Forces between kinks and antikinks with long-range tails

In a scalar field theory with a symmetric octic potential having a quartic minimum and two quadratic
minima, kink solutions have long-range tails. We calculate the force between two kinks and between a
kink and an antikink when their long-range tails overlap. This is a nonlinear problem, solved using
an adiabatic ansatz for the accelerating kinks that leads to a modified, first-order Bogomolny
equation. We find that the kink–kink force is repulsive and decays with the fourth power of the kink
separation. The kink–antikink force is attractive and decays similarly. Remarkably, the kink–kink
repulsion has four times the strength of the kink–antikink attraction.

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