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

The computation of the amplitude, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>α</a:mi></a:math>, 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, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>ε</c:mi><c:mn>2</c:mn></c:msup><c:mo>≪</c:mo><c:mn>1</c:mn></c:math>, a new derivation of the “beyond all orders in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>ε</e:mi></e:math>” amplitude, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>α</g:mi></g:math>, is presented. It is shown by asymptotic matching techniques, extended to higher orders in <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>ε</i:mi></i:math>, that the value of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>α</k:mi></k:math> 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 [], not only sheds new light on the computation of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>α</m:mi></m:math> but also greatly facilitates its numerical determination to a remarkable precision for so small values of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>ε</o:mi></o:math>, which are beyond the capabilities of standard numerical methods.