Nonexistence of asymptotically free solutions for nonlinear Schrödinger system

Yonghang Chang; Menglan Liao; Yonghang Chang; Menglan Liao

doi:10.3934/cam.2024014

Communications in Analysis and Mechanics

2024, Volume 16, Issue 2: 293-306. doi: 10.3934/cam.2024014

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Research article

Nonexistence of asymptotically free solutions for nonlinear Schrödinger system

Yonghang Chang ,
Menglan Liao ^,

School of Mathematics, Hohai University, Nanjing, 210098, China

Received: 15 December 2023 Revised: 23 January 2024 Accepted: 23 January 2024 Published: 08 April 2024
35Q55, 35P25, 35B40

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 $.
- nonlinear Schrödinger system,
- asymptotically free solutions,
- large time behavior,
- pseudo-conformal identity,
- scattering
Citation: Yonghang Chang, Menglan Liao. Nonexistence of asymptotically free solutions for nonlinear Schrödinger system[J]. Communications in Analysis and Mechanics, 2024, 16(2): 293-306. doi: 10.3934/cam.2024014

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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 $.

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