Dimensional sine-Gordon Equation Research Articles

Multiple Lax integrable higher dimensional AKNS(-1) equations and sine-Gordon equations.

Through the modified deformation algorithm related to conservation laws, the (1+1)-dimensional AKNS(-1) equations are extended to a (4+1)-dimensional AKNS(-1) system. When one, two, or three of the independent variables are removed, the (4+1)-dimensional AKNS(-1) system degenerates to some novel (3+1)-dimensional, (2+1)-dimensional, and (1+1)-dimensional AKNS(-1) systems, respectively. Under a simple dependent transformation, the (1+1)-dimensional AKNS(-1) equations turn into the classical sine-Gordon equation. Then using the same deformation procedure, the (1+1)-dimensional sine-Gordon equation is generalized to a (3+1)-dimensional version. By introducing the deformation operators to the Lax pairs of the original (1+1)-dimensional models, the Lax integrability of both the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation is proven. Finally, the traveling wave solutions of the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation are implicitly given and expressed by tanh function and incomplete elliptic integral, respectively. These results may enhance our understanding of the complex physical phenomena described by the nonlinear system discussed in this paper.

The dynamics of some exact solutions to a [formula omitted]-dimensional sine-Gordon equation

In this paper, a (3+1)-dimensional sine-Gordon equation is systematically investigated. Firstly, the integrability of the equation is demonstrated by Painlevé analysis. Secondly, based on the Hirota bilinear method, the N-soliton solution of the (3+1)-dimensional sine-Gordon equation is derived. Then, by selecting and establishing conjugate relationships between parameters, the kink solutions, the breather solutions and their hybrid solutions were obtained. Finally, the lump solutions of equation are derived by selecting appropriate functions in the solution. In addition, the dynamic behavior of these solutions is systematically analyzed by their respective density profile plots and three-dimensional diagrams.

Breather Transitions and Their Mechanisms of a (2+1)-Dimensional Sine-Gordon Equation and a Modified Boussinesq Equation in Nonlinear Dynamics

Breather Transitions and Their Mechanisms of a (2+1)-Dimensional Sine-Gordon Equation and a Modified Boussinesq Equation in Nonlinear Dynamics

New exact solutions of the (3+1)-dimensional double sine-Gordon equation by two analytical methods

The (3+1)-dimensional double sine-Gordon equation plays a crucial role in various physical phenomena, including nonlinear wave propagation, field theory, and condensed matter physics. However, obtaining exact solutions to this equation faces significant challenges. In this article, we successfully employ a modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( \\dfrac{G'}{G^2}\\right) $$\\end{document}-expansion and improved tanϕξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an \\left( \\dfrac{\\phi \\left( \\xi \\right) }{2}\\right) $$\\end{document}-expansion methods to construct new analytical solutions to the double sine-Gordon equation. These solutions can be divided into four categories like trigonometric function solutions, hyperbolic function solutions, exponential solutions, and rational solutions. Our key findings include a rich spectrum of soliton solutions, encompassing bright, dark, singular, periodic, and mixed types, showcasing the (3+1)-dimensional double sine-Gordon equation ability to model diverse wave behaviors. We uncover previously unreported complex wave structures, revealing the potential for complex nonlinear interactions within the (3+1)-dimensional double sine-Gordon equation framework. We demonstrate the modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( \\dfrac{G'}{G^2}\\right) $$\\end{document}-expansion and improved tanϕξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an \\left( \\dfrac{\\phi \\left( \\xi \\right) }{2}\\right) $$\\end{document}-expansion methods effectiveness in handling higher-dimensional nonlinear partial differential equations, expanding their applicability in mathematical physics. These method offers enhanced flexibility and broader solution categories compared to conventional approaches.

Multi-wave, breather and interaction solutions to (3+1) dimensional sine-Gordon equation arising in nonlinear physical sciences

Based on the Hirota bilinear forms, the multi-wave solutions of the (3 + 1) dimensional sine-Gordon equation with various physical properties were constructed with using symbolic computation. This equation is characterized by constant coefficients that can be integrated in the Painleve sense. Double exponential and homoclinic breather approaches are used to obtain solutions. In addition, the interaction phenomenon between the exponential function, sine function and the hyperbolic sine function are constructed. For a better understanding the dynamics of the underlying equation three dimensional, density and contour graphs are demonstrated for the appropriately selected parameters. Thus, it has become more possible to observe the physical behaviors of the solutions. The results obtained in this study will contribute to the expansion and enrichment of the solution forms available in the literature of the model under consideration. The results obtained could have important applications in nonlinear optics, solid state physics, optical solitons and quantum field theory.

