Differential and Integral Equations

Local well-posedness of the NLS equation with third order dispersion in negative Sobolev spaces

Tomoyuki Miyaji and Yoshio Tsutsumi

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We show the time local well-posedness in $H^s$ of the reduced NLS equation with third order dispersion (r3NLS) on $\mathbf{T}$ for $s > -1/6$. Our proof is based on the nonlinear smoothing effect, which is similar to that for mKdV. However, when (r3NLS) is considered in Sobolev spaces of negative indices, the unconditional uniqueness of solutions, that is, the uniqueness of solutions without auxiliary spaces breaks down in marked contrast to mKdV.

Article information

Differential Integral Equations, Volume 31, Number 1/2 (2018), 111-132.

First available in Project Euclid: 26 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35A01: Existence problems: global existence, local existence, non-existence 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness


Miyaji, Tomoyuki; Tsutsumi, Yoshio. Local well-posedness of the NLS equation with third order dispersion in negative Sobolev spaces. Differential Integral Equations 31 (2018), no. 1/2, 111--132. https://projecteuclid.org/euclid.die/1509041404

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