Large time asymptotics for the cylindrical Korteweg–de Vries equation. II

This paper is the second part of our study of asymptotic behavior for the solutions of the cylindrical Korteweg-de Vries equation (cKdV). In the first part (Nonlinearity, 33, no. 10, 5215–5245), we the large time asymptotics was calculated by using the method based on the a priori information about the behavior of the solution of the Cauchy problem for the cKdV equation. In this work, the well-known Deift–Zhou nonlinear steepest descent method for oscillatory Riemann–Hilbert problems will be employed (Annals of Math., II Series, 137, no. 2, 295–368). This method allows us to avoid the use of any a priori information and derive the asymptotics directly from the relevant Riemann–Hilbert problem. In the third part of this study, it is planned, by using again the Deift–Zhou method, to investigate the short time asymptotics of the solutions of the Cauchy problem (which is fixed at $t=1$) for the cylindrical Korteweg–de Vries equation and calculate the relationship of the coefficients that correspond to the small time and to the large time.