Consider then cubic defocusing nonlinear wave equation on three dimensional Euclidean space, with radial initial data. In this thesis, we construct a Gibbs measure for this system and prove its invariance under the flow of the NLW. In particular, we are in the infinite volume setting.

For the finite volume analogue, specifically on the unit ball with zero boundary values, an invariant Gibbs measure was constructed by Burq, Tvetkov, and de Suzzoni as a Borel measure on super-critical Sobolev spaces.

We first show that this finite volume Gibbs measure is supported on a space of weighted Holder continuous functions. Next, we show that the NLW is locally well-posed there, a counter-point to the Sobolev super-criticality noted by Burq and Tzvetkov. Furthermore, the flow of the NLW leaves this measure invariant.

We use a multi-time Feynman--Kac formula to construct the infinite volume limit measure by computing the asymptotics of the fundamental solution of an appropriate parabolic PDE. We use finite speed of propagation and results from descriptive set theory to establish invariance of the infinite volume measure.

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 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.

This thesis studies scaling critical Strichartz estimates for the Schrödinger flow on compact symmetric spaces. A general scaling critical Strichartz estimate (with an ε-loss, respectively) is given conditional on a conjectured dispersive estimate (with an ε-loss, respectively) on general compact symmetric spaces. The dispersive estimate is then proved for the special case of connected compact Lie groups. Slightly more generally, for products of connected compact Lie groups and spheres of odd dimension, the dispersive estimate is proved with an ε-loss.

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 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.