Long-time asymptotic behavior of a mixed Schrödinger 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)∈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)∈H2,2(R), we apply the ∂ 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 qt + iqxx and the error order O(t−3/4) comes from a ∂ equation.