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, that is, 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.