Deformed solitons of a typical set of (2+1)-dimensional complex modified Korteweg–de Vries equations

Deformed soliton solutions are studied in a typical set of (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) equations. Through constructing the determinant form of the n-fold Darboux transformation for these (2+1)-dimensional cmKdV equations, we obtain general order-n deformed soliton solutions using zero seeds. With no loss of generality, we focus on order-1 and order-2 deformed solitons. Three types of order-1 deformed solitons, namely, the polynomial type, the trigonometric type, and the hyperbolic type, are derived. Meanwhile, their dynamical behaviors, including amplitude, velocity, direction, periodicity, and symmetry, are also investigated in detail. In particular, the formulas of |q[1]| and trajectories are provided analytically, which are involved by an arbitrary smooth function f(y + 4λ2t). For order-2 cases, we obtain the general analytical expressions of deformed solitons. Two typical solitons, possessing different properties in temporal symmetry, are discussed.