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.

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