Exact traveling wave solutions for the (2+1)-dimensional double sine-Gordon equation using direct integral method

In this paper, the direct integral method and complete discrimination system for quartic polynomial in standard form are applied to the (2+1)-dimensional double sine-Gordon equation. All possible exact traveling wave solutions of different types are successfully obtained from nine subcases. These exact solutions include kink solutions, and periodic solutions. Different from traditional methods (the Jacobi elliptic function method, the Exp-function method, extended Tanh method and the algebraic mapping method, etc.), the direct integral method described in this work is very natural and simple, and gives a new idea to obtain exact solutions for the mathematical physics equations considered.

Dynamics of kink-soliton solutions of the $$(2+1)$$-dimensional sine-Gordon equation

In this paper we study the dynamics of explicit solutions of $2+1$-dimensional ($2$D) sine-Gordon equation. The Darboux transformation is applied to the associated linear eigenvalue problem to construct nontrivial solutions of $2$D sine-Gordon equation in terms of ratios of determinants. We obtained a generalized expression for $N$-fold transformed dynamical variable which enables us to calculate explicit expressions of nontrivial solutions. In order to explore the dynamics of kink soliton solutions explicit expressions one- and two-soliton solutions are derived for particular column solutions. Different profiles of kink-kink and, kink and anti-kink interactions are illustrated for a different parameters and arbitrary functions. First-order bound state solution is also displayed in our work.

Solving Two Dimensional Coupled Burger's Equations and Sine-Gordon Equation Using El-Zaki Transform-Variational Iteration Method

In this paper, the combined form of the Elzaki transform and variation iteration method is implemented efficiently in finding the analytical and numerical solutions of the two-dimensional nonlinear coupled Burger's partial differential equations and sine-Gordon partial differential equation. The obtained solutions were compared to the exact solutions and other existing methods. Illustrative examples show the efficiency and the power of the used method.

Multiple optical kink solutions for new Painlevé integrable (3+1)-dimensional sine-Gordon equations with constant and time-dependent coefficients

This work establishes entirely new Painlevé integrable (3+1)-dimensional sine-Gordon equations. The first equation is characterized with constant coefficients that nicely passes the Painlevé property to emphasize its integrability, the other model, with time-dependent coefficients, gives the compatibility conditions that exhibit the relations between the time-dependent coefficients. Multiple optical kink wave solutions will be reported for each equation.

New integrable (2+1)-dimensional sine-Gordon equations with constant and time-dependent coefficients: Multiple optical kink wave solutions

This work studies new integrable (2+1)-dimensional sine-Gordon equations, the first is characterized with constant coefficients, while the other is defined with time-dependent coefficients. We show that the first equation passes the Painlevé test for complete integrability. We also study the compatibility conditions that ensure the integrability of the second model with time-dependent coefficients. We show that the dispersion relations of the two equations are distinct, whereas the phase shifts are identical. We apply the simplified Hirota's method where multiple optical kink wave solutions will be reported for each equation.

Embedded solitons in the $$(2+1)$$-dimensional sine-Gordon equation

An effective and simple method to solve nonlinear evolution partial differential equations is the self-similarity transformation, in which one utilizes solutions of the known equation to find solutions of the unknown. In this paper, we employ an improved similarity transformation to transform the $$(2+1)$$-dimensional (D) sine-Gordon (SG) equation into the $$(1+1)$$-D SG equation and obtain non-rational solutions of the $$(2+1)$$-D SG equation by utilizing the known solutions of the $$(1+1)$$-D SG equation. Based on the solutions obtained, and with the help of special choices of the involved solution parameters, several localized structures of the $$(2+1)$$-D SG model are analyzed on a finite background, such as the embedded hourglass, split silo, dumbbell, and pie solitons. Their spatiotemporal profiles are displayed, and their properties are discussed.

A (2+1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions

In this paper, a (2+1)-dimensional sine-Gordon equation and a sinh-Gordon equation are derived from the well-known AKNS system. Based on the Hirota bilinear method and Lie symmetry analysis, kink wave solutions and traveling wave solutions of the (2+1)-dimensional sine-Gordon equation are constructed. The traveling wave solutions of the (2+1)-dimensional sinh-Gordon equation can also be provided in a similar manner. Meanwhile, conservation laws are derived.

A stable time–space Jacobi pseudospectral method for two-dimensional sine-Gordon equation

In this article, we present the stability analysis and approximate solutions of circular ring solitary solitons wave for (2 + 1)-dimensional nonlinear sine-Gordon equation using time–space pseudospectral method. Error bounds on discrete $$L_{2}$$-norm and Sobolev norm $$(H^{p})$$ are presented. The discretization of the problem using proposed method leads a system of nonlinear equations, which are solved using Newton–Raphson method. Further, a study is carried out to investigate the effects of dissipative term in sine-Gordon equation, which presence forms circular ring solitons. To support the theoretical results, the proposed method is tested for two problems, single circular ring solitary soliton waves and the collision of four circular ring solitary soliton waves and numerical results are presented.

