Then-component nonlinear Schrödinger equations: dark–bright mixedN- and high-order solitons and breathers, and dynamics

Abstract

The general n -component nonlinear Schrödinger equations are systematically investigated with the aid of the Darboux transformation method and its extension. Firstly, we explore the condition of the existence for dark–bright mixed soliton solutions and derive an explicit formula of dark–bright mixed multi-soliton solutions in terms of the determinant. Secondly, we present the formula of dark–bright mixed high-order semi-rational solitons, and predict their general N th-order wave structures. Thirdly, we investigate the wing-shaped structures of breather. Finally, we perform the numerical simulations for some representative solitons to study their dynamical behaviours.

Rational and semi‐rational solutions of a nonlocal (2 + 1)‐dimensional nonlinear Schrödinger equation

“…where i , b i , and ij are defined in (21). Substituting (27) into (19), the two-rogue waves can be derived.…”

“…where Γ 12 , b i , i (i = 1, 2, 3, 4) are given by (21), e A 34 , i , i (i = 3, 4) are given by (14). R s , P l , and Q l are all arbitrary real constants.…”

“…Recently, many attentions have been focused on the research of breathers, rogue waves, and lump waves. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] Lump and rogue waves, both are rational solutions, are observed in many physical phenomena, such as nonlinear optic media, superfluids and Bose-Einstein condensates, and so on. [22][23][24][25][26][27][28][29] In addition, searching for semi-rational solutions of NLEEs also has caused concern for scientists.…”

“…For instance, Hirota bilinear method, Darboux transformation (DT), inverse scattering transformation (IST), and so on. Recently, many attentions have been focused on the research of breathers, rogue waves, and lump waves . Lump and rogue waves, both are rational solutions, are observed in many physical phenomena, such as nonlinear optic media, superfluids and Bose‐Einstein condensates, and so on .…”

The general coupled Hirota equations: modulational instability and higher-order vector rogue wave and multi-dark soliton structures

“…Firstly, distinct kinds of localized wave solutions can coexist with each other. For example, the coexistence of rogue waves (RWs) and breathers or solitons can emerge, which are the so-called semirational RWs [2][3][4][5][6][7][8]. Secondly, in contrast to the usual eye-shaped RWs in the scalar equations [9,10], many novel excitation dynamical patterns for the vector RWs will appear.…”