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 Schrödinger equation. For the Schwartz initial data q 0 (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 q 0 (x) ∈ H 2 , 2 (R) , we apply the ∂ ̄ steepest descent method to obtain long-time asymptotics for the mixed Schrödinger 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 ∂ ̄ equation. [ABSTRACT FROM AUTHOR]