Nanopteron solution of the Korteweg-de Vries equation

TL;DR

This paper derives an explicit nanopteron solution for the KdV equation, revealing unique wave interactions where cnoidal waves experience a universal phase shift after soliton interaction.

Contribution

It provides the first analytical nanopteron solution for the KdV equation, connecting soliton-cnoidal wave solutions with nanopteron structures.

Findings

Nanopteron solutions exist for the KdV equation.

The soliton core approaches a classical soliton under certain parameters.

Cnoidal waves experience a universal half-wavelength phase shift after interaction.

Abstract

The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interesting topic in numerical study for many decades. However, analytical solution of such a special soliton is rarely considered. In this Letter, 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 the suitable choice of the wave parameters, the soliton core of the soliton-cnoidal wave trends to be a classical soliton of the KdV equation and the surrounded cnoidal periodic wave appears as small amplitude sinusoidal variations on both sides of the main core. Some interesting features of the wave propagation are revealed. In addition to the elastic interaction, it is surprising that the phase shift of the cnoidal periodic wave after the interaction with the soliton core is always half of its wavelength, and this conclusion is universal to soliton-cnoidal wave interactions.