This textbook introduces the well-posedness theory for initial-value problems of nonlinear,
dispersive partial differential equations, with special focus on two key models, the Korteweg …

We study the local Cauchy problem in time for the Zakharov system,(1.1) and (1.2),
governing Langmuir turbulence, with initial data (u (0), n (0),∂ tn (0))∈ Hk⊕ Hlscr;⊕ Hℓ− 1 …

1. Fourier multiplier, function space [symbol]. 1.1. Schwartz space, tempered distribution,
Fourier transform. 1.2. Fourier multiplier on L [symbol]. 1.3. Dyadic decomposition, Besov …

We show that the Benjamin–Ono equation is globally well-posed in Hs (R) for s≥ 1. This is
despite the presence of the derivative in the nonlinearity, which causes the solution map to …

We establish that the quadratic non-linear Schrödinger equation where u: R× R→ C, is
locally well-posed in Hs (R) when s⩾-1 and ill-posed when s<-1. Previous work in [C. Kenig …

Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on
â—š³ Page 1 Global Existence and Scattering for Rough Solutions of a …

We introduce low regularity exponential-type integrators for nonlinear Schrödinger
equations for which first-order convergence only requires the boundedness of one …

We consider semilinear Schr\" odinger equations with nonlinearity that is a polynomial in the
unknown function and its complex conjugate, on $\mathbb {R}^ d $ or on the torus. Norm …

We consider the NLS on spheres. We describe the nonlinear evolutions of the highest
weight spherical harmonics. As a consequence, in contrast with the flat torus, we obtain that …

We introduce a new non-resonant low-regularity integrator for the cubic nonlinear
Schrödinger equation (NLSE) allowing for long-time error estimates which are optimal in the …