Localized dynamical behavior in the (2+1)-dimensional sine-Gordon equation

In this paper, we investigate the (2+1)-dimensional (D) sine-Gordon (SG) equation, to describe the propagation of localized light waves in nonlinear media. The main purpose is to obtain a simple specific form of exact solutions of the (2+1)-D SG equation by Hirota’s bilinear method. The novel dynamical behavior is discussed systematically for some special types of localized excitations, such as multi-dromion, multi-lump, plateau and basin solitons, the quasi-tetragonal breather, and elliptical embedded solitons, which are derived by selecting two arbitrary functions in the solution appropriately. We find that the collision of (2+1)-D solitons could be elastic but also inelastic, depending on the functional form of the solitons.

Numerical solution of two-dimensional nonlinear sine-Gordon equation using localized method of approximate particular solutions

Sine-Gordon equation is one of the most famous nonlinear hyperbolic partial differential equations, it arises in many science and engineering fields. In the present work, we consider the numerical solution of two dimensional sine-Gordon equation using localized method of approximate particular solutions (LMAPS), in this technique, the method of approximate particular solutions (MAPS) occurs on some local domains, that greatly reduces the size of the collection matrix, and by combining the conditional positive radial basis function (RBF) generalized thin plate splines (GTPS) with additional low-order polynomial basis to avoid selecting shape parameters during localization. This method is effective compared with other existing methods and since this method is really meshless, it can be used to solve the nonlinear model with complicated computational domains. Several numerical examples are given to demonstrate the ability and accuracy of the present approach for solving nonlinear sine-Gordon equation.

Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

In this paper, a class of nonlinear algebraic differential equations (NADEs) is studied. The Painlevé analysis of the NADEs is considered. Abundant meromorphic solutions of the NADEs are obtained by means of the complex method. Then, meromorphic exact solutions of the Schamel‐Korteweg‐de Vries (S‐KdV) equation and (2 + 1)‐dimensional sine‐Gordon equation are derived via the applications of the NADEs.

Digging into the Elusive Localised Solutions of (2+1) Dimensional sine-Gordon Equation

Abstract In this paper, we revisit the (2+1) dimensional sine-Gordon equation analysed earlier [R. Radha and M. Lakshmanan, J. Phys. A Math. Gen. 29, 1551 (1996)] employing the Truncated Painlevé Approach. We then generate the solutions in terms of lower dimensional arbitrary functions of space and time. By suitably harnessing the arbitrary functions present in the closed form of the solution, we have constructed dromion solutions and studied their collisional dynamics. We have also constructed dromion pairs and shown that the dynamics of the dromion pairs can be turned ON or OFF desirably. In addition, we have also shown that the orientation of the dromion pairs can be changed. Apart from the above classes of solutions, we have also generated compactons, rogue waves and lumps and studied their dynamics.

A fourth-order AVF method for the numerical integration of sine-Gordon equation

In this paper, a new scheme, which has energy-preserving property, is proposed for solving the sine-Gordon equation with periodic boundary conditions. It is obtained by the Fourier pseudo-spectral method and the fourth order average vector field method. In numerical experiments, the new high order energy-preserving scheme is compared with a number of existing numerical schemes for the one dimensional sine-Gordon equation. The new high order energy-preserving scheme for the two dimensional sine-Gordon equation is also investigated. Numerical results are addressed to further illustrate the conservation of energy and the evolutional behaviors of solitons.

Analytical soliton solutions of the (2 $$+$$ + 1)-dimensional sine-Gordon equation

In this letter, we investigated a new (2\(+\)1)-dimensional sine-Gordon equation. By the subsidiary ordinary differential equation method, some new explicit solutions are given. These solutions include hyperbolic function solutions and trigonometric function solutions amongst others. In particular, a topological 1-soliton solution is derived.

Numerical study of the process of nonlinear supratransmission in Riesz space-fractional sine-Gordon equations

In this work, we consider a (1+1)-dimensional Riesz space-fractional damped sine-Gordon equation defined on a bounded spatial interval. Sinusoidal Dirichlet boundary data are imposed at one end of the interval and homogeneous Neumann conditions at the other. The system is initially at rest in the equilibrium position, and is discretized to simulate its complex dynamics. The method employed in this work is a finite-difference discretization of the mathematical model of interest. Our scheme is throughly validated against simulations on the dynamics of the classical and the space-fractional sine-Gordon equations, which are available in the literature. As the main result of this manuscript, we have found numerical evidence on the presence of the phenomenon of nonlinear supratransmission in Riesz space-fractional sine-Gordon systems. Simulations have been conducted in order to predict its occurrence for some values of the fractional order of the spatial derivative, and a wide range of values of the frequency of the sinusoidal perturbation at the boundary. As far as the author knows, this may be one of the first numerical reports on the existence of nonlinear supratransmission in sine-Gordon systems of Riesz space-fractional order.