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arXiv · 1708.05759

Construction of 2-peakon Solutions and Ill-Posedness for the Novikov equation

A. Alexandrou Himonas·Curtis Holliman·Carlos Kenig
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Abstract

For the Novikov equation, on both the line and the circle, we construct a 2-peakon solution with an asymmetric antipeakon-peakon initial profile whose $H^s$-norm for $s<3/2$ is arbitrarily small. Immediately after the initial time, both the antipeakon and peakon move in the positive direction, and a collision occurs in arbitrarily small time. Moreover, at the collision time the $H^s$-norm of the solution becomes arbitrarily large when $5/4 3/2$, these results put together establish $3/2$ as the critical index of well-posedness for this equation. The case $s=3/2$ remains an open question.

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A. Alexandrou Himonas, Curtis Holliman, Carlos Kenig. 2017-08-18. Construction of 2-peakon Solutions and Ill-Posedness for the Novikov equation. https://arxiv.org/abs/1708.05759

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