Traveling pulses for the nonlocal and lattice Klein -Gordon equations
The thesis includes two parts. In the first part, we study a nonlinear nonlocal Klein-Gordon equation on the whole real line. We first prove the existence of traveling pulses and then study the spatial asymptotic properties of these pulses and their instability. In the second part, we study a nonlinear lattice Klein-Gordon equation, which is actually a system of infinitely many ordinary differential equations. We show the existence and determine the spatial asymptotic properties of the traveling pulses using a similar method to that in the first part and then study the instability by numerical simulations. For both the nonlinear nonlocal and nonlinear lattice Klein-Gordon equations we illustrate that they share some similar properties with the limit equation, the classical Klein-Gordon equation, as a parameter ϵ approaches 0.