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In this work, the double and triple-pole solutions for the third-order flowequation of Kaup-Newell system (TOFKN) with zero boundary conditions (ZBCs) andnon-zero boundary conditions (NZBCs) are investigated by means of theRiemann-Hilbert (RH) approach stemming from the inverse scatteringtransformation. Starting from spectral problem of the TOFKN, the analyticity,symmetries, asymptotic behavior of the Jost function and scattering matrix, thematrix RH problem with ZBCs and NZBCs are constructed. Then the obtained RHproblem with ZBCs and NZBCs can be solved in the case of scatteringcoefficients with double or triple zeros, and the reconstruction formula ofpotential, trace formula as well as theta condition are also derivedcorrespondingly. Specifically, the general formulas of $N$-double and$N$-triple poles solutions with ZBCs and NZBCs are derived systematically bymeans of determinants. The vivid plots and dynamics analysis for double andtriple-pole soliton solutions with the ZBCs as well as double and triple-poleinteraction solutions with the NZBCs are exhibited in details. Compared withthe most classical second-order flow Kaup-Newell system, we find thethird-order dispersion and quintic nonlinear term of the Kaup-Newell systemchange the trajectory and velocity of solutions. Furthermore, the asymptoticstates of the 1-double poles soliton solution and the 1-triple poles solitonsolution are analyzed when $t$ tends to infinity.

Double and Triple-Pole Solutions for the Third-Order Flow Equation of the Kaup-Newell System with Zero/Nonzero Boundary Conditions