Backlund Transformation Research Articles

Fractional Fokas-Lenells equation: analyzing travelling waves via advanced analytical method

In this paper, we consider the fractional Fokas-Lenells equation, which allows us to analyze how a nonlinear optic pulse spreads in time as single-mode fiber produces higher-order nonlinear effects. We have computed perfectly accurate travelling wave solutions for the Fokas-Lenells equation using the Riccati-Bernoulli sub-Ode approach. For the corresponding equation, we have established three distinct classes of perfectly accurate travelling wave solutions with different free parameters; hyperbolic, trigonometric, and rational. A sophisticated Backlund transformation is implemented to the equation to change it to ordinary differential equation domain, leading to many extra exact solutions.

Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation

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.

Robust inverse scattering analysis of discrete high-order nonlinear Schrödinger equation

Abstract In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.

Analysis of solitary wave solutions in the fractional-order Kundu–Eckhaus system

The area of fractional partial differential equations has recently become prominent for its ability to accurately simulate complex physical events. The search for traveling wave solutions for fractional partial differential equations is a difficult task, which has led to the creation of numerous mathematical approaches to tackle this problem. The primary objective of this research work is to provide optical soliton solutions for the Frictional Kundu–Eckhaus equation (FKEe) by utilizing generalized coefficients. This strategy utilizes the Riccati–Bernoulli sub-ODE technique to effectively discover the most favorable traveling wave solutions for fractional partial differential equations. As a result, it facilitates the extraction of optical solitons and intricate wave solutions. The Backlund transformation is used to methodically construct a sequence of solutions for the specified equations. The study additionally showcases 3D and Density graphics that visually depict chosen solutions for certain parameter selections, hence improving the understanding of the outcomes.

Analysis of peakon-like soliton solutions: (3+1)-dimensional Fractional Klein-Gordon equation

<abstract><p>In this study, we investigate the fundamental properties of ($ 3+1 $)-$ D $ Fractional Klein-Gordon equation using the sophisticated techniques of Riccatti-Bornoulli sub-ODE approach with Backlund transformation. Using a more stringent criterion, our study reveals new soliton solutions that have peakon-like properties and unique cusp features. This research provides significant understanding of the dynamic behaviours and odd events related to these solutions. This work is important because it helps to elucidate the complex dynamics that exist within physical systems, which will benefit many different scientific fields. Our method is used to examine the existence and stability of compactons and kinks in the context of actual physical systems. Under a double-well on-site potential, these structures are made up of a network of connected nonlinear pendulums. Both $ 2D $ and contour plots produced by parameter changes provide as clear examples of the efficiency, simplicity, and conciseness of the computational method used. The results highlight how flexible this approach is, and demonstrate how symbolic calculations broaden its application to more complex events. This work offers a useful framework and studying intricate physical systems, as well as a flexible computational tool that may be used in a variety of scientific fields.</p></abstract>

Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions

<abstract><p>We examine a biological population model of fractional order (FBPM) in this paper using the Riccati-Bernoulli sub-ODE approach. Many scenarios in computational biology make use of this fundamental fractional model. Of particular note is that our study's FBPM uses fractional derivatives to track changes in the density populations. The study is concerned with the construction of new solitary wave solutions for the FBPM, a system of two nonlinear fractional ordinary differential equations. In this investigation, we use the conformable derivative as the fractional derivative. The Backlund transformation is the foundation of the solution process. We create a variety of families of soliton wave solutions and explain different physical behaviours that are inherent in the problems we explore. In particular, we apply the suggested methods to investigate rational, periodic, and hyperbolic solutions. The solutions found in various classes provide insightful information about the underlying physical mechanisms. To sum up, our current methods are superior instruments for analyzing different families of solutions in fractional-order issues.</p></abstract>

Perturbed Gerdjikov–Ivanov equation: Soliton solutions via Backlund transformation

The perturbed Gerdjikov–Ivanov (pGI) equation, a mathematical model that depicts the behaviour of optical pulses during propagation while accounting for perturbation influences, has been thoroughly studied by us. This equation has significant applications in the field of optical fibres, notably for photonic crystal fibres. In this paper, we combined the Riccati-Bernoulli sub-ODE method with the Backlund transformation to construct travelling wave solutions for the perturbed Gerdjikov–Ivanov (pGI) equation. This technique offered us families of solutions as well as a fresh way of dealing with nonlinear models. The results that emerge could greatly advance our knowledge of the physical consequences that are present in the nonlinear model that we are studying. The resulting solutions broaden the scope of earlier findings made using various approaches by including solitons, trigonometric functions, and rational expressions. The Backlund transformation used in this study is distinguished by its simplicity and brevity, and it produces results that are noticeably more thorough than those often obtained by alternative methods. Additionally, we create graphical representations of our results using the computational capabilities of the Maple software. In order to do this, the proper parameter values must be chosen, allowing for a visual study of the results. Through the display of two-dimensional graphical depictions, these effects have been clearly explained, permitting a thorough understanding of their physical implications.

