We employ the integrable structure of the Benjamin--Ono equation in order to study its rough solutions. For rough data, our most useful tools are the Lax pair formalism and, as in the inverse scattering transform, the structural information embedded in solutions to the scattering equation. Using these, we prove that Sobolev norms are conserved and locally smoothed for rough initial data. Using the integrable structure, we construct a Hamiltonian that usefully approximates the Benjamin--Ono Hamiltonian. With this we may provide a short new proof that the Cauchy problem is well-posed in L2.

We present several large-data results for nontranslation-invariant analogues of the energy-critical and mass-critical nonlinear Schrödinger equations, obtained by introducing external potentials or non-Euclidean geometries.

The main goal of this work is to prove a symplectic non-squeezing result for the Korteweg--de Vries (KdV) equation on the line $\R$. This is achieved via a finite-dimensional approximation argument. Our choice of finite-dimensional Hamiltonian system that effectively approximates the KdV flow is inspired by the recent breakthrough in the well-posedness theory of KdV in low regularity spaces \cite{KV18}, relying on its completely integrable structure. The employment of our methods also provides us with a new concise proof of symplectic non-squeezing for the same equation on the circle $\T$, recovering the result of \cite{CKSTT}.

This work explores the existence and behavior of solutions to the Korteweg–de Vries equation on the line for large perturbations of certain classical solutions. First, we show that given a suitable solution $V(t,x)$, KdV is globally well-posed for initial data $u(0,x) \in V (0,x) + H^{-1}(\mathbb{R})$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. In particular, we show that smooth periodic and step-like profiles $V(0,x)$ satisfy our hypotheses.

Our second main objective is to prove a variational characterization of KdV multisolitons. Maddocks and Sachs used that $n$-solitons are local constrained minimizers of the polynomial conserved quantities in order to prove that $n$-solitons are orbitally stable in $H^n(\mathbb{R})$. We show that multisolitons are the unique global constrained minimizers for this problem. We then use this characterization to provide a new proof of the Maddocks–Sachs orbital stability result via concentration compactness.

We study the critical initial-value problem for defocusing nonlinear Schr\"odinger equations.

We adapt techniques that were originally developed to treat the mass- and energy-critical equations to the case of `non-conserved' critical regularity. In particular, we follow the minimal counterexample approach to the induction on energy technique of Bourgain.

For a range of dimensions and critical regularities, we prove that any solution that remains bounded in the critical Sobolev space must exist globally in time, obey spacetime bounds, and scatter to a free solution. In certain cases, the main result applies only to radial solutions. An equivalent formulation of the main result is the statement that any solution that fails to scatter must blow up its critical Sobolev norm.