@article{22044, 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.}, author = {Killip, Rowan and Murphy, Jason and Visan, Monica}, issn = {1361-6544}, journal = {Nonlinearity}, number = {1}, publisher = {IOP Publishing}, title = {{Determination of Schrödinger nonlinearities from the scattering map}}, doi = {10.1088/1361-6544/ada1bf}, volume = {38}, year = {2025}, } @article{22083, abstract = {We prove global well-posedness for the cubic nonlinear Schrödinger equation with nonlinearity concentrated on a homogeneous Poisson process.}, author = {Harrop-Griffiths, Benjamin and Killip, Rowan and Visan, Monica}, issn = {1361-6544}, journal = {Nonlinearity}, number = {9}, publisher = {IOP Publishing}, title = {{The nonlinear Schrödinger equation with sprinkled nonlinearity}}, doi = {10.1088/1361-6544/ae022e}, volume = {38}, 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{11701, abstract = {In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an Lp-setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers' equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1361-6544}, journal = {Nonlinearity}, number = {8}, pages = {4100--4210}, publisher = {IOP Publishing}, title = {{Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence}}, doi = {10.1088/1361-6544/abd613}, volume = {35}, year = {2022}, } @article{7637, abstract = {The evolution of finitely many particles obeying Langevin dynamics is described by Dean–Kawasaki equations, a class of stochastic equations featuring a non-Lipschitz multiplicative noise in divergence form. We derive a regularised Dean–Kawasaki model based on second order Langevin dynamics by analysing a system of particles interacting via a pairwise potential. Key tools of our analysis are the propagation of chaos and Simon's compactness criterion. The model we obtain is a small-noise stochastic perturbation of the undamped McKean–Vlasov equation. We also provide a high-probability result for existence and uniqueness for our model.}, author = {Cornalba, Federico and Shardlow, Tony and Zimmer, Johannes}, issn = {1361-6544}, journal = {Nonlinearity}, number = {2}, pages = {864--891}, publisher = {IOP Publishing}, title = {{From weakly interacting particles to a regularised Dean-Kawasaki model}}, doi = {10.1088/1361-6544/ab5174}, volume = {33}, year = {2020}, } @article{8697, abstract = {In the computation of the material properties of random alloys, the method of 'special quasirandom structures' attempts to approximate the properties of the alloy on a finite volume with higher accuracy by replicating certain statistics of the random atomic lattice in the finite volume as accurately as possible. In the present work, we provide a rigorous justification for a variant of this method in the framework of the Thomas–Fermi–von Weizsäcker (TFW) model. Our approach is based on a recent analysis of a related variance reduction method in stochastic homogenization of linear elliptic PDEs and the locality properties of the TFW model. Concerning the latter, we extend an exponential locality result by Nazar and Ortner to include point charges, a result that may be of independent interest.}, author = {Fischer, Julian L and Kniely, Michael}, issn = {1361-6544}, journal = {Nonlinearity}, number = {11}, pages = {5733--5772}, publisher = {IOP Publishing}, title = {{Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model}}, doi = {10.1088/1361-6544/ab9728}, volume = {33}, year = {2020}, }