@article{22021, abstract = {We establish global well-posedness for both the defocusing and focusing complex-valued modified Korteweg–de Vries equations on the real line in modulation spaces Ms,2p (R), for all 1  p < 1 and 0  s < 3/2 − 1/p. We will also show that such solutions admit global-in-time bounds in these spaces and that equicontinuous sets of initial data lead to equicontinuous ensembles of orbits. Indeed, such information forms a crucial part of our well-posedness argument.}, author = {Haque, Saikatul and Killip, Rowan and Visan, Monica and Zhang, Yunfeng}, issn = {2578-5885}, journal = {Pure and Applied Analysis}, number = {3}, pages = {615--637}, publisher = {Mathematical Sciences Publishers}, title = {{ Global well-posedness and equicontinuity for modified Korteweg–de Vries equations in modulation spaces}}, doi = {10.2140/paa.2025.7.615}, volume = {7}, year = {2025}, } @article{22046, abstract = {We show that solutions to the Ablowitz–Ladik system converge to solutions of the cubic nonlinear Schrödinger equation for merely L2 initial data. Furthermore, we consider initial data for this lattice model that excites Fourier modes near both critical points of the discrete dispersion relation and demonstrate convergence to a decoupled system of nonlinear Schrödinger equations.}, author = {Killip, Rowan and Ouyang, Zhimeng and Visan, Monica and Wu, Lei}, issn = {1361-6544}, journal = {Nonlinearity}, keywords = {Ablowitz–Ladik, continuum limit, cubic NLS}, number = {7}, pages = {3751--3775}, publisher = {IOP Publishing}, title = {{Continuum limit for the Ablowitz–Ladik system}}, doi = {10.1088/1361-6544/acd978}, volume = {36}, year = {2023}, } @article{22042, abstract = {We study the L p-theory for the Schrödinger operatorLa with inverse-square potential a|x|^−2. Our main result describes when L p-based Sobolev spaces defined in terms of the operator (La)^s/2 agree with those defined via (−)^s/2.We consider all regularities 0 < s < 2. In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood–Paley theory, and Hardy-type inequalities associated to the operator La.}, author = {Killip, R. and Miao, C. and Visan, Monica and Zhang, J. and Zheng, J.}, issn = {1432-1823}, journal = {Mathematische Zeitschrift}, number = {3-4}, pages = {1273--1298}, publisher = {Springer Nature}, title = {{Sobolev spaces adapted to the Schrödinger operator with inverse-square potential}}, doi = {10.1007/s00209-017-1934-8}, volume = {288}, year = {2018}, } @article{22045, abstract = {We consider the initial-value problem for the cubic-quintic nonlinear Schrödinger equation (𝑖𝜕𝑡+Δ)⁢𝜓 =𝛼1⁢𝜓 −𝛼3⁢|𝜓|2⁢𝜓 +𝛼5⁢|𝜓|4⁢𝜓 in three spatial dimensions in the class of solutions with |𝜓⁡(𝑥)| →𝑐 >0 as |𝑥| →∞. Here 𝛼1, 𝛼3, 𝛼5, and 𝑐 are such that 𝜓⁡(𝑥) ≡𝑐 is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to 𝜓⁡(𝑥) →1 as |𝑥| →∞, we study the associated initial-value problem for 𝑢 =𝜓 −1 and prove a scattering result for small initial data in a weighted Sobolev space.}, author = {Killip, Rowan and Murphy, Jason and Visan, Monica}, issn = {0036-1410}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {cubic-quintic NLS, nonvanishing boundary conditions, space-time resonances, scattering}, number = {3}, pages = {2681--2739}, publisher = {Society for Industrial & Applied Mathematics}, title = {{The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions}}, doi = {10.1137/17m1116702}, volume = {50}, year = {2018}, } @article{22051, abstract = {We construct solutions with prescribed scattering state to the cubic-quintic NLS (mathematical formular)in three spatial dimensions in the class of solutions with (mathematical formular). This models disturbances in an infinite expanse of (quantum) fluid in its quiescent state— the limiting modulus c corresponds to a local minimum in the energy density. Our arguments build on work of Gustafson, Nakanishi, and Tsai on the (defocusing) Gross–Pitaevskii equation. The presence of an energy-critical nonlinearity and changes in the geometry of the energy functional add several new complexities. One new ingredient in our argument is a demonstration that solutions of such (perturbed) energy-critical equations exhibit continuous dependence on the initial data with respect to the weak topology on H1/x.