Boussinesq’s equation (BE) was the first model for the propagation of surface waves over shallow inviscid fluid layer. He found an analytical solution of his equation and thus proved that the balance between the steepening effect of the nonlinearity and the flattening effect of the dispersion maintains the shape of the wave. This discovery can be properly termed ‘Boussinesq Paradigm.’ Apart from the significance for the shallow water flows, this paradigm is very important for understanding the particle-like behavior of nonlinear localized waves. As it should have been expected, most of the physical systems are not fully integrable (even in one spatial dimension) and only a numerical approach can lead to unearthing the pertinent physical mechanisms of the interactions. A different approach to removing the incorrectness is by changing the spatial fourth derivative to a mixed fourth derivative, which resulted into an equation know nowadays as the Regularized Long Wave Equation (RLWE) or Benjamin–Bona–Mahony equation (BBME) In fact, the mixed derivative occurs naturally in Boussinesq derivation and was changed by Boussinesq to a fourth spatial derivative under an assumption that second temporal derivative is approximately equal to the second partial derivative,which is currently known as the ‘Linear Impedance Relation’ (or LIA). The LIA has produced innumerable instances of unphysical results.