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arXiv:2607.19649

On the global well-posedness for the nonlocal Fokas-Lenells equation with the weighted Sobolev initial data on the line

Zhenya Yan, Guoqiang Zhang, Guancheng Zhu

nlin.SImath-phmath.APphysics.opticsquant-phSubmitted Jul 22, 2026• Announced Jul 23, 2026
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Abstract

We establish the global well-posedness of the Cauchy problem for the reverse space-time nonlocal Fokas-Lenells equation with the weighted Sobolev initial data $q_0(x)\in H^{3}(\mathbb{R}) \cap H^{2,1}(\mathbb{R})$ on the line. We develop the inverse scattering transform formulated via the associated Riemann-Hilbert problems to study this issue. A spectral uniformization transform is introduced to resolve the singular behavior inherent in the KN-type negative flow spectral problem. Owing to the reverse space-time reduction, reflection coefficients no longer satisfy the usual Hermitian conjugation symmetry, and the coercivity of the jump matrix is therefore not available a priori. The quantitative smallness condition on the initial data yields uniform bounds on the reflection coefficients and ensures the uniform positive definiteness of the Hermitian part of the associated jump matrix. The resulting coercivity allows us to establish the bounded invertibility of the associated singular integral operator through a Fredholm and vanishing-lemma argument. Under this condition, we prove an $L^{2}$-Sobolev bijective correspondence between the potential and scattering data, exclude spectral singularities on continuous spectra, and obtain the global existence and uniqueness of solutions. Moreover, the associated solution map is Lipschitz continuous on the admissible initial-data class.

Comments
60 pages, 2 figures

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We investigate the global well-posedness and modified scattering for the one-dimensional Schrödinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we establish global existence of solution together with persistence of the localization of the associated profile. We further provide a rigorous derivation of the asymptotic expansion at arbitrary order of such solutions, taking into account long-range effects induced by the cubic component of the nonlinearity. Our analysis relies on the space-time resonance method.
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