Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation

Abstract

Abstract In this article, we mainly prove low regularity conservation laws for the Fokas-Lenells equation in Besov spaces with small initial data both on the line and on the circle. We develop a new technique in Fourier analysis and complex analysis to obtain the a priori estimates. It is based on the perturbation determinant associated with the Lax pair introduced by Killip, Vişan, and Zhang for completely integrable dispersive partial differential equations. Additionally, we also utilize the perturbation determinant to derive the global a priori estimates for the Schwartz solutions to the Camassa-Holm (CH) equation in H 1 {H}^{1} . Even though the energy conservation law of the CH equation is a fact known to all, the perturbation determinant method indicates that we cannot get any conserved quantities for the CH equation in H k {H}^{k} except k = 1 k=1 .