doi.org/10.3390/math13060921 - 11 Mar 2025

Viewed by 653

Abstract

This paper explores the third-order nonlinear coupled KdV equation utilizing prolongation structure theory and gauge transformation. By applying the prolongation structure method, we obtained an extended version of the equation. Starting from the Lax pairs of the equation, we successfully derived the corresponding [...] Read more.

This paper explores the third-order nonlinear coupled KdV equation utilizing prolongation structure theory and gauge transformation. By applying the prolongation structure method, we obtained an extended version of the equation. Starting from the Lax pairs of the equation, we successfully derived the corresponding Darboux transformation and Bäcklund transformation for this equation, which are fundamental to our solving process. Subsequently, we constructed and calculated the recursive operator for this equation, providing an effective approach to tackling complex problems within this domain. These results are crucial for advancing our understanding of the underlying principles of soliton theory and their implications on related natural phenomena. Our findings not only enrich the theoretical framework but also offer practical tools for further research in nonlinear wave dynamics. Full article

(This article belongs to the Special Issue Modern Methods and Applications Related to Integrable Systems)

13 pages, 3037 KB

Open AccessArticle

The Multi-Soliton Solutions for the (2+1)-Dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

by Li-Jun Xu, Zheng-Yi Ma, Jin-Xi Fei, Hui-Ling Wu and Li Cheng

Mathematics 2025, 13(2), 236; https://doi.org/10.3390/math13020236 - 12 Jan 2025

Cited by 1 | Viewed by 1106

Abstract

The (2+1)-dimensional integrable Caudrey–Dodd–Gibbon–Kotera–Sawada equation is a higher-order generalization of the Kadomtsev–Petviashvili equation, which can be applied in some physical branches such as the nonlinear dispersive phenomenon. In this paper, we first present the bilinear form for this equation after constructing one Bäcklund [...] Read more.

The (2+1)-dimensional integrable Caudrey–Dodd–Gibbon–Kotera–Sawada equation is a higher-order generalization of the Kadomtsev–Petviashvili equation, which can be applied in some physical branches such as the nonlinear dispersive phenomenon. In this paper, we first present the bilinear form for this equation after constructing one Bäcklund transformation. As a result, the one-soliton solution, two-soliton solution, and three-soliton solution are shown successively and the corresponding soliton structures are constructed. These solitons and their interactions illustrate that the obtained solutions have powerful applications. Full article

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17 pages, 2004 KB

Open AccessArticle

Applications of Riccati–Bernoulli and Bäcklund Methods to the Kuralay-II System in Nonlinear Sciences

by Khudhayr A. Rashedi, Musawa Yahya Almusawa, Hassan Almusawa, Tariq S. Alshammari and Adel Almarashi

Mathematics 2025, 13(1), 84; https://doi.org/10.3390/math13010084 - 29 Dec 2024

Cited by 4 | Viewed by 779

Abstract

The Kuralay-II system (K-IIS) plays a pivotal role in modeling sophisticated nonlinear wave processes, particularly in the field of optics. This study introduces novel soliton solutions for the K-IIS, derived using the Riccati–Bernoulli sub-ODE method combined with Bäcklund transformation and conformable fractional derivatives. [...] Read more.

The Kuralay-II system (K-IIS) plays a pivotal role in modeling sophisticated nonlinear wave processes, particularly in the field of optics. This study introduces novel soliton solutions for the K-IIS, derived using the Riccati–Bernoulli sub-ODE method combined with Bäcklund transformation and conformable fractional derivatives. The obtained solutions are expressed in trigonometric, hyperbolic, and rational forms, highlighting the adaptability and efficacy of the proposed approach. To enhance the understanding of the results, the solutions are visualized using 2D representations for fractional-order variations and 3D plots for integer-type solutions, supported by detailed contour plots. The findings contribute to a deeper understanding of nonlinear wave–wave interactions and the underlying dynamics governed by fractional-order derivatives. This work underscores the significance of fractional calculus in analyzing complex wave phenomena and provides a robust framework for further exploration in nonlinear sciences and optical wave modeling. Full article

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25 pages, 408 KB

Open AccessArticle

Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions

by Henrik Aratyn, José Francisco Gomes, Gabriel Vieira Lobo and Abraham Hirsz Zimerman

Mathematics 2024, 12(23), 3701; https://doi.org/10.3390/math12233701 - 26 Nov 2024

Viewed by 743

Abstract

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group

{\hat{A}}_{N - 1}^{(1)}

contains [...] Read more.

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group

{\hat{A}}_{N - 1}^{(1)}

contains the conventional Bäcklund transformations

s_{j}, j = 1, \dots, N

, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of N points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to the existence of degenerated solutions, and for

N = 4

, we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of

{\hat{A}}_{3}^{(1)}

. We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials. Full article

(This article belongs to the Special Issue Advanced Research in Mathematical Physics Models with Painlevé Property and Affine Weyl Group Symmetry)

20 pages, 1576 KB

Open AccessArticle

A New (3+1)-Dimensional Extension of the Kadomtsev–Petviashvili–Boussinesq-like Equation: Multiple-Soliton Solutions and Other Particular Solutions

by Xiaojian Li and Lianzhong Li

Symmetry 2024, 16(10), 1345; https://doi.org/10.3390/sym16101345 - 11 Oct 2024

Cited by 1 | Viewed by 1364

Abstract

In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). [...] Read more.

