This paper presents a new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems, particularly focusing on the modified Korteweg-de Vries (MKdV) equation. The method involves deforming contours using the classical method of steepest descent to extract the leading asymptotics. For the MKdV equation, the authors deform the Riemann-Hilbert problem onto a contour that includes two crosses, where the jump matrices converge rapidly to the identity as \( t \to \infty \). This reduction allows for explicit solutions in terms of parabolic cylinder functions. The leading asymptotics of the solution \( y(x, t) \) are described in six regions, each with specific decay rates and behaviors. The method is shown to extend to other nonlinear wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, as well as "integrable" ordinary differential equations like Painlevé transcendents. The paper also discusses the historical context and previous work on the long-time behavior of these equations, including contributions from Manakov, Ablowitz, Newell, Zakharov, and others.This paper presents a new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems, particularly focusing on the modified Korteweg-de Vries (MKdV) equation. The method involves deforming contours using the classical method of steepest descent to extract the leading asymptotics. For the MKdV equation, the authors deform the Riemann-Hilbert problem onto a contour that includes two crosses, where the jump matrices converge rapidly to the identity as \( t \to \infty \). This reduction allows for explicit solutions in terms of parabolic cylinder functions. The leading asymptotics of the solution \( y(x, t) \) are described in six regions, each with specific decay rates and behaviors. The method is shown to extend to other nonlinear wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, as well as "integrable" ordinary differential equations like Painlevé transcendents. The paper also discusses the historical context and previous work on the long-time behavior of these equations, including contributions from Manakov, Ablowitz, Newell, Zakharov, and others.