}, author = {Killip, Rowan and Murphy, Jason and Visan, Monica}, issn = {1948-206X}, journal = {Analysis & PDE}, keywords = {final-state problem, wave operators, cubic-quintic NLS, nonvanishing boundary conditions}, number = {7}, pages = {1523--1574}, publisher = {Mathematical Sciences Publishers}, title = {{The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions}}, doi = {10.2140/apde.2016.9.1523}, volume = {9}, year = {2016}, } @article{22053, abstract = {We consider the Gross–Pitaevskii equation on R^4 and the cubic-quintic nonlinear Schrödinger equation (NLS) on R^3 with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.}, author = {Killip, Rowan and Oh, Tadahiro and Pocovnicu, Oana and Visan, Monica}, issn = {1945-001X}, journal = {Mathematical Research Letters}, keywords = {NLS, Gross–Pitaevskii equation, non-vanishing boundary condition}, number = {5}, pages = {969--986}, publisher = {International Press of Boston}, title = {{Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions}}, doi = {10.4310/mrl.2012.v19.n5.a1}, volume = {19}, year = {2013}, } @article{22049, abstract = {We consider the minimal mass m0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + Δu = μ|u|^4/d u to blow up. If m0 is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235 (2006), 171–192], in dimensions 1, 2 and Begout and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the L2-critical nonlinear Schrödinger equation, preprint], in dimensions d ≥ 3 for the mass-critical NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, preprint], in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in for the defocusing NLS in three and higher dimensions with spherically symmetric data.}, author = {Tao, Terence and Visan, Monica and Zhang, Xiaoyi}, issn = {1435-5337}, journal = {Forum Mathematicum}, number = {5}, pages = {881--919}, publisher = {De Gruyter}, title = {{Minimal-mass blowup solutions of the mass-critical NLS}}, doi = {10.1515/forum.2008.042}, volume = {20}, year = {2008}, } @article{22047, abstract = {We undertake a comprehensive study of the nonlinear Schrödinger equation (mathematical formular) where u(t, x) is a complex-valued function in spacetime R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H^1(ℝ n ) and in the pseudoconformal space Σ := {f ∈ H^1(ℝ^n); xf ∈ L^2(ℝ^n)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical formular) . The results at the endpoint p1= 4/n are conditional on a conjectured global existence and spacetime estimate for the L2/x-critical nonlinear Schrödinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint). As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H1/x for solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n < p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985).}, author = {Tao, Terence and Visan, Monica and Zhang, Xiaoyi}, issn = {1532-4133}, journal = {Communications in Partial Differential Equations}, keywords = {Energy-critical, Mass-critical, Nonlinear Schrödinger equation, Wellposedness}, number = {8}, pages = {1281--1343}, publisher = {Informa UK Limited}, title = {{The nonlinear Schrödinger equation with combined power-type nonlinearities}}, doi = {10.1080/03605300701588805}, volume = {32}, year = {2007}, } @article{22050, abstract = {We obtain global well-posedness, scattering, and global L2(n+2)/(n−2)/t,x space-time bounds for energy-space solutions to the energy-critical nonlinear Schrodinger (NLS) ¨ equation in Rt × Rn/x , n ≥ 5.}, author = {Visan, Monica}, issn = {0012-7094}, journal = {Duke Mathematical Journal}, number = {2}, pages = {281--374}, publisher = {Duke University Press}, title = {{The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions}}, doi = {10.1215/s0012-7094-07-13825-0}, volume = {138}, year = {2007}, }