We revisit the scattering result of Holmer and Roudenko (2008) on radial focusing cubic NLS in three space dimensions. Using Sobolev embedding a virial/Morawetz estimate, we give simple proof below ground state that avoids use concentration compactness.

We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. identify a sharp threshold between scattering and blowup, establishing result analogous to that of Duyckaerts, Holmer, Roudenko for standard NLS [7, 11]. also prove failure uniform space-time bounds at

We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for non-radial $\dot H^{1/2}$-critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof below ground state in dimensions $d\geq 3$ that avoids use concentration compactness.

The conduction-electron spin susceptibility ${\ensuremath{\chi}}_{P}$ for Li and Na was measured as a function of pressure. By combining the present data with previously volume dependence Knight shift these metals, amplitude electron wave functions at nucleus, ${P}_{F}(V)$ is deduced first time. A comparison theoretical predictions indicates that they all agree qualitatively some numerically experimentally obtained results. None theories, however, correctly predict experimental results Li....

We prove almost sure global existence and scattering for the energy-critical nonlinear Schrödinger equation with randomized spherically symmetric initial data in Hs(R4) 56<s<1. were inspired to consider this problem by recent work of Dodson–Lührmann–Mendelson, which treated analogous wave equation.

We extend the result of Farah and Guzm\'an on scattering for $3d$ cubic inhomogeneous NLS to non-radial setting. The key new ingredient is a construction solutions corresponding initial data living far from origin.

We study the defocusing nonlinear Schrödinger equation in three space dimensions. prove that any radial solution remains bounded critical Sobolev must be global and scatter. In energy-supercritical setting we employ a space-localized Lin–Strauss Morawetz inequality of Bourgain. intercritical regime long-time Strichartz estimates frequency-localized inequalities.

We consider a class of power-type nonlinear Schrödinger equations for which the power nonlinearity lies between mass- and energy-critical exponents. Following concentration-compactness approach, we prove that if solution $u$ is bounded in critical Sobolev space throughout its lifespan, is, $u\in L_t^\infty \dot{H}_x^{s_c}$, then global scatters.

We revisit the problem of scattering below ground state threshold for mass-supercritical focusing nonlinear Schrödinger equation in two space dimensions. present a simple new proof that treats case radial initial data. The key ingredient is localized virial/Morawetz estimate; assumption aids controlling error terms resulting from spatial localization.

Introduction A variety of psychological factors may influence weight gain among undergraduates. As one the that might such gain, this research introduces food conscientiousness, a behavioral tendency toward making healthier choices. Methods In Phase 1 study, we developed conscientiousness scale. 2, examined whether undergraduates demonstrated and it was smaller those high in conscientiousness. Results The results indicated college students (2 lbs, on average) during fall 2020 semester....

We study the final-state problem for mass-subcritical NLS above Strauss exponent. For $u_+\in L^2$, we perform a physical-space randomization, yielding random final states $u_+^\omega \in L^2$. show that almost every $\omega$, there exists unique, global solution to scatters $u_+^\omega$. This complements deterministic result of Nakanishi, which proved existence (but not necessarily uniqueness) solutions scattering prescribed $L^2$ states.

Abstract We consider the initial-value problem for one-dimensional cubic nonlinear Schrödinger equation with a repulsive delta potential. prove that small initial data in weighted Sobolev space lead to global solutions decay $L^{\infty }$ and exhibit modified scattering.

We consider the initial-value problem for one-dimensional nonlinear Schrödinger equation in presence of an attractive delta potential.We show that sufficiently small initial data, corresponding global solution decomposes into a solitary wave plus radiation term decays and scatters as t → ∞.In particular, we establish asymptotic stability family waves.

AbstractWe follow up on work of Strauss, Weder, and Watanabe concerning scattering inverse for nonlinear Schrödinger equations with nonlinearities the form α(x)|u|pu.KEYWORDS: NLSscatteringinverse problem AcknowledgmentsI am grateful to Rowan Killip, Monica Visan, Michiyuki Watanabe, John Singler helpful discussions related this work. I would also like thank anonymous referees their comments suggestions.Additional informationFundingThe author was supported in part by NSF DMS-2137217.

We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in dimensions $d\geq 4.$ In spirit of Kenig and Merle [10], we combine a concentration-compactness approach with Lin--Strauss Morawetz inequality to prove that if solution $u$ is bounded $\dot{H}^{1/2}$ throughout its lifespan, then global scatters.