New derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons

The computation of the amplitude, $\alpha$, 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, ${\epsilon }^2\ll 1$, a new derivation of the “beyond all orders in $\epsilon$” amplitude, $\alpha$, is presented. It is shown by asymptotic matching techniques, extended to higher orders in $\epsilon$, that the value of $\alpha$ 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. A 424, 19 (1989)], not only sheds new light on the computation of $\alpha$ but also greatly facilitates its numerical determination to a remarkable precision for so small values of $\epsilon$, which are beyond the capabilities of standard numerical methods.