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The integrable KdV equation is a model for shallow water. However, another integrable equation, the Camassa Holm equation is also claimed to be a model for long wave shallow water. The Camassa Holm equation was derived from the Green Naghdi equations which itself is a model for water. Robin lohnson found that the derivation of the Camassa Holm equation from the Green Naghdi model was inconsistent in the reduction to right moving waves. Dullin, Gottwald and Holm then derived the Camassa Holm equation from the shallow water wave equations. Since the publication of this paper Camassa Holm equation was ubiquitously used as model for water waves. In chapter 2 it is shown that the Camassa Holm equation is inconsistent with the long wave asymptotic expansion. Symbolic representation is used to find the approximate symmetries of perturbed integrable equations. The use of symbolic representation in integrable systems is well developed. In this thesis it is used to find near identity transformations of perturbed integrable equations to integrable equations. The question arises whether the conditions necessary for there to exist approximate symmetries are sufficient for there to exist such a transformation. In chapter 5 we study reciprocal coordinate transformations of perturbed integrable equations and asymptotic expansions of discrete Lax operators.