Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modified KdV Equation

We consider the integrable extended complex modified Korteweg–de Vries equation, which is generalized modified KdV equation. The first part of the article considers the construction of solutions via the Darboux transformation. We obtain some exact solutions, such as soliton solution, soliton molecules, positon solution, rational positon solution, rational solution, periodic solution and rogue waves solution. The second part of the article analyzes the dynamics of rogue waves. By means of the numerical analysis, under the standard decomposition, we divide the rogue waves into three patterns: fundamental patterns, triangular patterns and ring patterns. For the fundamental patterns, we define the length and width of the rogue waves and discuss the effect of different parameters on rogue waves.