Volume 108, Number 2, October 2014
|Number of page(s)||6|
|Published online||16 October 2014|
Nanopteron solution of the Korteweg-de Vries equation
1 Department of Physics and Astronomy, Shanghai Jiao Tong University - Shanghai 200240, China
2 Department of Mathematics and Physics, Quzhou University - Quzhou 324000, China
3 Institute of Systems Science, East China Normal University - Shanghai 200241, China
4 Faculty of Science, Ningbo University - Ningbo 315211, China
5 Shanghai Key Laboratory of Trustworthy Computing, East China Normal University - Shanghai 200062, China
Received: 1 July 2014
Accepted: 24 September 2014
The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interesting topic in numerical studies for many decades. However, the 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 its wavelength, and this conclusion is universal to soliton-cnoidal wave interactions.
PACS: 02.30.Ik – Integrable systems / 05.45.Yv – Solitons / 52.35.Mw – Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)
© EPLA, 2014
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