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7 pages, 273 KiB

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 130

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 KiB

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 244

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 Symmetries of Integrable Systems, 2nd Edition)

13 pages, 3037 KiB

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

Viewed by 704

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 KiB

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 1 | Viewed by 508

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 KiB

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 574

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 KiB

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

Viewed by 1115

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 KiB

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 1 | Viewed by 1052

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 KiB

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 928

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 KiB

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 1 | Viewed by 1731

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 KiB

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 1293

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)

36 pages, 506 KiB

Open AccessFeature PaperArticle

On Rational Solutions of Dressing Chains of Even Periodicity

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

Symmetry 2023, 15(1), 249; https://doi.org/10.3390/sym15010249 - 16 Jan 2023

Cited by 3 | Viewed by 1678

Abstract

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under

$A_{N - 1}^{(1)}$ symmetry. This formalism identifies rational solutions (as well [...] Read more.

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under

$A_{N - 1}^{(1)}$ symmetry. This formalism identifies rational solutions (as well as special function solutions) with points on orbits of fundamental shift operators of

$A_{N - 1}^{(1)}$ affine Weyl groups acting on seed configurations defined as first-order polynomial solutions of the underlying dressing chains. This approach clarifies the structure of rational solutions and establishes an explicit and systematic method towards their construction. For the special case of the

$N = 4$ dressing chain equations, the method yields all the known rational (and special function) solutions of the Painlevé V equation. The formalism naturally extends to

$N = 6$ and beyond as shown in the paper. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

9 pages, 254 KiB

Open AccessFeature PaperArticle

Novel Bäcklund Transformations for Integrable Equations

by Pilar Ruiz Gordoa and Andrew Pickering

Mathematics 2022, 10(19), 3565; https://doi.org/10.3390/math10193565 - 29 Sep 2022

Cited by 1 | Viewed by 1159

Abstract

In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation [...] Read more.

In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation analogous to that of this matrix fourth Painlevé equation. Such auto-Bäcklund transformations, in appearance similar to those for Painlevé equations, are quite novel, having been little studied in the case of partial differential equations. Our work here shows the importance of the underlying structure of differential equations, whether ordinary or partial, in the derivation of such results. The starting point for the results in this paper is the construction of a new completely integrable equation, namely, an inverse matrix dispersive water wave equation. Full article

(This article belongs to the Special Issue Completely Integrable Equations: Algebraic Aspects and Applications)

9 pages, 478 KiB

Open AccessArticle

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

by Xue-Ke Liu and Xiao-Yong Wen

Symmetry 2022, 14(5), 920; https://doi.org/10.3390/sym14050920 - 30 Apr 2022

Cited by 2 | Viewed by 1601

Abstract

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of [...] Read more.

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to

$3 \times 3$ matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known

$3 \times 3$ Lax pair of this equation, the discrete generalized

$(m, 3 N - m)$ -fold Darboux transformation was constructed for the first time and extended from the

$2 \times 2$ Lax pair to the

$3 \times 3$ Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

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7 pages, 354 KiB

Open AccessArticle

Interaction Behaviours between Soliton and Cnoidal Periodic Waves for Nonlocal Complex Modified Korteweg–de Vries Equation

by Junda Peng, Bo Ren, Shoufeng Shen and Guofang Wang

Mathematics 2022, 10(9), 1429; https://doi.org/10.3390/math10091429 - 23 Apr 2022

Cited by 2 | Viewed by 1616

Abstract

The reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to the CTE method, a nonauto-Bäcklund transformation theorem of nonlocal complex mKdV is obtained. The interactions between one kink soliton and other [...] Read more.

The reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to the CTE method, a nonauto-Bäcklund transformation theorem of nonlocal complex mKdV is obtained. The interactions between one kink soliton and other different nonlinear excitations are constructed via the nonauto-Bäcklund transformation theorem. By selecting cnoidal periodic waves, the interaction between one kink soliton and the cnoidal periodic waves is derived. The specific Jacobi function-type solution and graphs of its analysis are provided in this paper. Full article

(This article belongs to the Special Issue Variational Problems and Applications)

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10 pages, 2639 KiB

Open AccessArticle

Painlevé Test and Exact Solutions for (1 + 1)-Dimensional Generalized Broer–Kaup Equations

by Sheng Zhang and Bo Xu

Mathematics 2022, 10(3), 486; https://doi.org/10.3390/math10030486 - 2 Feb 2022

Cited by 4 | Viewed by 2059

Abstract

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund [...] Read more.

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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