Soliton Molecules and Some Hybrid Solutions for the Nonlinear Schrödinger Equation^*

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.

Propagation and interaction between special fractional soliton and soliton molecules in the inhomogeneous fiber

“…However, theoretical investigation on soliton molecules was less carried out. Until fairly recently, the formation mechanism of soliton molecules was theoretically proposed [3] , [19] , [20] , [21] . Soliton molecules based on fractional nonlinear models(FNMs) are hardly reported although many fractional soliton structures have been studied [22] , [23] .…”

Linear stability of exact solutions for the generalized Kaup-Boussinesq equation and their dynamical evolutions

“…In the same way, there are also many other trivial degenerations for the periodic wave solution (31). For example, when the roots satisfy u 1 = u 2 and u 3 = u 4 , the periodic wave solution (31) is reduced to the constant solution u = u 2 , while it degenerates into the constant solution u = u 3 for u 1 = u 2 = u 3 = u 4 .…”

“…In this sense, the soliton theory [13] provides efficient tools for studying integrable [1] and nonintegrable systems [32]. In particular, as an important branch of nonlinear science, soliton theory has important applications in nonlinear optics [14], fluid mechanics [16,7,8,6], Bose-Einstein condensates [2] and plasma physics [24,33,31].…”

“…When the "Potential" curve Q(u) has four mutually distinct real roots shown in Figure 2, integrating the ordinary differential equation ( 19) yields the periodic wave solution in term of the Jacobi elliptic function as where W 2 = (ξ −ξ 0 ) (u 1 − u 3 )(u 2 − u 4 )/2 and the modulus of the Jacobian elliptic sn function is m 2 = (u1−u2)(u3−u4) (u1−u3)(u2−u4) , 0 ≤ m 2 ≤ 1. When the roots of the "Potential" curve Q(u) satisfy u 1 = u 2 and u 3 = u 4 , the periodic wave solution (31) can degenerate into the following periodic wave solution in term of the trigonometric function…”

“…When the roots u 2 = u 3 , the periodic wave solution (31) degenerates into the soliton solution of the form…”