Local existence for the semilinear Schrödinger equations in one space dimension

Large Time Behavior of Solutions for Derivative Cubic Nonlinear Schrödinger Equations

“…This last method is used in the present paper. It relies on some algebraic properties of the nonlinearity (similar to that of papers [1], [6], [12]) and is subtle. Another difficulty in the study of the large time asymptotic behavior of solutions to the Cauchy problem (1.1) is that the cubic nonlinear term of (1.1) is critical for large time values, because it does not satisfy the so called null gauge condition introduced in [13].…”

On quadratic derivative Schrödinger equations in one space dimension

“…These equations have been studied extensively by many authors; for the most recent developments, we refer to [2], [3], [14], [15]. Most of these works were inspired by a pseudodifferential calculus approach pioneered by Doi, [7], [8].…”

The Initial Value Problem for Cubic Semilinear Schrödinger Equations

“…Then studies on the initial value problem (l.l)-(1.2) have been mainly concerned with the case of (1.3). Recently, however, several researchers have been studying (l. l) -(1.2) without the condition (1.3) ( [1] , [2] , [3] , [4] , [7] , [8] , [10] , [11] , [18] , [20] ) . Except for [11] , these works are the application of the theory of linear Schrodinger type equations (see S. Mizohata [14, Lecture VII] and S. Doi [5] for instance).…”