Quasilinear Schrodinger equations

Quasilinear NLS (QNLS)

These are general equations of the form

$u_{t}=ia(x,t,u,Du)D^{2}u+b_{1}(x,t,u,Du)Du+b_{2}(x,t,u,Du)D{\underline {u}}+firstorderterms\,$,

where $a,b_{1},b_{2}\,,$ and the lower order terms are smooth functions of all variables.These general systems arise in certain physical models (see e.g. BdHaSau1997).Also under certain conditions they can be derived from fully nonlinear Schrodinger equations by differentiating the equation.

In order to qualify as a quasilinear NLS, we require that the quadratic form a is real and elliptic.It is also natural to assume that the metric structure induced by a obeys a non-trapping condition (all geodesics eventually reach spatial infinity), as this is what is necessary for the optimal local smoothing estimate to occur.For a similar reason it is useful to assume that the magnetic field b_1 (or more precisely, the imaginary part of this field) is uniformly integrable along lines in space in the time independent case (for the time dependent case the criterion involves the bicharacteristic flow and is more complicated, see Ic1984); without this condition even the linear equation can be ill-posed.

A model example of QNLS is the equation

$\partial _{t}u=i(\Delta -V(x))u-2iuh'(|u|^{2})\Delta h(|u|^{2})+iug(|u|^{2})\,$

for smooth functions $h,g\,$.

When V=0 local existence for small data is known in $H^{6}(R^{n})\,$ for $n=1,2,3\,$ BdHaSau1997

Under certain conditions on the initial data the LWP can be extended to GWP for $n=2,3\,$ BdHaSau1997.

For large data, LWP is known in $H^{s}(R^{n})\,$ for any n and any sufficiently large $s>s(n)\,$Col2002

For suitable choices of V LWP is also known for $H^{\infty }(R^{n})\,$ for any $n\,$ Pop2001; this uses the Nash-Moser iteration method.

In one dimension, the fully nonlinear Schrodinger equation has LWP in $H^{\infty }(R^{n})\,$ assuming a cubic nonlinearity Pop2001b.Other LWP results for the one-dimensional QNLS have been obtained by [LimPo-p] using gauge transform arguments.

In general dimension, LWP for data in $H^{s,2}\,$ for sufficiently large s has been obtained in KnPoVe-p assuming non-trapping, and asymptotic flatness of the metric a and of the magnetic field $Imb_{1}\,$ (both decaying like $1/|x|^{2}\,$ or better up to derivatives of second order).