  We theoretically analyse the equation of topological solitons in a chain of
particles interacting via a repulsive power-law potential and confined by a
periodic lattice. Starting from the discrete model, we perform a gradient
expansion and obtain the kink equation in the continuum limit for a power law
exponent $n \ge 1$. The power-law interaction modifies the sine-Gordon
equation, giving rise to a rescaling of the coefficient multiplying the second
derivative (the kink width) and to an additional integral term. We argue that
the integral term does not affect the local properties of the kink, but it
governs the behaviour at the asymptotics. The kink behaviour at the center is
dominated by a sine-Gordon equation and its width tends to increase with the
power law exponent. When the interaction is the Coulomb repulsion, in
particular, the kink width depends logarithmically on the chain size. We define
an appropriate thermodynamic limit and compare our results with existing
studies performed for infinite chains. Our formalism allows one to
systematically take into account the finite-size effects and also slowly
varying external potentials, such as for instance the curvature in an ion trap.
