Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

Abstract

We derive an asymptotic solution of the vacuum Einstein equations that describes the propagation and diffraction of a localized, largeamplitude, rapidly-varying gravitational wave. We compare and contrast the resulting theory of strongly nonlinear geometrical optics for the Einstein equations with nonlinear geometrical optics theories for variational wave equations.

Orientation waves in a director field with rotational inertia

“…On the other hand, general relativity leads to a form of nonlinearity that is analogous to that of the general director-field equations: the wave operator in the Einstein equations acts on the metric and has coefficients that are functions of the metric. The effects of this nonlinearity in the Einstein equations are, however, much more degenerate than in the director-field equations [17].…”

Weak solutions to a nonlinear variational wave equation with general data

“…for some process h. In the inviscid (ν = 0), additive noise case, [32] used the choice h(t) = 0 in their well-posedness arguments. Suppose we replaced the viscosity ν∂ 2 xx R in the R-equation by ν∂ x (c(u)∂ x R) and ν∂ 2 xx S by ν∂ x (c(u)∂ x S) in the S-equation. Physically, these viscous terms model greater dissipation at higher wave-speeds.…”

“…Theorem 4.6 (Pathwise uniqueness). Let (R 1 , S 1 ) and (R 2 , S 2 ) be pathwise solutions to (1.5), both with initial conditions (R 0 , S 0 ) ∈ L 2p0 (Ω; L 2 (T)) 2 . Then…”

“…The equation (1.1) arises as the Euler-Lagrange equation of variational princple for the energy ´T 0 ´T |∂ t u| 2 + |c(u) ∂ x u| 2 dx dt. They are pertinent in systems such as wave maps from 4-dimensional Minkowski space to S 2 , in geometric optics, in orientation waves of the director fields of nematic liquid crystals with Oseen-Franck potential energy; see, e.g., [30,Section 2], [2,40], and references there for a more complete discussion. Mathematically, like the related Camassa-Holm and Hunter-Saxton equations, the variational wave equation exhibits supercritical behaviour.…”

“…2 in the R-equation and Qε (S) replaces S 2 in the Sequation in(1.3) [58,60]. We shall have need for similar linearisation in a two-level Galerkin approximation in Section 2.1.Viscous approximations, adding ν∂ 2 xx R to the R-equation and ν∂ 2 xx S to the S equation was used to study local classical(H k …”