Quadratic NLS

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Quadratic NLS
Description
Equation iu_t + \Delta u = Q(u, \overline{u})
Fields u: \R \times \R^d \to \mathbb{C}
Data class u(0) \in H^s(\R^d)
Basic characteristics
Structure non-Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity \dot H^{d/2 - 2}(\R^d)
Criticality N/A
Covariance N/A
Theoretical results
LWP varies
GWP -
Related equations
Parent class NLS
Special cases Quadratic NLS on R, T, R^2, T^2, R^3, T^3
Other related -


Contents

Quadratic NLS

Equations of the form

i \partial_t u + \Delta u = Q(u, \overline{u})

which Q(u, \overline{u}) a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.


Quadratic NLS on R

  • Scaling is s_c=-3/2\,.
  • For any quadratic non-linearity one can obtain LWP for s \ge 0\, CaWe1990, Ts1987.
  • If the quadratic non-linearity is of \underline{uu}\, or uu\, type then one can push LWP to s > -3/4.\, KnPoVe1996b.
  • If the quadratic non-linearity is of \underline{u}u\, type then one can push LWP to s > -1/4.\, KnPoVe1996b.
  • Since these equations do not have L^2\, conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • If the non-linearity is |u|u\, then there is GWP in L^2\, thanks to L^2\, conservation, and ill-posedness below L^2\, by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on T

  • For any quadratic non-linearity one can obtain LWP for s \ge 0\, Bo1993. In the Hamiltonian case (|u| u\,) this is sharp by Gallilean invariance considerations KnPoVe-p
  • If the quadratic non-linearity is of \underline{uu}\, or uu\, type then one can push LWP to s > -1/2.\, KnPoVe1996b.
  • In the Hamiltonian case (a non-linearity of type |u| u\,) we have GWP for s \ge 0\, by L^2\, conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.

Quadratic NLS on R2

  • Scaling s_c = -1.\,
  • For any quadratic non-linearity one can obtain LWP for s \ge 0\, CaWe1990, Ts1987.
    • In the Hamiltonian case (|u| u\,) this is sharp by Gallilean invariance considerations KnPoVe-p
  • If the quadratic non-linearity is of \underline{uu}\, or u u\, type then one can push LWP to s > -3/4.\, St1997, CoDeKnSt2001.
    • This can be improved to the Besov space B^{-3/4}_{2,1}\, MurTao2004.
  • If the quadratic non-linearity is of u \underline{u}\, type then one can push LWP to s > -1/4.\, Ta2001.
  • In the Hamiltonian case (a non-linearity of type |u| u\,) we have GWP for s \ge 0\, by L^2\, conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
    • Below L^2\, we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on T^2

  • If the quadratic non-linearity is of \underline{uu}\, type then one can obtain LWP for s > -1/2\, Gr-p2

Quadratic NLS on R3

  • Scaling is s_c = -1/2.\,
  • For any quadratic non-linearity one can obtain LWP for s \ge 0\, CaWe1990, Ts1987.
  • If the quadratic non-linearity is of \underline{uu}\, or u u\, type then one can push LWP to s > -1/2.\, St1997, CoDeKnSt2001.
  • If the quadratic non-linearity is of u \underline{u}\, type then one can push LWP to s > -1/4.\, Ta2001.
  • In the Hamiltonian case (a non-linearity of type |u| u\,) we have GWP for s \ge 0\, by L^2\, conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
    • Below L^2\, we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on T3

  • If the quadratic non-linearity is of \underline{uu}\, type then one can obtain LWP for s > -3/10\, Gr-p2.
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