Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

Lie Symmetry Analysis and Dynamics of Exact Solutions of the (2+1)-Dimensional Nonlinear Sharma–Tasso–Olver Equation

In soliton theory, the dynamics of solitary wave solutions may play a crucial role in the fields of mathematical physics, plasma physics, biology, fluid dynamics, nonlinear optics, condensed matter physics, and many others. The main concern of this present article is to obtain symmetry reductions and some new explicit exact solutions of the (2 + 1)-dimensional Sharma–Tasso–Olver (STO) equation by using the Lie symmetry analysis method. The infinitesimals for the STO equation were achieved under the invariance criteria of Lie groups. Then, the two stages of symmetry reductions of the governing equation are obtained with the help of an optimal system. Meanwhile, this Lie symmetry method will reduce the STO equation into new partial differential equations (PDEs) which contain a lesser number of independent variables. Based on numerical simulation, the dynamical characteristics of the solitary wave solutions illustrate multiple-front wave profiles, solitary wave solutions, kink wave solitons, oscillating periodic solitons, and annihilation of parabolic wave structures via 3D plots.

Coloring variants of the Re3B type

The Re3B type, space group Cmcm, has boron-centered trigonal prisms as central building units and is one of the basic structure types with numerous binary and ternary representatives. The coloring of different atoms on the two crystallographically independent rhenium sites leads to a manifold of compounds with different bonding peculiarities that are rather isopointal than isotypic with the prototype. Typical compounds are the S-phase precipitate MgCuAl2, the silicide ScPt2Si or the iodide Th0.5Pb0.5I3 (PuBr3 type). Differences in size or composition might force symmetry reductions. This is discussed for YZn3 (space group Pnma) and the different coloring variants ScRhSi2 and TaNi2P which show different twists of the trigonal prisms. Striking singular representatives with lower symmetry structures are BaThBr6 (Pmma) and NbCo2B (P21/c) which allow different ordering/distortion patterns for the prisms. All these crystal chemical details are discussed on the basis of group subgroup schemes (Bärnighausen trees).

Lie symmetry reductions, abound exact solutions and localized wave structures of solitons for a (2 + 1)-dimensional Bogoyavlenskii equation

By applying the two efficient mathematical methods particularly with regard to the classical Lie symmetry approach and generalized exponential rational function method, numerous exact solutions are constructed for a (2 + 1)-dimensional Bogoyavlenskii equation, which describes the interaction of Riemann wave propagation along the spatial axes. Moreover, we obtain the infinitesimals, all the possible vector fields, optimal system, and Lie symmetry reductions. The governing Bogoyavlenskii equation is converted into various nonlinear ordinary differential equations through two stages of Lie symmetry reductions. Accordingly, abundant exact closed-form solutions are obtained explicitly in terms of independent arbitrary functions, rational functions, trigonometric functions, and hyperbolic functions with arbitrary free parameters. The dynamical behavior of the resulting soliton solutions is presented through 3D-plots via numerical simulation. Eventually, single solitons, multi-solitons with oscillations, kink wave with breather-type solitons, and single lump-type solitons are obtained. The proposed mathematical techniques are effective, trustworthy, and reliable mathematical tools to work out new exact closed-form solutions of various types of nonlinear evolution equations in mathematical physics and engineering sciences.