The modified KdV equation with higher dispersion in Sobolev and analytic spaces on the line

Abstract

This paper studies the Cauchy problem on the line for the modified Korteweg–deVries equation whose dispersion is of order $$m=2j+1$$ , where $$j\ge 2$$ is a positive integer. Trilinear estimates in Bourgain spaces are proved and are used to show that local in time well-posedness holds in Sobolev spaces $$H^s$$ for negative values of s. Then, using the analytic version of the trilinear estimates, well-posedness in a class of analytic functions on the line that can be extended holomorphically in a symmetric strip around the x-axis is proved. Finally, existence of global analytic solutions and a lower bound for the uniform radius of spatial analyticity are established.