This paper presents a new method for analyzing the asymptotics of oscillatory Riemann-Hilbert problems, which arise in the study of the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. The method is applied to the modified Korteweg-de Vries (MKdV) equation, but it is shown to extend naturally to other equations such as the KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, as well as to integrable ordinary differential equations. The inverse scattering method for the MKdV equation leads to a Riemann-Hilbert factorization problem for a 2x2 matrix-valued function m(z; x, t), which is analytic in C \ R. The solution of the inverse problem is given by an integral involving a function μ(z; x, t) and a weight function w_{x,t}(z). The long-time behavior of solutions to nonlinear wave equations was first studied by Manakov and Ablowitz and Newell in 1973, and later rigorously justified by Buslaev and Sukhanov for the KdV equation and by Novokshenov for the NLS equation. The method of Zakharov and Manakov, which involves an ansatz for the asymptotic form of the solution and techniques from the theory of isomonodromic deformations, was later refined by Its, who used a parametrix to conjugate the Riemann-Hilbert problem to a simpler one. This approach allows for the extraction of leading asymptotics of the MKdV equation by deforming contours in the spirit of the classical method of steepest descent. The paper presents a detailed analysis of the asymptotic behavior of the MKdV equation in six regions, showing that the solution can be described by different asymptotic forms in each region. The results are supported by error estimates and are consistent with previous studies. The method is also shown to extend naturally to other nonlinear wave equations solvable by the inverse scattering method.This paper presents a new method for analyzing the asymptotics of oscillatory Riemann-Hilbert problems, which arise in the study of the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. The method is applied to the modified Korteweg-de Vries (MKdV) equation, but it is shown to extend naturally to other equations such as the KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, as well as to integrable ordinary differential equations. The inverse scattering method for the MKdV equation leads to a Riemann-Hilbert factorization problem for a 2x2 matrix-valued function m(z; x, t), which is analytic in C \ R. The solution of the inverse problem is given by an integral involving a function μ(z; x, t) and a weight function w_{x,t}(z). The long-time behavior of solutions to nonlinear wave equations was first studied by Manakov and Ablowitz and Newell in 1973, and later rigorously justified by Buslaev and Sukhanov for the KdV equation and by Novokshenov for the NLS equation. The method of Zakharov and Manakov, which involves an ansatz for the asymptotic form of the solution and techniques from the theory of isomonodromic deformations, was later refined by Its, who used a parametrix to conjugate the Riemann-Hilbert problem to a simpler one. This approach allows for the extraction of leading asymptotics of the MKdV equation by deforming contours in the spirit of the classical method of steepest descent. The paper presents a detailed analysis of the asymptotic behavior of the MKdV equation in six regions, showing that the solution can be described by different asymptotic forms in each region. The results are supported by error estimates and are consistent with previous studies. The method is also shown to extend naturally to other nonlinear wave equations solvable by the inverse scattering method.