Conservative solutions to a nonlinear variational sine-Gordon equation

This paper is concerned with a nonlinear variational sine-Gordon equation u t t − c ( u ) [ c ( u ) u x ] x + λ 2 2 s i n ( 2 u ) = 0 which describes the motion of long waves on a neutral dipole chain in the continuum limit and a few other physical phenomena. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. To deal with the possible blowup of solutions, we introduce a new set of variables depending on the energy, whereby all singularities are resolved.