Inverse scattering solutions of the hunter-saxton equation

Abstract

The nonlinear partial differential equation 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 ux 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.

Keywords:

• Continued fractions
• Stieltjes; Inverse scattering;
• Harry Dym flows

^*Dedicated to Robert Carroll on his seventieth birthday.

^*Dedicated to Robert Carroll on his seventieth birthday.

Notes

^*Dedicated to Robert Carroll on his seventieth birthday.