  We investigate the Cauchy problem of a new higher-order nonlinear
Schr\"{o}dinger equation (NHNSE) with weighted Sobolev initial data which is
derived by ourselves. By applying $\bar{\partial}$-steepest descent method, we
derive the long-time asymptotics of the NHNSE. Explicit steps are as follows:
first of all, based on the spectral analysis of a Lax pair and scattering
matrice, the solution of the NHNSE is exhibted through solving the
corresponding Riemann-Hilbert problem. Secondly, by applying some properties of
the Riemann-Hilbert problem, we obtain the long-time asymptotics of the
solution to the NHNSE. As we know that the properties of the NHNSE presented in
the paper have not been found in any scholar journals.
