Asymptotics of the multipositon–soliton τ function of the Korteweg–de Vries equation and the supertransparency

Abstract

The asymptotics of the τ function generating the N‐positon–M‐soliton solution of the Korteweg–de Vries equation is calculated. This allows to prove that solitons do not experience any phase shift in a collision with positons. The positons themselves survive mutual collisions unchanged. This phenomenon is called the supertransparency or super‐reflectionless property of the multipositon solutions. The linear aspects of this phenomenon are also discussed. It is demonstrated that positons acquire two additional but always finite phase shifts in collision with solitons. This result admits a natural extension to any number of solitons and positons in interaction.