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Inverse scattering solutions of the Hunter-Saxton equation. (English) Zbl 1020.35092
Summary: The nonlinear partial differential equation (u t +uu x ) xx =1 2(u x 2 ) 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:
35Q53KdV-like (Korteweg-de Vries) equations
37K15Integration of completely integrable systems by inverse spectral and scattering methods
35C10Series solutions of PDE
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q51Soliton-like equations
35P20Asymptotic distribution of eigenvalues and eigenfunctions for PD operators
35P25Scattering theory (PDE)