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On a completely integrable nonlinear hyperbolic variational equation. (English) Zbl 0900.35387
Summary: We show that the nonlinear partial differential equation, ${\left({u}_{t}+u{u}_{x}\right)}_{xx}=1/2{\left({u}_{x}^{2}\right)}_{x}$, is a completely integrable, bi-variational, bi-Hamiltonian system. The corresponding equation for $w={u}_{xx}$ belongs to the Harry Dym hierarchy. This equation arises in two different physical contexts in two nonequivalent variational forms. It describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field, and it is the high-frequency limit of the Camassa-Holm equation, which is an integrable model equation for shallow water waves. Using the bi-Hamiltonian structure, we derive a recursion operator, a Lax pair, and an infinite family commuting Hamiltonian flows, together with the associated conservation laws. We also give the transformation to action-angle coordinates. Smooth solutions of the partial differential equation break down because their derivative blows up in finite time. Nevertheless, the Hamiltonian structure and complete integrability appear to remain valid globally in time, even after smooth solutions break down. We show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions. We compute the bi-Hamiltonian structure on this invariant manifold, which is obtained by restricting the bi-Hamiltonian structure of the partial differential equation.
##### MSC:
 35Q58 Other completely integrable PDE (MSC2000) 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 76A15 Liquid crystals (fluid mechanics)