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A novel type of exact rogue wave is found for the (1+1)-dimensional Ito equation, which is generated by the interaction solution between an algebraic localized soliton (named “lump”) and an exponentially localized twin soliton. In addition, the interaction solution among triangular periodic wave and twin soliton is also proposed. Three special interaction phenomenons are displayed by some visual figures, respectively.

The paper aims to explore the existence of diverse lump and interaction solutions to the (1+1)-dimensional Sharma–Tasso–Olver–Burgers equation. Through the Cole–Hopf transformation with the nonzero seed solution, the remarkable richness of exact solutions are exhibited, including lump, lump-periodic, lump-multiple solitary wave and periodic-multiple solitary wave. Some specific interaction phenomena are analyzed by the limit behavior. The generated rogue wave, half periodic kink (HPK) and breather-like solutions are displayed by some visual figures, respectively.

The truncated Painleve expansion is developed to construct Backlund transformations and nonlocal symmetries of the Bogoyavlenskii coupled KdV (BcKdV) system. The Schwarzian form of BcKdV system is found while the nonlocal symmetry is localized to offer the corresponding nonlocal group. Furthermore, the BcKdV system is verified to have a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, the soliton–cnoidal wave solutions are explicitly studied.

In this paper, a systematic method is provided to formally derive the multiple- cosh solutions of the (1+1)-dimension Ito equation. By virtue of the bilinear form and matrix theory, the remarkable richness of exact solutions to Ito equation will be exhibited through Maple symbolic computations, which yields exact multiple- cosh , multiple- cos and lump-periodic solutions.