  We consider the Cauchy problem of the higher-order KdV-type equation: \[
\partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x
u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The
nonlinearity is critical in the sense of long-time behavior. Using the method
of testing by wave packets, we prove that there exists a unique global solution
of the Cauchy problem satisfying the same time decay estimate as that of linear
solutions. Moreover, we divide the long-time behavior of the solution into
three distinct regions.
