Nonexistence of asymptotically free solutions for nonlinear Schrödinger system

Abstract

In this paper, the Cauchy problem for the nonlinear Schrödinger system $ \begin{equation*} \begin{cases} i\partial_tu_1(x, t) = \Delta u_1(x, t)-|u_1(x, t)|^{p-1}u_1(x, t)-|u_2(x, t)|^{p-1}u_1(x, t), \\ i\partial_tu_2(x, t) = \Delta u_2(x, t)-|u_2(x, t)|^{p-1}u_2(x, t)-|u_1(x, t)|^{p-1}u_2(x, t), \end{cases} \end{equation*} $ was investigated in $ d $ space dimensions. For $ 1 < p\le 1+2/d $, the nonexistence of asymptotically free solutions for the nonlinear Schrödinger system was proved based on mathematical analysis and scattering theory methods. The novelty of this paper was to give the proof of pseudo-conformal identity on the nonlinear Schrödinger system. The present results improved and complemented these of Bisognin, Sepúlveda, and Vera(Appl. Numer. Math. 59(9)(2009): 2285–2302), in which they only proved the nonexistence of asymptotically free solutions when $ d = 1, \; p = 3 $.

Keywords

• nonlinear schrödinger system
• asymptotically free solutions
• large time behavior
• pseudo-conformal identity
• scattering