Spectral analysis and long-time asymptotics for the Harry Dym-type equation with the Schwartz initial data

We study the Cauchy problem of the Harry Dym-type equation associated with the 3 x 3 matrix spectral problem by establishing the corresponding Riemann-Hilbert problem with the initial value lies in Schwartz space. Based on the nonlinear steepest descent method, we give the detailed contour deformation process to reduce the basic Riemann-Hilbert problem to a model Riemann-Hilbert problem, by which the long-time asymptotics for the Harry Dym-type equation is obtained with the aid of a series of precise and uniform error estimates. (c) 2023 Elsevier Inc. All rights reserved.