Long-time asymptotic behavior of a mixed Schrodinger equation with weighted Sobolev initial data

In this paper, we consider the initial value problem for the mixed Schrodinger equation. For the Schwartz initial data q0(x)& ISIN;S(R), by defining a general analytical domain and two reflection coefficients, we ever found an unified long-time asymptotic formula via the Deift-Zhou nonlinear steepest descent method. In this paper, under essentially minimal regularity assumptions on initial data in a much weak weighted Sobolev space q0(x)& ISIN;H2,2(R), we apply the partial differential steepest descent method to obtain long-time asymptotics for the mixed Schrodinger equation. In the asymptotic expression, the leading order term O(t-1/2) comes from the dispersive part q(t) + iq(xx) and the error order O(t-3/4) comes from a partial differential equation.