Soliton molecules may exists in both experimental and theotetical aspects. In this work, we investigate the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation, which can be used to describe weakly dispersive waves propagating in the quasi media and fluid mechanics. Soliton molecules are generated by N-soliton solution and a new velocity resonance condition. Furthermore, soliton molecules can become to asymmetric solitons when the distance between two solitons of the molecule is small enough. Based on the N-soliton solution, we obtain some novel interaction solutions which component of soliton molecules, breather waves and lump waves by deal with part of parameters by applying velocity resonance, module resonance and long wave limit method, and the interactions are elastic. Finally, some graphic analysis are discussed to understand the propagation phenomena of these solutions.

In this letter, we investigate multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas–Lenells equation over a nonzero background. First, we obtain 2n-soliton solutions with a nonzero background via n-fold Darboux transformation, and find that these soliton solutions will appear in pairs. Particularly, 2n-soliton solutions consist of n ‘bright’ solitons and n ‘dark’ solitons. This phenomenon implies a new form of integrability: even integrability. Then interactions between solitons with even numbers and breathers are studied in detail. To our best knowledge, a novel nonlinear superposition between a kink and 2n-soliton is also generated for the first time. Finally, interactions between some different smooth positons with a nonzero background are derived.

In this paper, we investigate breather positons and higher-order rogue waves for the nonlocal Fokas-Lenells equation. In this nonlocal optical system, rogue waves can be generated when periods of breather positons go to infinity. In addition, we find two very interesting phenomena: one is that rogue waves sitting on a periodic line wave background are derived; the other is that a hybrid of rogue waves and a periodic kink wave is also constructed. We believe that these interesting findings exist in the optical system corresponding to the nonlocal Fokas-Lenells equation.

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