The sine-Gordon and Klein-Gordon equations

The sine-Gordon equation has been around since the 1860s, regarding the study of surfaces of constant curvature. The Klein-Gordon equation was set up in 1926 to study the electron in the context of Einstein's theory of relativity. A wide range of applications of these two equations to mathematics and to the physical sciences has evolved over the years, including applications to black holes, which we mention only in passing as the aim to maintain a very elementary level presentation. Some concrete solutions of these two famous equations are presented. The material is for students with some interest in differential equations, and/ or applications of hyperbolic functions. A review of the latter functions will be given to render the talk and material expository and self-contained.