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We introduce and analyze an $hp$-version $C^1$-continuous Petrov-Galerkin(CPG) method for nonlinear initial value problems of second-order ordinarydifferential equations. We derive a-priori error estimates in the $L^2$-,$L^\infty$-, $H^1$- and $H^2$-norms that are completely explicit in the localtime steps and local approximation degrees. Moreover, we show that the$hp$-version $C^1$-CPG method superconverges at the nodal points of the timepartition with regard to the time steps and approximation degrees. As anapplication, we apply the $hp$-version $C^1$-CPG method to time discretizationof nonlinear wave equations. Several numerical examples are presented to verifythe theoretical results.

$hp$-version $C^1$-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations