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Dynamics of director fields. (English) Zbl 0761.35063

The authors study the quasilinear wave equation ${\psi }_{tt}=c\left(\psi \right){\left[c\left(\psi \right){\psi }_{x}\right]}_{x}$, which is a simplified equation for the director field of a nematic liquid crystal. Making the usual Ansatz of weakly nonlinear geometrical optics, they obtain for the first order term the quasilinear equation ${\left({u}_{t}+u{u}_{x}\right)}_{x}=\left(1/2\right){u}_{x}^{2}$.

An essential part of the paper consists in studying solutions of this equation. It is shown that in general smooth solutions do not exist globally, weak solutions are defined, and admissibility criteria are suggested. Simple examples of continuous, weak solutions with corners are given, which show that weak solutions are not unique. The asymptotic solutions are compared with numerical solutions.

These results are used to construct asymptotic solutions of the original equation.

##### MSC:
 35L70 Nonlinear second-order hyperbolic equations 35C20 Asymptotic expansions of solutions of PDE 35L85 Linear hyperbolic unilateral problems; linear hyperbolic variational inequalities 76A99 Foundations, constitutive equations, rheology