On Asymptotic Stability of the Sine-Gordon Kink in the Energy Space

We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H^1times L^2. It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space. In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in Hoffman and Wayne (Differ Int Equ 26(3–4):303–320, 2013), Muñoz and Palacios (Ann. IHP C Analyse Nonlinéaire 36(4):977–1034, 2019) and Virial estimates around odd perturbations of the vacuum solution, in the spirit of Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and phi ^4 at the level of linear Bäcklund transformations.

Consistency of the Bäcklund Transformation for the Spin Calogero–Moser System

We prove the consistency of the Bäcklund transformation (BT) for the spin Calogero–Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ordinary differential equations; to establish our result, we construct and analyze certain functions that measure the departure of this overdetermined system from consistency. We show that these functions are identically zero and that this allows for a unique solution to the initial value problem for the overdetermined system.

Vector nonlinear Schrödinger equation with an integrable defect and new integrable boundary conditions

The main purpose of the present paper is to study integrable defects and new integrable boundaries associated with the vector nonlinear Schrödinger (NLS) equation. For the first issue, we present a type I defect condition for the vector NLS equation by using a Bäcklund transformation (BT) frozen at defect location. Based on the canonical property of the BT, the integrability of our defect model is proved by virtue of the classical r-matrix method. The generating function of the infinite set of conserved quantities to the defect system is also constructed. For the second issue, we present several new integrable boundary conditions associated with the K(λ) matrices that are characterized by the presence of the vector NLS fields. These boundary conditions are constructed by imposing U(N) symmetry reduction on our defect condition and by dressing a boundary with our defect condition. The integrability of our new boundary conditions is proved by virtue of the Sklyanin’s approach. The Hamiltonian description of the vector NLS equation in the presence of our new boundary conditions is also discussed.

Complexitons, Bilinear forms and Bilinear Bäcklund transformation of a [formula omitted]-dimensional Boiti–Leon–Manna–Pempinelli model describing incompressible fluid

In this work, we have presented bilinear form and two separate Bäcklund transformations (BT) for a (2+1)-Dimensional Bilti–Leon–Manna–Pempinelli equation (BLMPE), which characterizes the flow of an incompressible fluid and demonstrates the development of the horizontal velocity component of water waves that propagate in the XY plane in an infinite narrow passage of constant depth. The first form includes six arbitrary parameters, but the second version has only two. Based on the suggested bilinear (BT), rational traveling wave solutions with random wave numbers and classifications of exponential are determined. Furthermore, the extended transformed rational function approach, using the Hirota bilinear version of the governing model, is employed to provide complexions. We have also given the 3D and 2D graphics of the obtained solutions to understand the physical dynamics of the BLMPE.

On The Bäcklund Transformations for Cosgrove's Equation

In this paper we study Bäcklund transformations (BTs) for Cosgrove’s equation F-XVIII. We use the generalization of Fokas and Ablowitz method to derive BT between F-XVIII and new fourth-order ordinary differential equations of Painlevé type. Moreover we derive auto-BT and give special solutions for F-XVIII.

Bäcklund transformation and nonlinear superposition formula for the two-component short pulse equation

By means of the reciprocal transformation and the associated equation, a Bäcklund transformation (BT) containing both independent and dependent variables is worked out and studied for the two-component short pulse (2SP) equation. Based on it, the corresponding nonlinear superposition formula (NSF) is also constructed. Through reducing the BT and NSF of the 2SP equation, the BTs and NSFs for the SP equation, the focusing complex short pulse (cSP) equation and the defocusing cSP equation are obtained respectively. By analyzing the range of parameters in the solution, various solutions including soliton solitons, cuspon solitons, loop solitons, breather solutions are divided and their interaction are considered.

