Generating mechanism and dynamic of the smooth positons for the derivative nonlinear Schrödinger equation

Abstract

Based on the degenerate Darboux transformation, the n-order smooth positon solutions for the derivative nonlinear Schrödinger equation are generated by means of the general determinant expression of the N -soliton solution, and interesting dynamic behaviors of the smooth positons are shown by the corresponding three dimensional plots in this paper. Furthermore, the decomposition process, bent trajectory and the change of the phase shift for the positon solutions are discussed in detail. Additional, three kinds of mixed solutions, namely (1) the hybrid of one-positon and two-positon solutions, (2) the hybrid of two-positon and two-positon solutions, and (3) the hybrid of onesoliton and three-positon solutions are presented and their rather complicated dynamics are revealed.

Breather Positons and Rogue Waves for the Nonlocal Fokas-Lenells Equation

“…It is always a difficult problem to find the trajectories of breather positons [10,16,17]. Not to mention, let us study the dynamic properties of breather positons like decomposing smooth positons [7][8][9].…”

“…Similarly, their team applied this mechanism to other integrable equations, such as the modified KdV equation (5) and Fokas-Lenells (FL) equation (7), to obtain higher-order rogue waves. Since Matveev [5,6] proposed singular positon solutions for the Korteweg-de Vries equation in 1992, many experts [7][8][9][10][11] have been working in this field. Among the numerous studies on positons, Wang et al [10] elaborate the connection between breather positons and rogue waves for a local integrable system.…”

Nth order smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger equation

“…Non-singular positon solutions on vanishing background are named as smooth positons or degenerate soliton solutions [26][27][28][29][30][31][32]. This kind of solution has also been constructed for several nonlinear partial differential equations including nonlinear Schrödinger (NLS) equation [25,26], Bogoyavlensky-Konoplechenko equation [27], coupled KdV [28] and mKdV equations [29], derivative NLS equation [30], nonlocal Kundu-NLS equation [31], complex mKdV equation [32], Wadati-Konno-Ichikawa equation [33] and higher-order Chen-Lee-Liu equation [34]. Recently positons on nonvanishing background for the NLS equation have also been constructed and they have been coined as breather-positon (B-P) solutions.…”

Higher order smooth positon and breather positon solutions of an extended nonlinear Schrödinger equation with the cubic and quartic nonlinearity

“…The singular and smooth positon solutions have been constructed for a class of integrable equations [21,22,23,24,25,26,27,28,29,30,31,32]. Breather-positons (b-p) are equal amplitude breathers (localized periodic waves on constant background) and they travel with equal speed [33,34,35,36,37,38].…”