We will look at the Soliton solutions in the sine-Gordon equation (which also shares correspondence with the (massive) Thirring model in perturbation theory). Let us first see a standard example of soliton in field theory. We take a non-linear scalar field theory $\phi$ with Lagrangian

$$\mathcal{L} = -\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - U(\phi)$$

where potential $U(\phi)$ is described by

$$U(\phi)=\lambda\left(\phi^2-m^2/\lambda\right)^2/4$$

and the dimensionless coupling constant is $g=\lambda/m^2$. Here, $m$ is the mass of the elementary solutions of $\phi$. Then we define the  topological current

$$j_u = \frac{\sqrt{g}}{2}\epsilon_{\mu \nu}\partial^\nu \phi$$

$$Q = \int_{-\infty}^{\infty}dx\ j_0$$

integrating it becomes

$$Q = \frac{\sqrt{g}}{2}\left(\phi(\infty)-\phi(-\infty)\right)$$

where the $\infty$ is for a kink solution and $-\infty$ is for an anti-kink solution. These kinks deserve our attention here. $\phi$ varies from the minimum of $U(\phi)$ at $\phi = \mp 1/\sqrt{g}$ at $x=\infty$ to the minimum of $U(\phi)$ at $\phi = \pm 1/\sqrt{g}$ at $x=-\infty$. We can write a solution to this equation, which follows

$$\phi^{''}=\frac{\partial U}{\partial \phi}$$

integrating this with $U$ with $\phi'$ vanishing at infinity we get

$$\frac{1}{2}(\phi^{'})^{2} = U(\phi).$$

Integrating this now over our choice of $U$ will give us the kink (k) and anti-kink (k') solution

$$\phi(x)_{k(k')} = \pm \frac{m}{\sqrt{\lambda}} tanh\left[m(x-x_0)/\sqrt{2}\right].$$

The rest mass for the soliton is given by

$$E = \int dx \frac{1}{2}\left(\phi^{'}\right)^{2} + U(\phi) = \frac{2\sqrt{2}}{3}\frac{m}{g}$$

which clearly states that kink (rest) mass divided by the $m$ is proportional to $1/g$. This is also an indication that solitons are non-perturbative physics.

Anyway, the previous example was about solutions of just one theory, where kink and elementary solutions shared a relation. The nature of these kinks will be apparent in the next post. Now, what about a duality between two sectors of different theories. For this, we will turn to the massive Thirring model, which shares a correspondence with the sine-Gordon theory, in a next post.