评述“广义非局域非线性薛定谔方程的空时平移孤子与解相互作用”
Comment on "Space-time shifted solitons and solution interactions for a generalized nonlocal nonlinear Schrödinger equation"
- Hacettepe University（哈塞特佩大学）
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AI总结：
本文审视Zhou和Yu关于空时平移非局域NLS方程可积性的论断，指出仅当无时间反演（$\lambda=1, t_0=0$）时相容，$\lambda=-1$情形不能由标准NLS系统导出，并讨论平移等价性及修正Hirota方程。
AI中文摘要：
Zhou和Yu [Appl. Math. Lett. 178 (2026) 109937] 研究了空时平移非局域非线性薛定谔（NLS）方程 $iq_t(x,t)=q_{xx}(x,t)-2\sigma q^2(x,t) \bar{q}(x_0-x,t_0+\lambda t),\\, \lambda=\pm1$，并声称对于 $\lambda$ 的两种选择该方程都是可积的。这里我们从标准可积NLS系统的约化角度审视这一论断。我们发现，该方程与NLS系统的相容性仅在自变量中不存在时间反演时才成立。换言之，对于该方程，这要求 $\lambda=1$ 且 $t_0=0$。$\lambda=-1$ 的情形是一个有效的非局域方程，但不能通过相应的复共轭平移非局域约化从标准NLS系统导出，因此其可积性并不源于该可积NLS系统。我们还讨论了平移（具有实平移）与非平移非局域约化之间的等价性，并给出了Hirota双线性方程的修正形式。
英文摘要：
Zhou and Yu [Appl. Math. Lett. 178 (2026) 109937] studied the space-time shifted nonlocal nonlinear Schrödinger (NLS) equation $iq_t(x,t)=q_{xx}(x,t)-2σq^2(x,t) \bar{q}(x_0-x,t_0+λt),\,\, λ=\pm1$, and claimed that for both choices of $λ$ the equation is integrable. Here we examine this claim from the perspective of reductions of the standard integrable NLS system. We find that the compatibility of the equation with the NLS system is true only when there is no time reversal in the argument. In other words, for this equation it requires $λ=1$ and $t_0=0$. The case with $λ=-1$ is a valid nonlocal equation, but cannot be derived from the standard NLS system by the corresponding complex conjugate shifted nonlocal reduction, and its integrability therefore does not follow from that integrable NLS system. We also discuss the equivalence between shifted (with real shifts) and unshifted nonlocal reductions and give the adjusted form of the Hirota bilinear equation.
