Construction of the wave operator for non-linear dispersive equations

In this thesis, we will study non-linear dispersive equations. The primary focus will be on the construction of the positive-time wave operator for such equations. The positive-time wave operator problem arises in the study of the asymptotics of a partial differential equation. It is a map from a space of initial data X into itself, and is loosely defined as follows: Suppose that for a solution Ψ_lin to the dispersive equation with no non-linearity and initial data Ψ₊ there exists a unique solution Ψ to the non-linear equation with initial data Ψ_O such that Ψ behaves as Ψ_lin as t→ ∞. Then the wave operator is the map W ₊ that takes Ψ₊/sub; to Ψ₀.

By its definition, W₊ is injective. An important additional question is whether or not the map is also surjective. If so, then every non-linear solution emanating from X behaves, in some sense, linearly as it evolves (this is known as asymptotic completeness). Thus, there is some justification for treating these solutions as their much simpler linear counterparts.

The main results presented in this thesis revolve around the construction of the wave operator(s) at critical non-linearities. We will study the #8220; semi-relativistic ” Schrëdinger equation as well as the Klein-Gordon-Schrëdinger system on R². In both cases, we will impose fairly general quadratic non-linearities for which conservation laws cannot be relied upon. These non-linearities fall below the scaling required to employ such tools as the Strichartz estimates. We instead adapt the "first iteration method" of Jang, Li, and Zhang to our setting which depends crucially on the critical decay of the non-linear interaction of the linear evolution. To see the critical decay in our problem, careful analysis is needed to treat the regime where one has spatial and/or time resonance.