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Inverse scattering solutions of the Hunter-Saxton equation. (English) Zbl 1020.35092
Summary: The nonlinear partial differential equation ${\left({u}_{t}+u{u}_{x}\right)}_{xx}=\frac{1}{2}{\left({u}_{x}^{2}\right)}_{x}$ was proposed by Hunter and Saxton as an asymptotic model equation for nematic liquid crystals. Hunter and Zheng showed that it is a member of the Harry Dym hierarchy of integrable flows, and solved the equation explicitly for a family of finite-dimensional, piecewise linear functions in the case when ${u}_{x}$ has compact support. In this note, the associated inverse scattering problem is used to obtain the explicit solutions of the finite-dimensional flows in both the compact and non-compact case.
MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 37K15 Integration of completely integrable systems by inverse spectral and scattering methods 35C10 Series solutions of PDE 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35Q51 Soliton-like equations 35P20 Asymptotic distribution of eigenvalues and eigenfunctions for PD operators 35P25 Scattering theory (PDE)