Invariance transformations, invariance group transformations, and invariance groups of the sine-Gordon equations

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Abstract
We investigate a structure of continuous invariance transformations connected to the identity transformation. The transformations considered do not necessarily form a group. We clarify the relationship between the infinitesimal invariance transformation and the finite invariance transformation by showing explicitly how the infinitesimal transformations are woven into the finite one. The analysis leads to a new method of finding generators of the invariance group transformation. The results are useful in the study of symmetry properties, or group theoretic structure, of differential equations. We use the results in studying the group properties of the sine‐Gordon equation uxt=sinu, and indicate that the equation is invariant under an infinite number of one‐parameter groups; the groups obtained are of a more general type than that dealt with by Lie. These findings are used to prove the group theoretic origin of the well‐known conservation laws associated with the sine‐Gordon equation.