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1. Breakdown of Regularity of Scattering for Mass-Subcritical NLS

   https://doi.org/10.1093/imrn/rnaa072  

   Lee, Gyu Eun (April 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the scattering problem for the nonlinear Schrödinger equation $$i\partial _t u + \Delta u = |u|^p u$$ on $$\mathbb{R}^d$$, $$d\geq 1$$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $$\Sigma$$ holds and the wave operator is well defined on $$\Sigma$$. We show that there exists $$0<\beta <p$$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $$L^2\to L^2$$ of class $$C^{1+\beta }$$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology. 

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2. Determination of Schrödinger nonlinearities from the scattering map

   https://doi.org/10.1088/1361-6544/ada1bf  

   Killip, Rowan; Murphy, Jason; Visan, Monica (December 2024, Nonlinearity)

   Abstract We prove that the small-data scattering map uniquely determines the nonlinearity for a wide class of gauge-invariant, intercritical nonlinear Schrödinger equations. We use the Born approximation to reduce the analysis to a deconvolution problem involving the distribution function for linear Schrödinger solutions. We then solve this deconvolution problem using the Beurling–Lax Theorem. 

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3. On an inverse problem of nonlinear imaging with fractional damping

   https://doi.org/10.1090/mcom/3683  

   Kaltenbacher, Barbara; Rundell, William (January 2022, Mathematics of Computation)

   This paper considers the attenuated Westervelt equation in pressure formulation. The attenuation is by various models proposed in the literature and characterised by the inclusion of non-local operators that give power law damping as opposed to the exponential of classical models. The goal is the inverse problem of recovering a spatially dependent coefficient in the equation, the parameter of nonlinearity κ ( x ) \kappa (x) , in what becomes a nonlinear hyperbolic equation with non-local terms. The overposed measured data is a time trace taken on a subset of the domain or its boundary. We shall show injectivity of the linearised map from κ \kappa to the overposed data and from this basis develop and analyse Newton-type schemes for its effective recovery. 

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4. On the well-posedness of the periodic fractional Schrödinger equation

   https://doi.org/10.1007/s42985-025-00327-0  

   Sanchez, Beckett; Riaño, Oscar; Roudenko, Svetlana (April 2025, Partial Differential Equations and Applications)

   We consider the periodic fractional nonlinear Schroedinger equation, where the nonlinearity is expressed in two ways either as a derivative of a polynomial (including negative powers, such including logarithmic nonlinearities), or given as a sum of powers, possibly infinite. We obtain the local well-posedness for the Cauchy problem of this equation in weighted Sobolev spaces and with non-vanishing initial data. 

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5. Well-posedness in weighted spaces for the generalized Hartree equation with p < 2

   https://doi.org/10.1142/S0219199721500747  

   Arora, Anudeep K.; Riaño, Oscar; Roudenko, Svetlana (November 2022, Communications in Contemporary Mathematics)

   We investigate the well-posedness in the generalized Hartree equation [Formula: see text], [Formula: see text], [Formula: see text], for low powers of nonlinearity, [Formula: see text]. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the [Formula: see text]-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time. 

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