Abstract
Slower-than-light multi-front solutions of the Sine-Gordon in (1+2) dimensions, constructed through the Hirota algorithm, are mapped onto spatially localized structures, which emulate free, spatially extended, massive relativistic particles. A localized structure is an image of the junctions at which the fronts intersect. It propagates together with the multi-front solution at the velocity of the latter. The profile of the localized structure obeys the linear wave equation in (1+2) dimensions, to which a term that represents interaction with a slower-than-light, Sine-Gordon-multi-front solution has been added. This result can be also formulated in terms of a (1+2)-dimensional Lagrangian system, in which the Sine-Gordon and wave equations are coupled. Expanding the Euler-Lagrange equations in powers of the coupling constant, the zero-order part of the solution reproduces the (1+2)-dimensional Sine-Gordon fronts. The first-order part is the spatially localized structure. PACS: 02.30.Ik, 03.65.Pm, 05.45.Yv, 02.30.Ik.
Highlights
Data Availability Statement: All relevant data are within the paper
A dynamical variable, ρ, obeys the (1+2)-dimensional linear wave equation, which is driven by an “external field” generated from a multi-front solution of the (1 +2)-dimensional Sine-Gordon equation
In the case of the (1+2)-dimensional Sine-Gordon equation discussed here, this definition allows for the interpretation of the localized structures as emulating free, spatially extended, massive relativistic particles
Summary
Still, localized structures, which emulate spatially extended particles, can be generated from such solutions in two or three space dimensions by a procedure that is a natural consequence of the evolution equation considered. A dynamical variable, ρ, obeys the (1+2)-dimensional linear wave equation, which is driven by an “external field” generated from a multi-front solution of the (1 +2)-dimensional Sine-Gordon equation. [26] (KP II equation) and in this paper (Sine-Gordon equation in (1+2) dimensions), the definition of “particle” mass as the space integral of the profile of a localized structure bears fruits of physical significance. In the case of the (1+2)-dimensional Sine-Gordon equation discussed here, this definition allows for the interpretation of the localized structures as emulating free, spatially extended, massive relativistic particles. For the forms assigned to the functionals F[w] and G[w] in this paper, and with appropriate boundary conditions, the system of Eq (5) admits a solution, which, through first order in the coupling constant, g, describes the (1+2)-dimensional front solutions of the Sine-Gordon equation, and a positive definite, localized solution of the driven wave equation
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