A new derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons
The computation of the amplitude, $α$, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg-de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, $ε^2\ll1$, a new derivation of the ``beyond all orders in $ε$'' amplitude, $α$, is presented. It is shown by asymptotic matching techniques, extended to higher orders in $ε$, that the value of $α$ can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [Proc. R. Soc. Lond. A 424, 19 (1989)], not only sheds new light on the computation of $α$, but also greatly facilitates its numerical determination to a remarkable precision for so small values of $ε$, which are beyond the capabilities of standard numerical methods.