Instruction on the construction of coherent structures based on variable separation solutions of (2+1)-dimensional nonlinear evolution equations in fluid mechanics

At first, we focus on the projective Riccati equation method again and find that many solutions of the projective Riccati equation in the literatures are not independent. We derive an exponential form solution of the projective Riccati equation, which includes many solutions reported in the literatures. Then, via the projective Riccati equation method with the exponential form solution firstly reported in this paper, we can obtain variable separation solutions of (2+1)-dimensional nonlinear evolution equations in physics and engineering. For example, we get variable separation solutions of (2+1)-dimensional generalized Burgers equation and Boiti–Leon–Pempinelli equation in fluid mechanics. At last, based on these variable separation solutions of two equations, we study coherent structure including the interaction between four-dromion structures and collision between complex wave with dromion pair structures. We find that for the same physical coherent structure of the potential, coherent structures constructed from the original field quantities exhibit physical property (for Burgers equation) and un-physical property (for Boiti–Leon–Pempinelli equation). Therefore, we must be careful with the construction of coherent structures based on variable separation solutions, and we need consider the construction of coherent structures for all field quantities at the same time lest so-called new coherent structures found by researchers which are un-physical and illusive.