The Defocusing Nonlinear Schrödinger Equation with a Nonzero Background: Painlevé Asymptotics in Two Transition Regions

In this paper, we address the Painlevé asymptotics of the solution in two transition regions for the defocusing nonlinear Schrödinger (NLS) equation with finite density initial data iq_t+q_xx-2(|q|^2-1)q=0, q(x,0)=q_0(x) ∼± 1, x→±∞ . The key to prove this result is the formulation and analysis of a Riemann–Hilbert problem associated with the Cauchy problem for the defocusing NLS equation. With the ∂̅ -generalization of the Deift–Zhou nonlinear steepest descent method and double scaling limit technique, in two transition regions defined by 𝒫_± 1:={ (x,t) ∈ℝ×ℝ^+: 0<| x/2t-(± 1)| t^2/3≤ C} , where C>0 is a constant, we find that the leading order approximation to the solution of the defocusing NLS equation can be expressed in terms of the Hastings–McLeod solution of the Painlevé II equation in the generic case, while Ablowitz–Segur solution in the non-generic case.