Miura transformations and large-time behaviors of the Hirota-Satsuma equation

Abstract

The good Boussinesq equation has several modified versions, such as the modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma equation. This work builds the full relations among these equations by Miura transformation and invertible linear transformations and draws a pyramid diagram to demonstrate such relations. The direct and inverse spectral analysis shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation has a simple pole at origin, the solution of Riemann-Hilbert problem for the good Boussinesq equation has double pole at origin, while the solution of Riemann-Hilbert problem for the modified Boussinesq equation and Mikhailov-Lenells equation doesn't have singularity at origin. Further, the large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz class initial value are studied by Deift-Zhou nonlinear steepest descent analysis. In such initial conditions, the asymptotic expressions away from the origin are derived and it is shown that the leading term of asymptotic formulas matches well with the direct numerical simulations.