Negaton and positon solutions of the KdV and mKdV hierarchy

Abstract

We give a systematic classification and a detailed discussion of the structure, motion and scattering of the recently discovered negaton and positon solutions of the Korteweg-de Vries hierarchy. There are two distinct types of negaton solutions which we label [Sⁿ] and [Cⁿ], where (n + 1) is the order of the Wronskian used in the derivation. For negatons, the number of singularities and zeros is finite and they show very interesting time dependence. The general motion is in the positive x direction, except for certain negatons which exhibit one oscillation around the origin. In contrast, there is just one type of positon solution, which we label [C^~n] . For positons, one gets a finite number of singularities for n odd, but an infinite number for even values of n. The general motion of positons is in the negative x direction with periodic oscillations. Negatons and positons retain their identities in a scattering process and their phase shifts are discussed. We obtain a simple explanation of all phase shifts by generalizing the notions of 'mass' and 'centre of mass' to singular solutions. Finally, it is shown that negaton and positon solutions of the KdV hierarchy can be used to obtain corresponding new solutions of the modified KdV hierarchy.