Resonant collisions among two-dimensional localized waves in the Mel’nikov equation

Abstract

We study the resonant collisions among different types of localized solitary waves in the Mel'nikov equation, which are described by exact solutions constructed using Hirota's direct method . The elastic collisions among different solitary waves can be transformed into resonant collisions when the phase shifts of these solitary waves tend to infinity . First, we study the resonant collision among a breather and a dark line soliton. We obtain two collision scenarios: (i) the breather is semi-localized in space and is not localized in time when it obliquely intersects with the dark line soliton, and (ii) the breather is semi-localized in time and is not localized in space when it parallelly intersects with the dark line soliton. The resonant collision of a lump and a dark line soliton, as the limit case of resonant collision of a breather and a dark line soliton, shows the fusing process of the lump into the dark line soliton. Then we investigate the resonant collision among a breather and two dark line solitons. In this evolution process we also obtain two dynamical behaviors: (iii) when the breather and the two dark line solitons obliquely intersect each other we get that the breather is completely localized in space and is not localized in time, and (iv) when the breather and the two dark line solitons are parallel to each other, we get that the breather is completely localized in time and is not localized in space. The resonant collision of a lump and two dark line solitons is obtained as the limit case of the resonant collision among a breather and two dark line solitons. In this special case the lump first detaches from a dark line soliton and then disappears into the other dark line soliton. Eventually, we

Resonant interactions among two-dimensional nonlinear localized waves and lump molecules for the (2+1)-dimensional elliptic Toda equation

“…It should be noted that K 2 ≥ 0 when α 2 + α 3 = 0, so the nonsingular condition of the hybrid solution (38)…”

“…By introducing the resonant condition (40) into the hybrid solution (38), we derive the resonant solution as follows:…”

“…The resonant interaction between the breathers and line solitons in Mel'nikov equation, KP-I equations and (2 + 1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation as well as the resonant interaction between breathers in two-dimensional multi-component longwave-short-wave resonant interaction system were studied in Refs. [38][39][40][41]. In this paper, we investigate the resonant interaction among the line solitons, breathers and lumps for the (2 + 1)-dimensional elliptic Toda equation.…”

Nonlinear interactions of two-kink-breather solution in Yu-Toda-Sasa-Fukuyama equation by modulated phase shift

“…In 1983, Ohkuma and Wadati derived the N-soliton solution of the Kadomtsev-Petviashvili (KP) equation with negative dispersion by the trace method and gave some examples of resonant interactions [31]. Later, some researchers have published some works on the resonant interactions of some exact solutions, such as the resonance stripe solitons, the collisions between lumps and breathers, the resonant interactions of lumps and line solitons [32][33][34][35]. These above examples of resonant interactions were studied in the case where the phase shift is singular, and it is pleasing to note that Tajiri et al considered the interactions of periodic solitons in the case where the phase shift is non-singular, i.e., repulsive and attractive interactions [36,37].…”

Dynamics of degenerate and nondegenerate solitons in the two-component nonlinear Schrödinger equations coupled to Boussinesq equation

“…The vector coupled NLSE and NLSE coupled with other integrable equations are important generalizations of the NLSE in mathematics and physics, such as the M -coupled NLSEs (M-CNLSEs) and the NLSE coupled the Maxwell-Bloch equations, and the NLSE coupled to Boussinesq equation (NLS-Boussinesq). This is due to the fact that the generalizations of the NLSE have a lot of applications in many physical settings, spanning from nonlinear optics [5] to water waves [6,7], plasma physics [8], biophysics [9], Bose-Einstein condensates [10], metamaterial technologies [11], and other research areas [12][13][14][15]. The vector generalizations of the NLSE or their variants are integrable when the corresponding nonlinear coefficients satisfy particular restrictions [16,17].…”