Spectral Analysis and Long-Time Asymptotics for the Reverse Space-Time Nonlocal Complex Nonlinear Transverse Oscillation Equation

In this paper, a nonlocal integrable WKI-type equation is proposed, which is referred to as reverse space-time nonlocal complex nonlinear transverse oscillation equation. The long-time asymptotic behavior of the solution of the Cauchy problem for this nonlocal equation is studied by utilizing the nonlinear steepest descent method. It is difficult to obtain the basic Riemann–Hilbert problem for the reverse space-time nonlocal complex nonlinear transverse oscillation equation due to the corresponding analytical uncertainty of the eigenfunction. The basic Riemann–Hilbert problem can be obtained under the condition of $$$ m>0 $$$ and the potential function is also expressed. With the aid of the parabolic cylinder functions, the long-time asymptotics of the solution of the reverse space-time nonlocal complex nonlinear transverse oscillation equation is finally obtained.