Higher-order solitons in the N-wave system

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.