I have discovered that there was an error in one of the second-order equations (the evolution equation for $\phi _b$) given by (5.12b) in Choi (Reference Choi2022), which will be hereinafter referred to as C22. The correct evolution equation for $\phi _b$ is given by

(0.1)\begin{align} &\left [ \left \{ 1-\frac{\eta^2}{2!} + \frac{\eta^4}{4!} (\nabla^2)^2\right\} \phi_b\right ]_t +g\zeta+\frac{1}{2}\boldsymbol{\nabla}\phi_b\boldsymbol{\cdot}\boldsymbol{\nabla}\phi_b \nonumber\\ &\quad= \boldsymbol{\nabla}\boldsymbol{\cdot} \left ( \frac{\eta^2}{2!} \nabla^2\phi_b\boldsymbol{\nabla} \phi_b\right ) -\boldsymbol{\nabla}\boldsymbol{\cdot} \left [ \frac{\eta^4}{4!} \boldsymbol{\nabla}\phi_b (\nabla^2)^2\phi_b +\frac{\eta^4}{16} \boldsymbol{\nabla} (\nabla^2\phi_b)^2 \right ]. \end{align}

While the recursive formulas in C22 are correct, a mistake was made when the second-order model was simplified. As a result, the second-order evolution equation for $v={\phi _b}_x$ for one-dimensional waves given by (6.1b) in C22 is also incorrect. The evolution equation should read

(0.2)\begin{equation} \left [ v-\left (\frac{\eta^2}{2!}v_{x}-\frac{\eta^4}{4!}v_{xxx}\right )_x\right ]_t +g\zeta_x+vv_x = \left [ \frac{\eta^2}{2!}vv_x-\frac{\eta^4}{4!} ( vv_{xxx} +3v_xv_{xx} )\right ]_{xx}. \end{equation}

Notice that the coefficient for $v_xv_{xx}$ on the right-hand side has been changed to 3 from 5 in C22.

The numerical results presented in figures 9, 11, and 12 in C22 are recomputed with the corrected second-order system, but the new numerical solutions are found close to the previous solutions as the error was introduced in the high-order dispersive terms of $O(\beta ^4)\ll 1$. Therefore, no new numerical solutions are presented here although they are made available at https://web.njit.edu/~wychoi/pub/Choi22~newfigures.pdf. It should be remarked that the maximum difference between the new and old solutions, for example, for the head-on collision of two-counter propagating solitary waves presented in figure 12 in C22 is found about 3.76 %. Therefore, the discussion about the second-order model in C22 remains valid. In the meantime, the loss of the truncated energy for the second-order model is found 0.179 % at $t/(h/g)^{1/2}=200$ for the propagation of a single solitary wave presented in figure 9 and 0.198 % at $t/(h/g)^{1/2}=20$ for the head-on collision of two solitary waves in figure 11, instead of 0.230 % and 0.130 %, respectively, reported in C22.