Multitime sine-Gordon solitons via geometric characteristics (original) (raw)
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On multi soliton solutions of the Sine-Gordon equation in more than one space dimension
2013
The (1+1)-dimensional Sine-Gordon equation passes integrability tests commonly applied to nonlinear evolution equations. Its soliton solutions are obtained by a Hirota algorithm. In higher space-dimensions, the equation does not pass these tests. In this paper, using no more than the relativistic kinematics of the tachyonic momentum vectors, from which the soliton solutions are constructed through the Hirota algorithm, the existence and classification of N-soliton solutions of the (1+2)- and (1+3)-dimensional equations for all N greater than or equal to 1 are presented. In (1+2) dimensions, each multisoliton solution propagates rigidly at one velocity. The solutions are divided into two subsets: Solutions whose velocities are lower than the speed of light (c = 1), or are greater than or equal to c. In (1+3)-dimensions, multisoliton solutions are characterized by spatial structure and velocity composition. The spatial structure is either planar (rotated (1+2)-dimensional solutions), ...
Dynamics of sine-Gordon solitons
After reviewing a few physical examples in which the sine-Gordon equation arises as the governing dynamical equation, we discuss various solutions exhibiting multisoliton dynamics. Interaction of solitons and the corresponding velocitydependent interaction potentials are derived and discussed. Numerical experiments are carried out in order to study kink dynamics in an inhomogeneous medium. Finally, we introduce two kinds of generalized sine-Gordon equations and discuss their properties.
arXiv: Exactly Solvable and Integrable Systems, 2013
The Sine-Gordon equation in (1+2) dimensions has N-soliton solutions that propagate at velocities that are lower than the speed of light (c = 1), for any N greater tha or equal to 1. A first integral of the equation, which vanishes identically on the single soliton solution, maps multisoliton solutions onto structures that are localized around soliton junctions. The profile of such a structure obeys the (1+2)-dimensional linear wave equation, driven by a source term, which is constructed from a multisoliton solution of the Sine-Gordon equation. If the localized solutions of the source-driven wave equation are interpreted as mass densities, they emulate free, spatially extended, massive relativistic particles. This physical picture is summarized in terms of a Lagrangian density for a dynamical system, in which the Sine-Gordon equation and the linear wave equation are coupled by a small coupling term. The Euler-Lagrange equations of motion allow for solutions, which, in lowest order i...
Contrary to the common understanding, the sine-Gordon equation in (1 + 2) dimensions does have N-soliton solutions for any N. The Hirota algorithm allows for the construction of static N-soliton solutions (i.e., solutions that do not depend on time) of that equation for any N. Lorentz transforming the static solutions yields N-soliton solutions in any moving frame. They are scalar functions under Lorentz transformations. In an N-soliton solution in a moving frame, (N-2) of the (1 + 2)-dimensional momentum vectors of the solitons are linear combinations of the two remaining vectors. C 2013 American Institute of Physics. [http://dx.
Soliton Solutions in a Modified Double and Triple Sine-Gordon Models
2016
We modify both the double sine-Gordon (DSG) and triple sine-Gordon (TSG) model in (1,1) dimensions by the addition of an extra kinetic term and a potential term to their Lagrangian density and present a modified DSG (MDSG) and a modified TSG (MTSG) models. We obtain soliton solutions of the presented modified models and find that both of them possesses the same solutions of the unmodified model with some extra conditions imposed on the parameters of the models. We study some properties of the modified models, in particular, we show that the corresponding governing equation has two solutions, a special ones, which are the exact solutions of the unmodified models and a general ones, and these two types of solutions are coincides in our presented models. We end the paper with conclusions and some features and comments.
Contrary to the decades-old understanding, SGn, the Sine-Gordon equation in (1+n) dimensions, has N-soliton solutions for any N >= 1, not only for n = 1, but also for n = 2 and 3. While SG1 solitons are confined to a line, SG2- and SG3-solitons are confined to a plane. An SG2-soliton solution moves rigidly with a constant velocity in the plane, and an SG3-solution moves rigidly with a constant velocity in the plane, and along the normal to the plane. A conservation law for the current density, obeyed by the single-SGn-soliton solution, is violated by all multi-soliton solutions. The violation manifests itself by generating vertices, structures that are localized around the soliton collision regions and decay exponentially in all directions in the (1+n)-dimensional space. In (1+1) dimensions, vertices evolve and then decay. In (1+2) and (1+3) dimensions, they move with the whole solution at its constant velocity, preserving their profiles, thereby emulating free, spatially extende...
3 soliton solution to Sine-Gordon equation on a space scale
Journal of Mathematical Physics
Hirota's direct method is extended on a variable time-space scale. Using this extension, we construct 3-soliton solutions of the Sine-Gordon equation on a variable space scale. The determinant form of this solution is given as well.
2011
Multitime evolution PDEs for Rayleigh waves are considered, using geometrical ingredients capable to build an ultra-parabolic-hyperbolic differential operator. Their soliton solutions are found based on appropriate hypotheses and Bernoulli ODEs. These multitime solitons develop complex behavior of deformation phenomena. Section 1 presents the single-time Rayleigh wave equations. Section 2 analyzes the geometric characteristics (fundamental tensor, linear connection, vector fields, tensor
Two-dimensional gravitation and sine-Gordon solitons
Physical Review D, 1995
Some aspects of two-dimensional gravity coupled to matter fields, especially to the Sine-Gordon-model are examined. General properties and boundary conditions of possible soliton-solutions are considered. Analytic soliton-solutions are discovered and the structure of the induced space-time geometry is discussed. These solutions have interesting features and may serve as a starting point for further investigations.
2013
The Sine-Gordon equation in (1+2) dimensions has N-soliton solutions that propagate at velocities that are lower than the speed of light (c = 1), for any N ≥ 1. A first integral of the equation, which vanishes identically on the single-soliton solution, maps multi-soliton solutions onto structures that are localized around soliton junctions. The profile of such a structure obeys the (1+2)-dimensional linear wave equation, driven by a source term, which is constructed from a multi-soliton solution of the Sine-Gordon equation. If the localized solutions of the source-driven wave equation are interpreted as mass densities, they emulate free, spatially extended, massive relativistic particles. This physical picture is summarized in terms of a Lagrangian density for a dynamical system, in which the Sine-Gordon equation and the linear wave equation are coupled by a small coupling term. The Euler-Lagrange equations of motion allow for solutions, which, in lowest order in the coupling constant are the soliton solutions of the Sine-Gordon equation, and the first-order component are the structures that emulate spatially extended relativistic particles.