Conservative solutions to a one-dimensional nonlinear variational wave equation

This paper is focused on a nonlinear variational wave equation which is the Euler–Lagrange equation of a variational principle whose action is a quadratic function of the derivatives of the field. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. The approach follows very closely the method of energy-dependent coordinates proposed by Bressan, Zhang and Zheng [6] and [7]. By introducing a new set of variables, which resolve all singularities due to the possible concentration of energy, the equation can be rewritten as a semilinear system. We construct the global weak solution by expressing the solution of the semilinear system in terms of the original variables.