GKdV-3 equation

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Quartic gKdV
Description
Equation u_t + u_{xxx} = \pm u^3 u_x
Fields u: \R \times \R \to \R
Data class u(0) \in H^s(\R)
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear with derivatives
Linear component Airy
Critical regularity \dot H^{-1/6}(\R)
Criticality mass-subcritical, energy-subcritical
Covariance -
Theoretical results
LWP H^s(\R) for s \geq -1/6
GWP H^s(\R) for s \geq -1/6, small norm
Related equations
Parent class gKdV
Special cases -
Other related -


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Non-periodic theory

The local and global well-posedness theory for the quartic generalized Korteweg-de Vries equation on the line and half-line is as follows.

  • Scaling is s_c = -1/6.
  • LWP for s >= -1/6 Ta2007
    • For s > -1/6 this is in Gr-p3
    • Was shown for s>=1/12 KnPoVe1993
    • Was shown for s>3/2 in GiTs1989
    • The result s >= 1/12 has also been established for the half-line CoKn-p, assuming boundary data is in H^{(s+1)/3} of course.
  • GWP in H^s for s >= 0 Gr-p3
    • For s>=1 this is in KnPoVe1993
    • Presumably one can use either the Fourier truncation method or the I-method to go below L^2. Even though the equation is not completely integrable, the one-dimensional nature of the equation suggests that correction term techniques will also be quite effective.
    • On the half-line GWP is known when s >= 1 and the boundary data is in H^{5/4}, assuming compatibility and small L^2 norm CoKn-p
  • Solitons are H^1-stable CaLo1982, Ws1986, BnSouSr1987 and asymptotically H^1 stable MtMe-p3, MtMe-p
    • If one also assumes the error is small in the critical space \dot H^{-1/6}(\R) then one has asymptotic stability Ta2007
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Periodic theory

The local and global well-posedness theory for the quartic generalized Korteweg-de Vries equation on the line and half-line is as follows.

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