Lie symmetries, modulation instability, conservation laws, and the dynamic waveform patterns of several invariant solutions to a (2+1)-dimensional Hirota bilinear equation

In this paper, we dealt for the first time with a highly nonlinear (2+1)-dimensional associated Hirota bilinear equation using the Lie symmetry approach and symbolic computation with Mathematica. The primary objective of this paper is to employ the Lie symmetry analysis for the purpose of finding newly generated explicit exact solutions, as well as conservation laws and investigating modulation instability within the context of the (2+1)-dimensional associated Hirota bilinear equation. Equivalence transformations, a commutator table, and an adjoint table are generated using Lie's invariance infinitesimal criterion. Applying the optimal algebra classification, the differential invariants are generated. By utilizing the optimal system and capitalizing on infinitesimal symmetries, we successfully obtain numerous symmetry reductions and invariant solutions through the implementation of the Lie group method. The obtained solutions include the time variable, space variables, arbitrary constants, as well as arbitrary functions. By taking advantage of symbolic computation work with Mathematica, the achieved outcomes are manifested with 3D, 2D, and contour graphics to interpret the physical meaning of the acquired results, which show traveling waves, solitary waves, and periodic wave structures. The solutions exhibit different physical structures concerning the involved parameters. All the attained results are novel and are absolutely distinct from the earlier findings. Moreover, we establish nonlinear self-adjointness and derive conservation laws for the provided equation. Finally, the linear stability analysis of the governing equation is presented to study the modulational instability.