  The soliton dressing matrices for the higher-order zeros of the
Riemann-Hilbert problem for the $N$-wave system are considered. For the
elementary higher-order zero, i.e. whose algebraic multiplicity is arbitrary
but the geometric multiplicity is 1, the general soliton dressing matrix is
derived. The theory is applied to the study of higher-order soliton solutions
in the three-wave interaction model. The simplest higher-order soliton solution
is presented. In the generic case, this solution describes the breakup of a
higher-order pumping wave into two higher-order elementary waves, and the
reverse process. In non-generic cases, this solution could describe ($i$) the
merger of a pumping sech wave and an elementary sech wave into two elementary
waves (one sech and the other one higher-order); ($ii)$ the breakup of a
higher-order pumping wave into two elementary sech waves and one pumping sech
wave; and the reverse processes. This solution could also reproduce fundamental
soliton solutions as a special case.
