Schrodinger:quadratic NLS

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Quadratic NLS

Equations of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \partial_t u + \Delta u = Q(u, \overline{u})}

which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q(u, \overline{u})} a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.


Quadratic NLS on R

  • Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c=-3/2\,.}
  • For any quadratic non-linearity one can obtain LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>=0\,} CaWe1990, Ts1987.
  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{uu}\,} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle uu\,} type then one can push LWP to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -3/4.\,} KnPoVe1996b.
    • This can be improved to the Besov space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B^{-3/4}_{2,1}\,} [MurTao-p]. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X^{s,b}\,} bilinear estimates fail for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{-3/4}\,} references:NaTkTs-p NaTkTs2001.
  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{u}u\,} type then one can push LWP to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -1/4.\,} KnPoVe1996b.
  • Since these equations do not have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |u|u\,} then there is GWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} thanks to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} conservation, and ill-posedness below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

Quadratic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T}

  • For any quadratic non-linearity one can obtain LWP for s ³ 0 Bo1993. In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
  • If the quadratic non-linearity is of u u or u u 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 ³ 0 by L2 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^2}

  • Scaling is sc = -1.
  • For any quadratic non-linearity one can obtain LWP for s ³ 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 u u or u u type then one can push LWP to s > -3/4. St1997, references:CoDeKnSt-p CoDeKnSt-p.
    • This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p].
  • If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. references:Ta-p2 Ta-p2.
  • In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s ³ 0 by L2 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

Quadratic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^3}

  • Scaling is sc = -1/2.
  • For any quadratic non-linearity one can obtain LWP for s ³ 0 CaWe1990, Ts1987.
  • If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -1/2. St1997, references:CoDeKnSt-p CoDeKnSt-p.
  • If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. references:Ta-p2 Ta-p2.
  • In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s ³ 0 by L2 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T^3}

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