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It is well-known that the optimal transport problem on the real line for theclassical distance cost may not have a unique solution. In this paper werecover uniqueness by considering the transport problems where the costs are apower smaller than one of the distance, and letting this parameter tend to one.A complete construction of this solution that we call excursion coupling isgiven. This is reminiscent to the one in the convex case. It is alsocharacterized as the solution of secondary transport problems. Moreover, acombinatoric/geometric characterization of the routes used for this transportplan is provided.

On a solution to the Monge transport problem on the real line arising from the strictly concave case