$\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation

Abstract

This work discusses the $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $-symmetry and conservation law of $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $ procedure for the nonlinear dispersive modified Benjamin-Bona-Mahony equation (NDMBBME). This equation models an approximation for surface long waves in nonlinear dispersive media. It can also describe the hydromagnetic waves in a cold plasma, acoustic waves in inharmonic crystals, and acoustic gravity waves in compressible fluids. First and foremost, we offer some essential pieces of information about the $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $-symmetry and the conservation law of $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $ concepts. In light of such information, $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $-symmetries are found. Using characteristic equations, the NDMBBME is reduced to ordinary differential equations (ODEs). We obtained the exact invariant solutions by solving the nonlinear ODEs. Furthermore, employing the variational problem procedure, we get the Lagrangian and the $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $-conservation laws. The exact solutions and conservation laws are new for the NDMBBME that are not reported by the other studies. We also demonstrate the properties with figures for these solutions.