Narrow Results

The $N$-soliton solution of the $(2+1)$-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt (gKDKK) equation is constructed through taking the Bäcklund transformation, the Hirota bilinear method and the auxiliary function $f$, which is combined by several elementary functions. The line soliton molecule, lump and breather solutions are first presented. The explicit expressions of the soliton molecules, including the line soliton mixed with breather/lump, can also be constructed. The interaction solutions among soliton molecule, lump and breather structures are finally achieved. These results mainly introduce the ideas of velocity resonance, module resonance of wavenumber and long-wave limit. In order to illustrate these phenomena, the analytical solutions are shown and the dynamical features are depicted through the Maple software.

The $N$-soliton solutions of the (2+1)-dimensional Kadomtsev–Petviashvili hierarchy are first constructed. One soliton molecule satisfies the velocity resonance condition, the breather with the periodic solitary wave, the lump soliton localized in all directions in the space are showed successively for $N=2$. Interaction of one soliton molecule and a line soliton, the soliton molecule hybrid a line soliton with the breather/lump soliton are presented for $N=3$. Moreover, the elastic interaction between two-soliton molecules, the interaction between one soliton molecule, and a breather and the elastic collision between the lump soliton and one soliton molecule are also derived for $N=4$ by applying the velocity resonance, the module resonance of wave number, and the long-wave limit ideas. Figures are presented to demonstrate these dynamics features.

The $N$-soliton solution of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation is constructed. The line soliton molecule, the breather and the lump soliton are presented successively for $N=2$. The three-soliton molecule structure, interaction of one-soliton molecule and a line soliton, the soliton molecules consisting of a line soliton and the breather/lump soliton of the solution $u$ are constructed for $N=3$. Moreover, the four-soliton molecule structure, interaction of the soliton molecule and a line soliton, the soliton molecule consisting of the line soliton molecule and a lump soliton, the elastic interaction between the line soliton molecule and a lump soliton, the soliton molecules consisting of the line soliton molecule and the breather, two breather solitons, the breather soliton and a lump of the variable $u$ for this equation are also derived for $N=4$ by applying the velocity resonance, the module resonance of wave number and the long-wave limit ideas. To illustrate these phenomena, the analysis explicit solutions are all given and their dynamics features are all displayed through figures.

In recent years, soliton molecules have received reinvigorating scientific interests in physics and other fields. Soliton molecules have been successfully found in optical experiments. In this paper, we attribute the solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation by employing the bilinear method. Based on the $N$-soliton solutions, we establish the soliton molecules, asymmetric solitons and some novel hybrid solutions of this equation by means of the velocity resonance mechanism and the long wave limit method. Finally, we give dynamic graphs of soliton molecules, asymmetric solitons and some novel hybrid solutions.

Traveling wave solution is one of the effective methods for solving nonlinear partial differential equations. D’Alembert solution is a special kind of traveling wave solution. There have been many studies about D’Alembert solution. In this paper, we will solve D’Alembert-type wave solutions for (2+1)-dimensional generalized Nizhnik–Novikov–Veselov equation. Based on the Hirota bilinear transformation and velocity resonance mechanism, the states of soliton molecules composed of two solitons, three solitons and four solitons are studied. It is concluded that D’Alembert-type wave is closely related to soliton molecules.

Modulating dark optical fiber vector soliton molecules are theoretically investigated in this work. An optical fiber modulation system that out of fiber laser cavity is employed as a simulation model, which can flexibly change orthogonal electric fields’ properties of original optical fiber vector solitons. We consider two cases for simulation, one is original orthogonal polarization modes have the same central wavelength in the frequency domain, the other is original orthogonal polarization modes have different central wavelengths. In the first case, modulated dark vector soliton molecules with two pulse peaks and two pulse dips, accompanied by two wavelengths can always be observed. While in the second case, optical spectra will split in orthogonal directions, and obvious temporal pulse oscillation will occur, when the projection angle changes. The results further explore modulating vector solitons that with unique characteristics, in out-cavity optical fiber modulation systems.