Johan Matheus Tuwankotta
We are dealing with an ordinary difference equation which can be seen as iteration of functions in some linear space. In addition to that, there exists enough number of functions (called integrals) which are functionally independent (the level sets of these functions for some regular values are intersecting in a transversal manner) which are preserved by these iterates. This is called integrable mapping. There are some examples in the literature, and one of the most famous one is the Quispel-Roberts-Thompson map, or the QRT-map. A concise mathematical exploration of this QRT-map can be found in a seminal books by J.J. Duistermaat [4] In this research, we are trying to understand the concept of duality in integrable map. Suppose a discrete mapping is gien, and the mapping is equipped with several integrals. Then it is known that linear combination of the integrals is still an integral of the corresponding mapping. in one factor out the original mapping from the equation that needs to be satisfied by this linear combination of the integrals, then the reminder defines the so-called dual map. To understand the concept of duality, in this research we will study a particular map, derived from the partial difference sine-Gordon equations (see [6] and the reference therein). TO derive the mapping, we use travelling wave reduction (which is also known as staircase method) which depends on two parameters: z_1 and z_2. We will study the case where (z_1,z_2) = (1,4), and also for (2,3). These two mappings are five-dimensional but we expect to see difficulty for the latter case. Furthermore, we will also study the situation where (z_1,z_2)= (2,4). In this situation, the difficulty is of different level as we encounter a two-coupled mapping in six dimensional linear space.