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arXiv · 2609.05483

Comment on "Space-time shifted solitons and solution interactions for a generalized nonlocal nonlinear Schr\"odinger equation"

Aslı Pekcan
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Abstract

Zhou and Yu [Appl. Math. Lett. 178 (2026) 109937] studied the space-time shifted nonlocal nonlinear Schr\"odinger (NLS) equation $iq_t(x,t)=q_{xx}(x,t)-2\sigma q^2(x,t) \bar{q}(x_0-x,t_0+\lambda t),\,\, \lambda=\pm1$, and claimed that for both choices of $\lambda$ 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 $\lambda=1$ and $t_0=0$. The case with $\lambda=-1$ 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.

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Aslı Pekcan. 2026-08-23. Comment on "Space-time shifted solitons and solution interactions for a generalized nonlocal nonlinear Schr\"odinger equation". https://arxiv.org/abs/2609.05483

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