Special solitonic localized structures for the (3+1)(3+1)-dimensional Burgers equation in water waves

•A modified mapping method for NPDEs.•Variable separation solution for the 3+13+1D Burgers equation.•Elastic interactions between rectangle and ring solitons on background.

With the help of a modified mapping method, we re-study the (3+1)(3+1)-dimensional Burgers equation and derive three families of variable separation solutions. By selecting appropriate functions in the variable separation solution, we discuss interaction behaviors among flat-top rectangle-soliton and ring-soliton, embedded rectangle-soliton and embedded ring-soliton in a periodic wave background. All interaction behaviors among them are completely elastic, and no phase shift appears after the interaction. These results might be helpful to the understanding of the propagation processes for nonlinear water waves in fluid mechanics such as diverse nonequilibrium, nonlinear phenomena in turbulence and inter-face dynamics.