Novel N-soliton and periodic wave solutions for the (2 + 1)-dimensional Calogero–Degasperis equation with variable coefficients

Purpose In this study, the (2 + 1)-dimensional variable coefficient Calogero–Degasperis equation (vCDE), which models various phenomena, including wave propagation in plasmas, soliton interactions and nonlinear dispersive effects, is studied by two methods to cover all types of solutions as we can. The first method is the direct bilinear method, which is generalized to find one, two and n-soliton solutions under conditions between the variable coefficients, which is considered novel and has not been obtained before. The second methodology is the direct similarity reduction method. By using it combined with the Jacobi elliptic expansion technique, we have obtained different types of wave solutions for the vCDE. Finally, by choosing suitable variable coefficient functions, we have discussed the effect of the different choices of the variable coefficients on the one, two and three soliton propagation. Design/methodology/approach In this manuscript, three different methods are used. The first method is the direct bilinear method, which is generalized to find one, two and n-soliton solutions under conditions between the variable coefficients that are considered new and not obtained before. The second methodology is the direct similarity reduction method. By using it combined with the Jacobi elliptic expansion technique, many novel periodic and solitary waves are obtained for the (2 + 1)-dimensional vCDE. Findings Novel n-soliton solutions are derived, along with a direct reduction of the (2 + 1)-dimensional vCDE to a nonlinear ordinary equation. In addition, several new wave-structure solutions are obtained, including periodic and solitary waves. Finally, the dynamical behavior of some results is discussed to illustrate the effect of the variable coefficients on the wave propagation. Originality/value Novel n-soliton wave solutions and periodic waves are obtained for the (2 + 1)-dimensional vCDE, which describes the interaction between a Riemann wave propagating along the y-axis and a long wave along the x-axis.