We are exploring solitons and other solutions describe new interaction between two solitons besides, new three soliton solutions are generated. we examine the commutative product between multi unknown Lie infinitesimals for the (2+1) dimensional variable-coefficients Bogoyavlensky Konopelchenko (VCBK) equation and this study result some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, (VCBK) equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Integration method, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solitons and other solutions for (VCBK). Some selected solutions illustrated through three-dimensional plots.

Aim of the presented study is to investigate the numerical solution of fifth-order nonlinear Lane-Emden (LE) based singular system at the origin with different shape factors developed on the analogous pattern of standard second order LE equations. The stochastic neuro-evolution computing is exploited for numerical outcomes by using the artificial neural networks (ANNs) for applicable mapping and learning of decision variables with integrated meta-heuristics of genetic algorithms (GAs) for global search aided with the rapid local search of active-set (AS) i.e., ANN-GA-AS algorithm. The designed numerical computing approach ANN-GA-AS implemented effectively for solving fifth-order nonlinear LE singular system and results of statistical assessments further authenticate the accuracy, convergence, and stability.