Nanopteron solution of the Korteweg-de Vries equation with an application in ion-acoustic waves

The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interesting topic in numerical study for more than three decades. However, analytical solution of such a special soliton is rarely considered. In this paper, we study the explicit nanopteron solution of the Korteweg-de Vries (KdV) equation. Starting from the soliton-cnoidal wave solution of the KdV equation, the nanopteron structure is shown to exist. It is found that for suitable choice of wave parameters the soliton core of the soliton-cnoidal wave trends to be the classical soliton of the KdV equation and the surrounded cnoidal periodic wave appears as small amplitude sinusoidal variations on both side of the soliton core. Some interesting features of the wave propagation are revealed. In addition to the elastic interactions, it is surprising to find that the collision-induced phase shift of the cnoidal periodic wave is always half of its wavelength and this conclusion is universal to the soliton-cnoidal wave interaction. In the end, the nanopteron structure of the KdV equation is revealed in a plasma physics system. It is confirmed that the influence of plasma parameters on the nanopteron structure is in agreement with the classical soliton.