  We derive the soliton matrices corresponding to an arbitrary number of
higher-order normal zeros for the matrix Riemann-Hilbert problem of arbitrary
matrix dimension, thus giving the complete solution to the problem of
higher-order solitons. Our soliton matrices explicitly give all higher-order
multi-soliton solutions to the nonlinear partial differential equations
integrable through the matrix Riemann-Hilbert problem. We have applied these
general results to the three-wave interaction system, and derived new classes
of higher-order soliton and two-soliton solutions, in complement to those from
our previous publication [Stud. Appl. Math. \textbf{110}, 297 (2003)], where
only the elementary higher-order zeros were considered. The higher-order
solitons corresponding to non-elementary zeros generically describe the
simultaneous breakup of a pumping wave $(u_3)$ into the other two components
($u_1$ and $u_2$) and merger of $u_1$ and $u_2$ waves into the pumping $u_3$
wave. The two-soliton solutions corresponding to two simple zeros generically
describe the breakup of the pumping $u_3$ wave into the $u_1$ and $u_2$
components, and the reverse process. In the non-generic cases, these
two-soliton solutions could describe the elastic interaction of the $u_1$ and
$u_2$ waves, thus reproducing previous results obtained by Zakharov and Manakov
[Zh. Eksp. Teor. Fiz. \textbf{69}, 1654 (1975)] and Kaup [Stud. Appl. Math.
\textbf{55}, 9 (1976)].
