Solitons in the fifth-order KdV equation with a perturbation

The fifth-order KdV equation with a perturbation term is more consistent with the shallow water wave model in nature. By using the multi-scale perturbation expansion method, the leading-order asymptotic solution of the perturbed fifth-order KdV equation is derived. Furthermore, the asymptotic solution is reconsidered from the perspective of conservation laws, and consistent results are obtained. In addition, the numerical solution of the slowly varying solitary wave asymptotic solution is obtained by the Fourier spectral method under appropriate initial conditions to better understand its dynamic behavior. These results can enrich the research on fluid dynamics.