In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). Exact analytical solutions were derived through the power series method, with a comprehensive proof of solution convergence. Employing the

(G^{'} / G^{2})

-expansion method enabled the identification of trigonometric, exponential, and rational solutions of the equation. Furthermore, we established the auto-Bäcklund transformation of the equation. Multiple-soliton solutions were identified by utilizing Hirota’s bilinear method. The fundamental properties of these solutions were elucidated through graphical representations. Our results are of certain value to the interpretation of nonlinear problems. Full article

(This article belongs to the Section Mathematics)

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12 pages, 2013 KB

Open AccessArticle

Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

by Safyan Mukhtar, Weaam Alhejaili, Mohammad Alqudah, Ali M. Mahnashi, Rasool Shah and Samir A. El-Tantawy

Axioms 2024, 13(10), 686; https://doi.org/10.3390/axioms13100686 - 2 Oct 2024

Cited by 2 | Viewed by 1316

Abstract

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated [...] Read more.

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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12 pages, 1090 KB

Open AccessArticle

Dynamics of the Traveling Wave Solutions of Fractional Date–Jimbo–Kashiwara–Miwa Equation via Riccati–Bernoulli Sub-ODE Method through Bäcklund Transformation

by M. Mossa Al-Sawalha, Saima Noor, Mohammad Alqudah, Musaad S. Aldhabani and Roman Ullah

Fractal Fract. 2024, 8(9), 497; https://doi.org/10.3390/fractalfract8090497 - 23 Aug 2024

Viewed by 1138

Abstract

The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement [...] Read more.

The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement of the characterization of dynamic waves while providing better modeling ability compared to integer types of derivatives. The solutions of the above-mentioned time–space fractional Date–Jimbo–Kashiwara–Miwa equation have tremendous importance in numerous scientific scenarios. The regular dynamical wave solutions of the aforementioned equation encompass three fundamental functions: trigonometric, hyperbolic, and rational functions will be among the topics covered. These solutions are graphically classified into three categories: compacton kink solitary wave solutions, kink soliton wave solutions and anti-kink soliton wave solutions. In addition, to explore the impact of the fractional parameter (

α

) on those solutions,

2 D

plots are utilized, while

3 D

plots are applied to present the solutions involving the integer-order derivatives. Full article

(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)

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14 pages, 661 KB

Open AccessArticle

Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation

by Mohammad Alshammari, Khaled Moaddy, Muhammad Naeem, Zainab Alsheekhhussain, Saleh Alshammari and M. Mossa Al-Sawalha

Fractal Fract. 2024, 8(4), 222; https://doi.org/10.3390/fractalfract8040222 - 11 Apr 2024

Cited by 2 | Viewed by 1925

Abstract

The Zoomeron equation plays a significant role in many fields of physics, especially in soliton theory, such as helping to reveal new distinctive properties in different physical phenomena such as fluid dynamics, laser physics, and nonlinear optics. By using the Riccati–Bernoulli sub-ODE approach [...] Read more.

The Zoomeron equation plays a significant role in many fields of physics, especially in soliton theory, such as helping to reveal new distinctive properties in different physical phenomena such as fluid dynamics, laser physics, and nonlinear optics. By using the Riccati–Bernoulli sub-ODE approach and the Backlund transformation, we search for soliton solutions of the fractional Zoomeron nonlinear equation. A number of solutions have been put forth, such as kink, anti-kink, cuspon kink, lump-type kink solitons, single solitons, and others defined in terms of pseudo almost periodic functions. The (2 + 1)-dimensional fractional Zoomeron equation given in a form undergoes precise dynamics. We use the computational software, Matlab 19, to express these solutions graphically by changing the value of various parameters involved. A detailed analysis of their dynamics allows us to obtain completely better insights necessarily with the elementary physical phenomena controlled by the fractional Zoomeron equation. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)

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20 pages, 288 KB

Open AccessArticle

Applications of the R-Matrix Method in Integrable Systems

by Binlu Feng, Yufeng Zhang and Hongyi Zhang

Symmetry 2023, 15(9), 1623; https://doi.org/10.3390/sym15091623 - 23 Aug 2023

Cited by 3 | Viewed by 1484

Abstract

Based on work related to the R-matrix theory, we first abstract the Lax pairs proposed by Blaszak and Sergyeyev into a unified form. Then, a generalized zero-curvature equation expressed by the Poisson bracket is exhibited. As an application of this theory, a generalized [...] Read more.

Based on work related to the R-matrix theory, we first abstract the Lax pairs proposed by Blaszak and Sergyeyev into a unified form. Then, a generalized zero-curvature equation expressed by the Poisson bracket is exhibited. As an application of this theory, a generalized (2+1)-dimensional integrable system is obtained, from which a resulting generalized Davey–Stewartson (DS) equation and a generalized Pavlov equation (gPe) are further obtained. Via the use of a nonisospectral zero-curvature-type equation, some (3+1) -dimensional integrable systems are produced. Next, we investigate the recursion operator of the gPe using an approach under the framework of the R-matrix theory. Furthermore, a type of solution for the resulting linearized equation of the gPe is produced by using its conserved densities. In addition, by applying a nonisospectral Lax pair, a (3+1)-dimensional integrable system is generated and reduced to a Boussinesq-type equation in which the recursion operators and the linearization are produced by using a Lie symmetry analysis; the resulting invertible mappings are presented as well. Finally, a Bäcklund transformation of the Boussinesq-type equation is constructed, which can be used to generate some exact solutions. Full article

(This article belongs to the Section Mathematics)

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