Difference between revisions of "Quadratic NLS"

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

Latest revision as of 01:30, 17 March 2007

Quadratic NLS
Description
Equation
Fields
Data class
Basic characteristics
Structure non-Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity
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 -


Quadratic NLS

Equations of the form

which a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.


Quadratic NLS on R

  • Scaling is
  • For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
  • If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
    • This can be improved to the Besov space MurTao2004. The bilinear estimates fail for NaTkTs2001.
  • If the quadratic non-linearity is of type then one can push LWP to KnPoVe1996b.
  • Since these equations do not have conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • If the non-linearity is then there is GWP in thanks to conservation, and ill-posedness below by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on

  • For any quadratic non-linearity one can obtain LWP for Bo1993. In the Hamiltonian case () this is sharp by Gallilean invariance considerations KnPoVe-p
  • If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
  • In the Hamiltonian case (a non-linearity of type ) we have GWP for by 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

  • Scaling
  • For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
    • In the Hamiltonian case () this is sharp by Gallilean invariance considerations KnPoVe-p
  • If the quadratic non-linearity is of or type then one can push LWP to St1997, CoDeKnSt2001.
    • This can be improved to the Besov space MurTao2004.
  • If the quadratic non-linearity is of type then one can push LWP to Ta2001.
  • In the Hamiltonian case (a non-linearity of type ) we have GWP for by conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
    • Below 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 type then one can obtain LWP for Gr-p2

Quadratic NLS on

  • Scaling is
  • For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
  • If the quadratic non-linearity is of or type then one can push LWP to St1997, CoDeKnSt2001.
  • If the quadratic non-linearity is of type then one can push LWP to Ta2001.
  • In the Hamiltonian case (a non-linearity of type ) we have GWP for by conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
    • Below we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on

  • If the quadratic non-linearity is of type then one can obtain LWP for Gr-p2.