We study a two-dimensional Fermi-Pasta-Ulam lattice in the long-amplitude, small-wavelength limit. The one-dimensional lattice has been thoroughly studied in this limit, where it has been established that the dynamics of the lattice is well-approximated by the Korteweg–De Vries (KdV) equation for timescales of the order ε^−3. Further it has been shown that solitary wave solutions of the FPU lattice in the one dimensional case are well approximated by solitary wave solutions of the KdV equation. A two-dimensional analogue of the KdV equation, the Kadomtsev–Petviashvili (KP-II) equation, is known to be a good approximation of certain two-dimensional FPU lattices for similar timescales, although no proof exists. In this thesis we present a rigorous justification that the KP-II equation is the long-amplitude, small-wavelength limit of a two-dimensional FPU model we introduce, analogous to the one-dimensional FPU system with quadratic nonlinearity. We also prove that the cubic KP-II equation is the limit of a model analogous to a one-dimensional FPU system with cubic nonlinearity. Further we study whether stability of line solitons in the KP-II equation extends to stability of one-dimensional FPU solitary waves in the two-dimensional lattices. / Thesis / Doctor of Philosophy (PhD)

In this thesis, we consider the planning problem for first-order mean-field games (MFG). These games degenerate into optimal transport when there is no coupling between players. Our aim is to extend the concept of displacement convexity from optimal transport to MFGs. This extension gives new estimates for solutions of MFGs. First, we introduce the Monge-Kantorovich problem and examine related results on rearrangement maps. Next, we present the concept of displacement convexity. Then, we derive first-order MFGs, which are given by a system of a Hamilton-Jacobi equation coupled with a transport equation. Finally, we identify a large class of functions, that depend on solutions of MFGs, which are convex in time. Among these, we find several norms. This convexity gives bounds for the density of solutions of the planning problem.