ABSTRACT
This chapter introduces the concepts of Lie point and Lie Backlund symmetry without going into the details, because the masterly written monograph of Olver explains the theory in a very lucid manner. Non-linear equations which are completely integrable have many properties which are not shared by other types of equations. The Lie-Backlund transformation is actually the higher-order contact transformation generalizing the Lie point transformation. Several methods were developed to obtain the recursion operator and a new concept was introduced by Fuchssteiner, known as the master symmetry. The master symmetry is used to generate all the finite number of symmetries of a given non-linear equation. The method of Fourier expansion and perturbation in the amplitude can lead to a host of important information, such as Lie-Backlund symmetries, recursion operators, bi-Hamiltonian structure, and much more. One type of symmetry which has been widely discussed only very recently is known as non-local symmetry.
