Soliton, breathers, positons and rogue waves for the vector complex modified Korteweg–de Vries equation

This paper we systematically investigate the general n-component vector complex modified Korteweg–de Vries (vcmKdV) equation, construct the N-fold Darboux transformation (DT) and presents its determinant representation. This paper is devoted to the study of the 3 + 1-dimensional vcmKdV equation. Utilizing the DT and multi-fold eigenvalue degeneracy, we derive globally bounded solutions for the vcmKdV equation, including N-bright-bright-bright solitons, N-dark-bright-bright solitons, N-breathers, N-positon solutions, and Nth-order rogue wave solutions. We primarily focus on deriving the specific forms and interaction scenarios for the first-order (N = 1) and second-order (N = 2) cases of these solutions. All these solutions are globally bounded. Graphical representations for the first- and second-order bright-bright-bright and dark-bright-bright soliton solutions are provided, illustrating phenomena where periodic oscillatory waves coexist or interact with solitons. The collision scenarios of the two-bright-bright-bright solution are investigated using asymptotic analysis. We also graphically present the bounded Akhmediev breather, the bounded breather with dark-bright soliton, and breather-breather mixed waves for the lower-order cases. Additionally, we show the graphs of the positon solution, the rogue wave, and their mixtures with dark-bright solitons and breathers for N = 1 and N = 2.