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We show how the famous soliton solution of the classical sine-Gordon field theory in (1 + 1)-dimensions may be obtained as a particular case of a solution expressed in terms of the Jacobi amplitude, which is the inverse function of the incomplete elliptic integral of the first kind.

The sine-Gordon field theory and the associated massive Thirring model [

In this work we present a simple and yet appealing step-by-step derivation of a more general solution for the classical sine-Gordon field theory in (1 + 1)-dimensions in terms of a special kind of elliptic function, namely the Jacobi amplitude, which has the famous sine-Gordon soliton solution as a particular case. Despite the fact that the connection between solitons and Jacobi elliptic functions has already been explored in [

We start by considering the following theory describing a real scalar field in (1 + 1)-dimensions

where the potential term is given by

with

The above Lagrangian gives rise, through the Euler-Lagrange equation,

Notice that since Equation (3) is invariant under Lorentz transformations

Indeed, by multiplying the above equation by

which, after an integration with respect to

By integrating both sides of the above equation, from

In order to compute the above integral, we must firstly notice that the potential, shown in Equation (2), may be rewritten as

Thus, by making the change of variables

The integral appearing in Equation (9) is called an incomplete elliptic integral of the first kind,

Notice that, from the above definition, we have

The solution of Equation (4) may be, finally, written as

Hence, from the above equation, we may notice that

as it should.

From the definition

may be obtained as a special case of the solution presented in Equation (11). Indeed, since

where

Last but not least, we must notice that by substituting the Equation (14) into Equation (3) and making the

change (Lorentz boost)

where

This result allows us to characterize the Lorentz boosted, and shifted by

as a generalization of the sine-Gordon soliton/anti-soliton solution for

We would like to make a few comments about the soliton solution, shown in Equation (16), and its generalized version, shown in Equation (17). Firstly, we may notice by comparing

Finally, let us observe that, as remarked in [

is finite, just as we should expect.

This work has been supported by University of Alberta’s Li Ka Shing Applied Virology Institute and CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brasil.