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<p>In this paper, we study the orbital stability of standing waves for one-dimensional nonlinear Schrödinger equations with potentials. We show that these standing waves are orbitally stable for all frequencies in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-subcritical and critical cases. Since the presence of potentials breaks the scale invariance of the equations, it is delicate to apply the abstract theory of Grillakis et al. (J Funct Anal 74:160–197, 1987) directly without resorting to perturbative arguments. For this reason, little is known about the orbital stability of standing waves for all frequencies in non-scale-invariant settings. McLeod, Stuart, and Troy (Differ Integral Equ 16:1025–1038, 2003) proved orbital stability for all frequencies in the case of nondecreasing and bounded potentials. In this paper, by employing the approach of Noris et al. (Anal PDE 7:1807–1838, 2014), we establish the orbital stability of standing waves for all frequencies in the presence of unbounded potentials.</p>

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Stability of standing waves for all frequencies to nonlinear Schrödinger equations with potentials in one dimension

  • 期刊论文

  • 发表日期：

    2026年3月31日

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  • Noriyoshi Fukaya,
  • Masahiro Ikeda,
  • Hiroaki Kikuchi

摘要

In this paper, we study the orbital stability of standing waves for one-dimensional nonlinear Schrödinger equations with potentials. We show that these standing waves are orbitally stable for all frequencies in the \(L^2\) L 2 -subcritical and critical cases. Since the presence of potentials breaks the scale invariance of the equations, it is delicate to apply the abstract theory of Grillakis et al. (J Funct Anal 74:160–197, 1987) directly without resorting to perturbative arguments. For this reason, little is known about the orbital stability of standing waves for all frequencies in non-scale-invariant settings. McLeod, Stuart, and Troy (Differ Integral Equ 16:1025–1038, 2003) proved orbital stability for all frequencies in the case of nondecreasing and bounded potentials. In this paper, by employing the approach of Noris et al. (Anal PDE 7:1807–1838, 2014), we establish the orbital stability of standing waves for all frequencies in the presence of unbounded potentials.