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

TL;DR

This paper constructs explicit 2-peakon solutions for the Novikov equation to demonstrate ill-posedness and norm inflation below the critical Sobolev index of 3/2, highlighting the equation's well-posedness threshold.

Contribution

It provides the first explicit construction of 2-peakon solutions illustrating ill-posedness and nonuniqueness for the Novikov equation below the critical Sobolev space.

Findings

Norm inflation occurs for s between 5/4 and 3/2.

Solutions are non-unique at the critical index s=5/4.

The equation is well-posed for s > 3/2.

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 $5/4<s<3/2$, thus resulting in norm inflation and ill-posedness. However, when $s<5/4$, the solution at the collision time coincides with a second solitary antipeakon solution. This scenario thus results in nonuniqueness and ill-posedness. Finally, when $s=5/4$ ill-posedness follows either from a failure of convergence or a failure of uniqueness. Considering that the Novikov equation is well-posed for $s>3/2$, 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.