  The extended nonlinear Schr\"{o}dinger (ENLS) equation with third-order term
and fourth-order term which describes the wave propagation in the optical
fibers is more accurate than the NLS equation. A study of high-order soliton
matrix is presented for an ENLS equation in the framework of the
Riemann-Hilbert problem (RHP). Through a standard dressing procedure and the
generalized Darboux transformation (gDT), soliton matrix for simple zeros and
elementary high-order zeros in the RHP for the ENLS equation are constructed.
Then the N-soliton solutions and high-order soliton solutions for the ENLS
equation can be determined. Moreover, collision dynamics along with the
asymptotic behavior for the two-solitons and long-time asymptotic estimations
for the high-order one-soliton are concretely analyzed. For the given spectral
parameters, we can control the propagation direction, velocity, width and other
physical quantities of solitons by adjusting the free parameters of ENLS
equation.

Exactly Solvable and Integrable Systems;Pattern Formation and Solitons