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Abstract We show that solutions to the Ablowitz–Ladik system converge to solutions of the cubic nonlinear Schrödinger equation for merely L 2 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. 

Abstract: We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as t→ - ∞ to asymptotic dynamics as t → + ∞. The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov–Poisson system. Via a pseudo-conformal inversion, we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function. 

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker–Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194–2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183–233, 2014) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi–Nash–Moser theory for the kinetic Fokker–Planck equations (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem.