Derivative non-linear Schrodinger equation

By derivative non-linear Schrodinger (D-NLS) equations, we refer to equations of the form

$i\partial _{t}u-i\Delta u=f(u,{\underline {u}},Du,{\underline {Du}})$

where $f\,$ is an analytic function of $u\,$, its spatial derivatives $Du\,$, and their complex conjugates which vanishes to at least second order at the origin. One particularly important class of such equations is the Schrodinger maps equation.

We often assume the natural gauge invariance condition

$f(e^{iq}{\underline {u}},e^{-iq}{\underline {u}},e^{iq}Du,e^{-iq}{\underline {Du}})=e^{iq}f(u,{\underline {u}},Du,{\underline {Du}})$.

The main new difficulty here is the loss of regularity of one derivative in the non-linearity, which causes standard techniques such as the energy method to fail (since the energy estimate does not recover any regularity in the case of the Schrodinger equation). However, there are other estimates which can recover a full derivative for the inhomogeneous Schrodinger equation providing there is sufficient decay in space, and so one can still obtain well-posedness results for sufficiently smooth and regular data. In the analytic category some existence results can be found in SnTl1985, Ha1990.

An alternative strategy is to apply a suitable transformation in order to place the non-linearity in a good form. For instance, a term such as ${\underline {uDu}}\,$ is preferable to $uDu\,$ (the Fourier transform is less likely to stay near the upper paraboloid, and these terms are more likely to disappear in energy estimates). One can often "gauge transform" the equation (in a manner dependent on the solution $u\,$) so that all bad terms are eliminated. In one dimension this can be done by fairly elementary methods (e.g. the method of integrating factors), but in higher dimensions one must use some pseudo-differential calculus.

In order to quantify the decay properties needed, we define $H^{s,m}\,$ denote the space of all functions $u\,$ for which $<x>^{m}D^{s}u\,$ is in $L^{2}\,$; thus $s\,$ measures regularity and $m\,$ measures decay. It is a well-known fact that the Schrodinger equation trades decay for regularity; for instance, data in $H^{m,m'}\,$ instantly evolves to a solution locally in $H^{m+m'}\,$ for the free Schrodinger equation and $m,m'\geq 0.\,$

• If $m\geq [d/2]+4\,$ is an integer then one has LWP in $H^{m}\cap H^{m-2,2}\,$ Ci1999; see also Ci1996, Ci1995, Ci1994.
  • If $f\,$ is cubic or better then one can improve this to LWP in $H^{m}\,$ Ci1999. Furthermore, if one also has gauge invariance then data in $H^{m,m'}\,$ evolves to a solution in $H^{m+m'}\,$ for all non-zero times and all positive integers $m'\,$ Ci1999.
  • If $d=1\,$ and $f\,$ is cubic or better then one has LWP in $H^{3}\,$ HaOz1994b.
    • For special types of cubic non-linearity one can in fact get GWP for small data in $H^{0,4}\cap H^{4,0}\,$ Oz1996.
  • LWP in $H^{s}\cap H^{0,m}\,$ for small data for sufficiently large $s\,$, $m\,$ was shown in KnPoVe1993c. The solution was also shown to have $s+1/2\,$ derivatives in $L_{t,x,loc}^{2}$.
    • If $f\,$ is cubic or better one can take $m=0\,$KnPoVe1993c.
    • If $f\,$ is quartic or better then one has GWP for small data in $H^{s}.\,$ KnPoVe1995
    • For large data one still has LWP for sufficiently large $s,m\,$ Ci1995, Ci1994.

If the non-linearity consists mostly of the conjugate wave ${\underline {u}}\,,$ then one can do much better. For instance [Gr-p], when $f=(D{\underline {u}})^{k}\,$ one can obtain LWP when $s>s_{c}=d/2+1-1/(k-1),s\geq 1\,$, and $k\geq 2\,$ is an integer; similarly when $f=D({\underline {u}}^{k})\,$ one has LWP when $s>s_{c}=d/2-1/(k-1),s\geq 0,\,$ and $k\geq 2\,$ is an integer. In particular one has GWP in $L^{2}\,$ when $d=1\,$ and $f=i({\underline {u}}^{2})_{x}\,$ and GWP in $H^{1}\,$ when $d=1\,$ and $f=i({\underline {u}}_{x^{2}})\,$. These results apply in both the periodic and non-periodic setting.

Non-linearities such as $t^{-\alpha }\,$ $|u_{x}|^{2}\,$ in one dimension have been studied in HaNm2001b, with some asymptotic behaviour obtained.

In $d=2\,$ one has GWP for small data when the nonlinearities are of the form ${\underline {uDu}}+uDu\,$ De2002.