On the long-time asymptotics to the focusing nonlocal mKdV equation with step-like initial data in curved wedges

We consider the Cauchy problem for the nonlocal modified Korteweg-de Vries (mKdV) equation with step-like boundary values: u(x,t)→0 as x→ -∞ and u(x,t)→ A as x→ ∞, where A is a positive constant. Recently, the long-time asymptotics of the solution of this problem along the rays ξ = x/12t = const≠0 has been studied, but the asymptotics for ξ→0 remains open. In the present work, we extend the asymptotics into the curved wedge regions, which are asymptotically closer to the ray $ x = 0 $ than any of these rays and imply ξ→ 0 as t→ ∞. The main techniques involve introducing the new scaled spectral parameter z = kx1-α and slow variable s = x3-2α12t with α∈(0,1), followed by using the nonlinear steepest descent method to analyze the basic Riemann-Hilbert problem. Finally, we derive the different long-time asymptotics of the solution of the Cauchy problem for the nonlocal mKdV equation in different curved wedges on $ (x,t) $-plane.