Asymptotic expansion of the solution to the nonlinear Schrödinger equation with nonlocal interaction (Q5929865)

The author studies the large time behaviour of the solution to the following nonlinear Schrödinger equation NEWLINE\[NEWLINE\begin{cases} iu_t+ (\tfrac 12)\Delta u=f(u)\\ u(0,x)=u_0(x).\end{cases} \tag{1}NEWLINE\]NEWLINE Here, \(u\) is a complex-valued function of \((t,x)\in\mathbb{R} \times\mathbb{R}^n\), \(u_t= {\partial u\over\partial t}\), \(\Delta\) denotes the Laplace operator in \(\mathbb{R}^n\), \(f\) is a nonlinear term Hartree type, \(n\geq 2\). Here the author determines the second term of the asymptotic expansion of the solution (1). The main goal of the author is to deal with the Coulomb-line limiting case. Moreover the author gives a simple proof for short range case, which is applicable to the power nonlinearity case.