Some new exact solutions of $(3+1)$-dimensional Burgers system via Lie symmetry analysis

In this paper, by using the Lie symmetry analysis, all of the geometric vector fields of the $(3+1)$ ( 3 + 1 ) -Burgers system are obtained. We find the 1, 2, and 3-dimensional optimal system of the Burger system and then by applying the 3-dimensional optimal system reduce the order of the system. Also the nonclassical symmetries of the $(3+1)$ ( 3 + 1 ) -Burgers system will be found by employing nonclassical methods. Finally, the ansatz solutions of BS equations with the aid of the tanh method has been presented. The achieved solutions are investigated through two- and three-dimensional plots for different values of parameters. The analytical simulations are presented to ensure the efficiency of the considered technique. The behavior of the obtained results for multiple cases of symmetries is captured in the present framework. The outcomes of the present investigation show that the considered scheme is efficient and powerful to solve nonlinear differential equations that arise in the sciences and technology.

Scaling and hetero-/auto-Bäcklund transformations with solitons of an extended coupled (2+1)-dimensional Burgers system for the wave processes in hydrodynamics and acoustics

The Burgers-type equations are applied to oceanography, hydrodynamic turbulence, gas dynamics, shock-wave formation, acoustic transmission structure, boundary-layer behavior, continuum-traffic simulation, convection-dominated diffusion, wave formation in the thermo-elastic media, vorticity transport, dispersion in the porous media, particle sedimentation in fluid suspension, colloid evolution, and so forth. Hereby, taking into account the wave processes in hydrodynamics and acoustics, we investigate an extended coupled (2+1)-dimensional Burgers system, and with symbolic computation, work out a scaling transformation, two hetero-Bäcklund transformations and two auto-Bäcklund transformations, with the soliton solutions. Our results are dependent on the coefficients in the system.