Quartic NLS

Quartic NLS on $R$

• Scaling is $s_{c}=-1/6\,$.
• For any quartic non-linearity one can obtain LWP for $s\geq 0\,$ CaWe1990
  • Below $L^{2}\,$ we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.
• If the quartic non-linearity is of ${\underline {uuuu}}\,$ type then one can obtain LWP for $s>-1/6\,.$ For $|u|^{4}\,$ one has LWP for $s>-1/8\,$, while for the other three types $u^{4}\,$, $uuu{\underline {u}}\,$, or $u{\underline {uuu}}\,$ one has LWP for $s>-1/6\,$ Gr-p2.
• In the Hamiltonian case (a non-linearity of type $|u|^{3}u\,$) 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.

Quartic NLS on $T$

• For any quartic non-linearity one has LWP for $s>0\,$ Bo1993.
• If the quartic non-linearity is of ${\underline {uuuu}}\,$ 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 > -1/6\,} , Gr-p2.
• If the nonlinearity 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|^3 u\,} type one has GWP for random data whose Fourier coefficients decay like 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 1/|k|\,} (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.

Quartic 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 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 = 1/3\,.}
• For any quartic 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 s_c\,} CaWe1990.
  • 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<s_c\,} we have ill-posedness, indeed 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 H^s\,} norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
• 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|^3 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 1\,} Ka1986.
  • This has been improved 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-e\,} in CoKeStTkTa2003c in the defocusing Hamiltonian case. This result can of course be improved further.
  • Scattering in the energy space Na1999c in the defocusing Hamiltonian case.
  • One also has GWP and scattering for small 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^{1/3}\,} data for any quintic non-linearity.