Inverse Scattering Method II.: Methodological part with an example: Breather solution of the Sine-Gordon Equation

Recently it was discussed the Inverse Scattering Method, Part I. (paper I.) . It was a methodological Part with an example - soliton (kink) solution of the Sine-Gordon Equation. The aim of the paper I. was to introduce the Inverse Scattering Method for later studies of some problems in nonlinear dynamics. As a methodological example we described how to solve the Sine-Gordon Equation using the Inverse Scattering Method to obtain a soliton. This soliton solution is well known. In this paper we discuss another methodological example: the solution of the Sine-Gordon Equation using the Inverse Scattering Method to obtain description of a breather. While this breather solution is also well known, we will discuss later breather solutions in some physical systems. Thus we will have in the Part I. and in this Part II. seeds for solving problems with the Inverse Scattering Method, and in case of the Sine-Gordon Equation solutions describing the soliton and the breather.