Asymptotic Variational Wave Equations

We investigate the equation (ut+(f(u))x)x=f′ ′(u) (ux)2/2 where f(u) is a given smooth function. Typically f(u)=u2/2 or u3/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation utt − c(u) (c(u)ux)x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.