A novel class of soliton solutions and conservation laws of the generalised BS equation by Lie symmetry method

The interaction between a Riemann wave propagating along the y-axis and a long wave along the x-axis results in a generalised breaking soliton (gBS) equation. Lie symmetries of the equation are generated in this article to derive some rarely available classes of invariant solutions. The presence of arbitrary functions in each solution opens up a broad class of solution profiles. 3D profiles are used to explore more properties of the solutions to the gBS equation. The profiles describe doubly solitons, annihilation of parabolic, periodic solitons, line solitons and solitons on curved surface types. Solution profiles are useful in optical fibre, acoustic waves in a crystal lattice, long waves in stratified oceans, long-distance transmission and shallow water waves. The Lie symmetry approach has future scope to provide more variety in solutions due to the capability of solutions to include functions and arbitrary constants. This research effectively demonstrates the uniqueness of the solutions when compared with the previously published result. Moreover, the adjoint equation and conserved vectors are determined using Noether’s theorem.