On asymptotic behavior of solutions to Korteweg–de Vries type equations related to vortex filament with axial flow

We study the global existence and asymptotic behavior in time of solutions to the Korteweg–de Vries type equation called as “Hirota” equation. This equation is a mixture of cubic nonlinear Schrödinger equation and modified Korteweg–de Vries equation. We show the unique existence of the solution for this equation which tends to the given “modified” free profile by using the two asymptotic formulae for some oscillatory integrals.