The soliton solutions and combined solutions of a high-dimensional wave soliton equation

In this paper, a high-dimensional wave soliton equation is considered and the simple Hirota method and bilinear backlund transformation are applied to construct the new soliton solutions, lump soliton solutions, breather solutions and their combined solutions. Then, through long-wave limit method and appropriate parameter constraints, one-soliton, one-breather, combined soliton-lump and lump-breather and soliton-breather solutions to nonlinear wave equation are obtained. Moreover, the physical interaction and frontal collision phenomena to the solutions of this equation are explored. In addition, a kind of hyperbolic and trigonometric ansatz is employed to derive other solutions such as kink soliton solution, periodic solutions for the high-dimensional wave equation.The obtained results verify the proposed solutions.

Integrable property, Lie symmetry analysis and explicit solutions to the generalized [formula omitted] model

In this paper, the combination of Painlevé analysis and Lie group method is performed, the Painlevé properties (PPs) and Bäcklund transformations (BTs) of the nonlinear ϕ4 types of equations are obtained under some conditions. Then, all of the point symmetries of the generalized ϕ4 model are given by Lie group classification method. Furthermore, the exact solutions to the equations are investigated by the symmetry reductions and analytic method.

The higher-order modified Korteweg-de Vries equation: Its soliton, breather and approximate solutions

In this study, the fifth-order modified Korteweg-de Vries (F-MKdV) equation is first addressed using Hirota’s bilinear method. Thereafter, the exact and approximative solutions of the generalized form of the F-MKdV equation are investigated using the modified Kudryashov method, the Riccati equation and its Backlund transformation method, the solitary wave ansatz method, and the homotopy perturbation transform method (HPTM). As a result, solitons, breather, and solitary wave solutions are derived from these methods. In particular, we obtain some new solutions such as the dark soliton, bright soliton, singular soliton, periodic trigonometric, exponential, hyperbolic, and rational solutions. The constraint conditions associated with the resulting solutions are also discussed in detail. The HPTM is employed to construct approximate solutions to the aforementioned generalized model due to its strong nonlinear terms and only a few terms are required to obtain accurate solutions. These findings may help to understand complex nonlinear phenomena.

The singular manifold method for a class of fractional-order diffusion equations

The present article presents the technique for finding exact wave solutions of nonlinear fractional-order partial differential equations (NLFPDEs). The singular manifold method (SMM) is applied by starting with the fractional complex transformation (FCT) to reduce the equations to ordinary differential equations (ODEs). Then using the Bäcklund transformation (BT) and the resulting Schwarzian derivatives of the Eigen functions, a new variety of exact solutions is obtained. The method is applied to nonlinear (2 + 1) time-fractional Zoomeron equation, nonlinear (3 + 1) conformable time-fractional Zakharov–Kuznetsov equation, and nonlinear fractional Fokas equation. Some resulting solutions are graphically presented, revealing different solitary wave solutions.

On Backlund transformation and motion of null Cartan curves

The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector [Formula: see text] In the case of [Formula: see text] we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve [Formula: see text]

A new Painlevé integrable Broer-Kaup system: symmetry analysis, analytic solutions and conservation laws

Purpose This study aims to find the symmetries and conservation laws of a new Painlevé integrable Broer-Kaup (BK) system with variable coefficients. This system is an extension of dispersive long wave equations. As the system is generalized and new, it is essential to explore some of its possible aspects such as conservation laws, symmetries, Painleve integrability, etc. Design/methodology/approach This paper opted for an exploratory study of a new Painleve integrable BK system with variable coefficients. Some analytic solutions are obtained by Lie classical method. Then the conservation laws are derived by multiplier method. Findings This paper presents a complete set of point symmetries without any restrictions on choices of coefficients, which subsequently yield analytic solutions of the series and solitary waves. Next, the authors derive every admitted non-trivial conservation law that emerges from multipliers. Research limitations/implications The authors have found that the considered system is likely to be integrable. So some other aspects such as Lax pair integrability, solitonic behavior and Backlund transformation can be analyzed to check the complete integrability further. Practical implications The authors develop a time-dependent Painleve integrable long water wave system. The model represents more specific data than the constant system. The authors presented analytic solutions and conservation laws. Originality/value The new time-dependent Painleve integrable long water wave system features some interesting results on symmetries and conservation laws.

Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation

Nonlocal symmetries related to the Bäcklund transformation (BT) for the modified KdV-sine-Gordon (mKdV-SG) equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations. Then through the parameter expansion procedure, an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived. Finally, several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.