We investigate the long-time asymptotics of the solution to the Cauchy problem for the nonlocal PT-symmetric derivative nonlinear Schrödinger equation (nPT-DNLS) \(\begin{aligned} \begin{aligned}&q_t(x, t)=i q_{x x}(x, t)+i \sigma (q^2(x, t) \bar{q}(-x, t))_x , \ x\in (-\infty ,\infty ),\ t>0,\\&q(x,0)=q_{0}(x), \end{aligned} \end{aligned}\) where \(q_{0}(x)\) belongs to the Schwartz class \(\mathcal {S}(R)\) with the assumption \(\Vert q_0(x)\Vert _{L^1(\mathbb {R})}<0.817\) .Beginning with the Lax pair, we define the corresponding Jost functions and scattering data, then formulate the Riemann–Hilbert problem related to the solution. Due to nonlocal effects, we introduce two distinct reflection coefficients \(r_1(\lambda )\) and \(r_2(\lambda )\) to address the broken space-inversion symmetry. Then we adapt the Deift–Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nPT-DNLS equation. Note that our main results are presented in two cases, corresponding to the distinct positions of the stationary phase points.In contrast with the local derivative nonlinear Schrödinger equation, we have some different results on the decay rate of the leading asymptotic term for the nPT-DNLS equation.
