Dynamics of dispersive long waves in fluids

We obtain many different variable separation solutions for (2+1)-dimensional variable coefficient dispersive long wave equation by means of five different methods, including the multilinear variable separation approach, the projective Ricatti equation method, the extended projective Ricatti equation method, the extended tanh-function method and the improved tanh-function method. However, by careful analysis, we find that variable separation solution obtained by the multilinear variable separation approach includes all variable separation solutions obtained by other four direct methods. Thus variable separation solution for (2+1)-dimensional variable coefficient dispersive long wave equation exists a uniform form. Based on this uniform variable separation solution, we discuss the completely elastic interaction between foldons, the non-completely elastic interaction between bell-like semi-foldon, peaked semi-foldon and foldon, and the completely non-elastic interaction between bell-like semi-foldon and peaked semi-foldon. These results are helpful to analyze more precisely nonlinear and dispersive long gravity waves traveling in two horizontal directions, such as the bubbles on (or under) a fluid surface and folded waves in various ocean waves.