Nonlinear Geometric Optics for Pulses with Nonlinear Phases

The nonlinear geometric optics approach is used to elucidate the form of pulse solutions to two semilinear hyperbolic PDE problems in which the characteristic phases are nonlinear. The first problem considers a pulse traveling freely through space, and the second considers the collision of two pulses. A key question we sought to address was whether the collision of two pulses produces only two or more than two pulses after collision, given that the phases are nonlinear. We show that one can construct an approximate solution to the collision problem in which there are precisely two pulses in the leading term of the approximate solution after collision. We also construct an approximate solution to the free space problem and prove the approximate solution is close to the exact solution at the rate O(epsilon) as epsilon approaches zero in an appropriate norm. The proof for this statement relies on construction of an exact solution with a uniform time of existence with respect to epsilon using an argument of Guy Metivier in a new context. The proof also relies on the analysis of the error term using some of Metivier's estimates. We anticipate that it can also be proved that the approximate solution to the collision problem is close to the collision problem's exact solution in an analogous way. The results we obtained suggest that the nonlinear geometric optics approach may successfully be applied to study problems involving interaction of pulses with nonlinear phases.Bachelor of Art