Singular matrix conjugacy problem with rapidly oscillating off-diagonal entries. Asymptotics of the solution in the case when a diagonal entry vanishes at a stationary point

The ( 2 × 2 ) (2\times 2) matrix conjugacy problem (the Riemann–Hilbert problem) with rapidly oscillating off-diagonal entries and quadratic phase function is considered, specifically, the case when one of the diagonal entries vanishes at a stationary point. For solutions of this problem, the leading term of the asymptotics is found. However, the method allows us to construct complete expansions in power orders. These asymptotics can be used, for example, to construct the asymptotics of solutions of the Cauchy problem for the nonlinear Schrödinger equation for large times in the case of the so-called collisionless shock region.