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Based on the degenerate Darboux transformation, the n-positon solution of the higher-order Chen–Lee–Liu (HOCLL) equation are obtained by the special limit $$\lambda _{j}\rightarrow \lambda _{1}$$ taking from the corresponding n-soliton solution, and using the higher-order Taylor expansion. Using the method of the modulus square decomposition, n-positon is decomposed into n single soliton solutions. The dynamic properties of smooth positon of the HOCLL equation are discussed in detail, and the corresponding trajectory, approximate trajectory and “phase shift” are obtained. In addition, the mixed solutions of soliton and positon are discussed, and the corresponding three-dimensional map are given.

In this paper, some degenerate solutions of the spatial discrete Hirota equation are constructed via the degenerate idea of positon solution. Under the zero seed solution, the n-positon is obtained by N-fold degenerate Darboux transformation (DT). The degenerate DT is taking the degenerate limit $$\lambda _{j}\rightarrow \lambda _{1}$$ for the eigenvalues $$\lambda _{j}(j=1,2, 3, \ldots , N)$$ of N-fold DT and then performing the high-order Taylor expansion near $$\lambda _{1}$$ . Considering the universal Darboux transformation, breather is obtained from the nonzero seed. Then, a new type of breather solution can be produced by using the same degenerated method and higher-order Taylor expansion for eigenvalues in determinant expression of breather solution. The explicit determinants of breather-positon solution and positon solution are constructed, respectively, and the complicated and significant dynamics of low-order solution are also revealed.

In this work, the $$(2+1)$$ -dimensional extended Kadomtsev–Petviashvili equation, which models the surface waves and internal waves in straits or channels, is investigated via the Hirota bilinear method. N-soliton and high-order breather solutions are obtained analytically. Furthermore, mixed solutions consisting of first-order breathers and solitons are also derived, and the corresponding dynamic behaviors are shown by three-dimensional plots. Additionally, based on the long-wave limit, we obtain line rogue waves, lumps and semi-rational solutions composed of lumps, line rogue waves and solitons. It is noteworthy that the semi-rational solutions derived in this paper exhibit elastic interactions.