Soliton Molecules and Some Hybrid Solutions for the Nonlinear Schrödinger Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11775121, Department of Education of Zhejiang Province under Grant No. Y201839043 and the K.C. Wong Magna Fund in Ningbo University.

Abstract

Based on velocity resonance and Darboux transformation, soliton molecules and hybrid solutions consisting of soliton molecules and smooth positons are derived. Two new interesting results are obtained: the first is that the relationship between soliton molecules and smooth positons is clearly pointed out, and the second is that we find two different interactions between smooth positons called strong interaction and weak interaction, respectively. The strong interaction will only disappear when t → ∞. This strong interaction can also excite some periodic phenomena.