Conservation laws and exact solutions of a class of non linear regularized long wave equations via double reduction theory and Lie symmetries

The conservation laws for (1+1)-dimensional non-linear generalized regularized long wave (GRLW) equation are derived via partial Noether approach after increasing its order. The GRLW equation is a third-order partial differential equation. We convert GRLW equation to fourth order equation by assuming new dependent variable v to be the derivative of original dependent variable u by setting either u = v x or u = v t . The partial Noether’s approach is then used to derive the conservation laws. The derived conserved vectors are adjusted to satisfy the divergence relationship. Finally, the conservation laws are expressed in the variable u and they constitute the conservation laws for the third-order GRLW equation. The Lie point symmetries for GRLW equation are computed. The double reduction theory based on symmetry and its associated conserved vector is utilized and two independent exact solutions are obtained. Moreover, the Lie symmetry method is used to derive an invariant solution. One of the solutions obtained by the double reduction method is the same as derived by Lie symmetry method. The second solution constructed by the double reduction theory is not obtained by the Lie symmetry method. A similar analysis is performed for regularized long wave (RLW) and modified Benjamin–Bona–Mahoney (MBBM) equations.