On the separability of the sine-Gordon equation and similar quasilinear partial differential equations. II. Dependent- and independent-variable transformations

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Abstract
In an earlier paper, we investigated the separability of the sine-Gordon equation (SGE), and of similar quasilinear partial differential equations, under transformations of the dependent variable (i.e, of the codomain). We found, in particular, that there is a general class of dependent-variable transformations which leads to separable forms of the SGE. In this paper, we extend our previous analysis to include independent- as well as dependent-variable transformations (i.e., transformations of both the domain and codomain) and treat, in detail, constant coefficient equations of the first and second orders. We illustrate our method by applying it to the SGE and find combinations of domain and codomain transformations which reduce the equation to separable forms. Some of these transformations lead to known solutions of the SGE, but others give new solutions expressible in terms of a fifth Painlevé transcendent. Our method can, in principle, be used to map out the space of separable solutions of the SGE and other similar second-order equations, but it does have limitations. A discussion of these limitations and suggestions for possible improvements are also given.