Classification of solutions for the (2+1)-dimensional Fokas–Lenells equations based on bilinear method and Wronskian technique

In this paper, we apply the bilinear method and Wronskian technique to the (2+1)-dimensional Fokas–Lenells (FL) equations for the first time, which simulate the propagation of richer pulses in the fibers. Specifically, based on the bilinear form of the above equations with parameters in the zero background, the double Wronskian solutions are provided and proved, and then various types of solutions for the local and nonlocal (2+1)-dimensional FL equations are obtained by using the reduction technique. This enables us to have a relatively complete classification of the solutions of the above two reduced equations as much as possible. Notably, we compare the solutions of these two reduced equations in detail, and find that the nonlocal equation has new characteristics that are different from the local ones, such as the N-order solutions of the nonlocal equation have \(\frac{(N+2)!}{N!2!}\) $\frac{(N+2)!}{N!2!}$ combinations in the cases of complex eigenvalues, which are much more complex than the local ones. In addition, the physical properties of the one-soliton and one-periodic solutions are investigated, the asymptotic behavior of the two-soliton solutions at the infinite time limit is analyzed, and then the coefficients of the equations controlling the rotation, separation and density of the solutions are discovered. Finally, we also talk about the periodic solutions, algebraically decayed solitary waves and mixed interaction solutions of the local (2+1)-dimensional FL equation that are not studied previously, which belong to real eigenvalues cases.