Schrodinger:quadratic NLS

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 $s\geq 0\,$ CaWe1990, Ts1987.
• If the quadratic non-linearity is of ${\underline {uu}}\,$ or $uu\,$ type then one can push LWP to $s>-3/4.\,$ KnPoVe1996b.
  • This can be improved to the Besov space $B_{2,1}^{-3/4}\,$ [MurTao-p]. The $X^{s,b}\,$ bilinear estimates fail for $H^{-3/4}\,$ references:NaTkTs-p NaTkTs2001.
• 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.

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Quadratic NLS on $T$

• For any quadratic non-linearity one can obtain LWP for $s\geq 0\,$ 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 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 > -1/2.\,} KnPoVe1996b.
• In the Hamiltonian case (a non-linearity of type $|u|u\,$) we have GWP 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 \ge 0\,} by 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. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.

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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 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 = -1.\,}
• 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 \ge 0\,} CaWe1990, Ts1987.
  • In the Hamiltonian case (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\,} ) this is sharp by Gallilean invariance considerations [KnPoVe-p]
• 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 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 > -3/4.\,} St1997, references:CoDeKnSt-p CoDeKnSt-p.
  • 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].
• 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 u \underline{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.\,} references:Ta-p2 Ta-p2.
• In the Hamiltonian case (a non-linearity of type 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\,} ) we have GWP 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 \ge 0\,} by 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. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • 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\,} we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

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Quadratic NLS on T^2

• 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}\,} type then 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 > -1/2\,} references#Gr-p2 Gr-p2

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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 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 = -1/2.\,}
• 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 \ge 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 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 > -1/2.\,} St1997, references:CoDeKnSt-p CoDeKnSt-p.
• 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 u \underline{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.\,} references:Ta-p2 Ta-p2.
• In the Hamiltonian case (a non-linearity of type $|u|u\,$) we have GWP for $s\geq 0\,$ 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.

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Quadratic NLS on $T^{3}$

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

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