| Abstract: |
| The sine-Gordon equation has slowly-modulated librational wave solutions that are approximated at leading-order by a Whitham averaging formalism. The Whitham modulation equations are an elliptic quasilinear system whose solutions develop singularities in finite time. We show that when the solution of the Whitham system develops a generic type of gradient catastrophe singularity, the solution of the sine-Gordon equation locally takes on a universal form, independent of initial data and described in terms of the real tritronqu\`ee solution of the Painlev\`e-I equation and a two-parameter family of exact solutions of sine-Gordon that represent space-time localized defects on an otherwise periodic background wave. |
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