Multi-vortex-like solutions of the sine-Gordon equation (original) (raw)
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New exact solutions of the (3+1)-dimensional double sine-Gordon equation by two analytical methods
Optical and quantum electronics, 2024
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 ′ G 2-expansion and improved tan () 2-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 ′ G 2-expansion and improved tan () 2-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. Keywords (3 + 1)-Dimensional double sine-Gordon equation • Solitons solutions • Modified G ′ G 2-expansion method • Improved tan
On Vortex Solutions and Links between the Weierstrass System and the Complex Sine-Gordon Equations
Journal of Nonlinear Mathematical Physics, 2003
The connection between the complex Sine and Sinh-Gordon equations associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlevé Test and analyse the possibility of deriving the Bäcklund transformation from the singularity analysis of the complex sine-Gordon equation. We make use of the analysis using the known relations for the Painlevé equations to construct explicit formulae in terms of the Umemura polynomials which are τ-functions for rational solutions of the third Painlevé equation. New classes of multivortex solutions of a Weierstrass system are obtained through the use of this proposed procedure. Some physical applications are mentioned in the area of the vortex Higgs model when the complex sine-Gordon equation is reduced to coupled Riccati equations.
Solution of the asymmetric double sine-Gordon equation
Arxiv preprint arXiv:0909.1872, 2009
We present solutions of asymmetric double sine-Gordon equation (DSGE) of an infinite system based on Möbius transformation and numerical exercise. This method is able to give the forms of the solutions for all the region on the ϕ − η parameter plane where ϕ is an additional ...
A study of the solutions of the combined sine–cosine-Gordon equation
Applied Mathematics and Computation, 2009
We have studied the solutions of the combined sine-cosine-Gordon Equation found by Wazwaz (App. Math. Comp. 177, 755 (2006)) using the variable separated ODE method. These solutions can be transformed into a new form. We have derived the relation between the phase of the combined sine-cosine-Gordon equation and the parameter in these solutions. Its applications in physical systems are also discussed.
New Travelling Wave Solutions for Sine-Gordon Equation
Journal of Applied Mathematics, 2014
We propose a method to deal with the general sine-Gordon equation. Several new exact travelling wave solutions with the form ofJacobiAmplitudefunction are derived for the general sine-Gordon equation by using some reasonable transformation. Compared with previous solutions, our solutions are more general than some of the previous.
Symmetries and Exact Solutions of a (2 + 1)-Dimensional Sine-Gordon System
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1996
We investigate the classical and nonclassical reductions of the 2+1-dimensional sine-Gordon system of Konopelchenko and Rogers, which is a strong generalisation of the sine-Gordon equation. A family of solutions obtained as a nonclassical reduction involves a decoupled sum of solutions of a generalised, real, pumped Maxwell-Bloch system. This implies the existence of families of solutions, all occurring as a decoupled sum, expressible in terms of the second, third and fifth Painlevé transcendents, and the sine-Gordon equation. Indeed, hierarchies of such solutions are found, and explicit transformations connecting members of each hierarchy are given. By applying a known Bäcklund transformation for the system to the new solutions found, we obtain further families of exact solutions, including some which are expressed as the argument and modulus of sums of products of Bessel functions with arbitrary coefficients. Finally, we prove the sine-Gordon system has the Painlevé property, which requires the usual test to be modified, and derive a non-isospectral Lax pair for the generalised, real, pumped Maxwell-Bloch system.
The elliptic sine-Gordon equation in a half plane The elliptic sine-Gordon equation in a half plane
We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case. We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197– 201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation. We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.
The (1+1)-dimensional Sine-Gordon equation passes integrability tests commonly applied to nonlinear evolution equations. Its kink solutions (one-dimensional fronts) are obtained by a Hirota algorithm. In higher space-dimensions, the equation does not pass these tests. Although it has been derived over the years for quite a few physical systems that have nothing to do with Special Relativity, the Sine-Gordon equation emerges as a non-linear relativistic wave equation. This opens the way for exploiting the tools of the Theory of Special Relativity. Using no more than the relativistic kinematics of tachyonic momentum vectors, from which the solutions are constructed through the Hirota algorithm, the existence and classification of N-moving-front solutions of the (1+2)-and (1+3)-dimensional equations for all N ! 1 are presented. In (1+2) dimensions, each multi-front solution propagates rigidly at one velocity. The solutions are divided into two subsets: Solutions whose velocities are lower than a limiting speed, c = 1, or are greater than or equal to c. To connect with concepts of the Theory of Special Relativity, c will be called "the speed of light." In (1+3)-dimensions, multifront solutions are characterized by spatial structure and by velocity composition. The spatial structure is either planar (rotated (1+2)-dimensional solutions), or genuinely three-dimensionalbranes. Planar solutions, propagate rigidly at one velocity, which is lower than, equal to, or higher than c. Branes must contain clusters of fronts whose speed exceeds c = 1. Some branes are "hybrids": different clusters of fronts propagate at different velocities. Some velocities may be lower than c but some must be equal to, or exceed, c. Finally, the speed of light cannot be approached from within the subset of slower-than-light solutions in both (1+2) and (1+3) dimensions.
Analytical Solution of Two-Dimensional Sine-Gordon Equation
Advances in Mathematical Physics, 2021
In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the ...