Schrodinger equations

Non-linear Schrodinger equations

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Overview

The free Schrodinger equation

$i\partial _{t}u+\Delta u=0$

where u is a complex-valued function in $R^{d+1}$, describes the evolution of a free non-relativistic quantum particle in d spatial dimensions. This equation can be modified in many ways, notably by adding a potential or an obstacle, but we shall be interested in non-linear perturbations such as

$i\partial _{t}u+\Delta u=f(u,{\overline {u}},Du,D{\overline {u}})$

where $D$ denotes spatial differentiation. In such full generality, we refer to this equation as a derivative non-linear Schrodinger equation (D-NLS). If the non-linearity does not contain derivatives then we refer to this equation as a [#nls semilinear Schrodinger equation] (NLS). These equations (particularly the cubic NLS) arise as model equations from several areas of physics.

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Semilinear Schrodinger (NLS)

[Many thanks to Kenji Nakanishi with valuable help with the scattering theory portion of this section. However, we are still missing many references and results, e.g. on NLS blowup. - Ed.]

The semilinear Schrodinger equation is

i u_t + D u + l |u|^{p-1} u = 0

for p>1. (One can also add a potential term, which leads to many physically interesting problems, however the field of Schrodinger operators with potential is far too vast to even attempt to summarize here). In order to consider this problem in H^s one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume

p is an odd integer, or p > [s]+1.

This is a Hamiltonian flow with the Hamiltonian

H(u) = ò |Ñ u|²/2 - l |u|^{p+1}/(p+1) dx

and symplectic form

{u, v} = Im ò u v dx.

From the phase invariance u -> exp(i q) u one also has conservation of the L² norm. The case l > 0 is focussing; l < 0 is defocussing.

The scaling regularity is s_c = d/2 - 2/(p-1). The most interesting values of p are the L²-critical or pseudoconformal power p=1+4/d and the H¹-critical power p=1+4/(d-2) for d>2 (for d=1,2 there is no H¹ conformal power). The power p = 1 + 2/d is also a key exponent for the scattering theory (as this is when the non-linearity |u|^{p-1} u has about equal strength with the decay t^{-d/2}). The cases p=3,5 are the most natural for physical applications since the non-linearity is then a polynomial. The cubic NLS in particular seems to appear naturally as a model equation for many different physical contexts, especially in dispersive, weakly non-linear perturbations of a plane wave. For instance, it arises as a model for dilute Bose-Einstein condensates.

Dimension d	Scattering power 1+2/d	L2-critical power 1+4/d	H1-critical power 1+4/(d-2)
1	3	5	N/A
2	2	3	infinity
3	5/3	7/3	5
4	3/2	2	3
5	7/5	9/5	7/3
6	4/3	5/3	2

The pseudoconformal transformation of the Hamiltonian gives that the time derivative of

|| (x + 2it Ñ) u ||²_2 - 8 l t²/(p+1) || u ||_{p+1}^{p+1}

is equal to

4dtl(p-(1+4/d))/(p+1) ||u||_{p+1}^{p+1}.

This law is useful for obtaining a priori spacetime estimates on the solution given sufficient decay in space (e.g. xu(0) in L²), especially in the L²-critical case p=1+4/d (when the above derivative is zero). The L² norm of xu(0) is sometimes known as the pseudoconformal charge.

The equation is invariant under Gallilean transformations

u(x,t) -> exp(i (v.x/2 - |v|² t)) u(x-vt, t).

This can be used to show ill-posedness below L² in the focusing case [KnPoVe-p], and also in the defocusing case [CtCoTa-p2]. (However if the non-linearity is replaced by a non-invariant expression such as u², then [#Quadratic_NLS one can go below L²]).

From scaling invariance one can obtain Morawetz inequalities, which usually estimate quantities of the form

ò ò |u|^{p+1}/|x| dx dt

in the defocussing case in terms of the H^{1/2} norm. These are useful for limiting the number of times the solution can concentrate at the origin; this is especially handy in the radially symmetric case.

In the other direction, one has LWP for s ³ 0, s_c CaWe1990; see also Ts1987; for the case s=1, see GiVl1979. In the L²-subcritical cases one has GWP for all s³0 by L² conservation; in all other cases one has GWP and scattering for small data in H^s, s ³ s_c. These results apply in both the focussing and defocussing cases. At the critical exponent one can prove Besov space refinements Pl2000, [Pl-p4]. This can then be used to obtain self-similar solutions, see CaWe1998, CaWe1998b, RiYou1998, [MiaZg-p1], [MiaZgZgx-p], [MiaZgZgx-p2], Fur2001.

Now suppose we remove the regularity assumption that p is either an odd integer or larger than [s]+1. Then some of the above results are still known to hold:

• ? In the H^1 subcritical case one has GWP in H^1, assuming the nonlinearity is smooth near the origin Ka1986
  • In R^6 one also has Lipschitz well-posedness [BuGdTz-p5]

In the periodic setting these results are much more difficult to obtain. On the one-dimensional torus T one has LWP for s > 0, s_c if p > 1, with the endpoint s=0 being attained when 1 <= p <= 4 Bo1993. In particular one has GWP in L^2 when p < 4, or when p=4 and the data is small norm.For 6 > p ³ 4 one also has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) Bo1995c. (For p=6 one needs to impose a smallness condition on the L² norm or assume defocusing; for p>6 one needs to assume defocusing).

• For the defocussing case, one has GWP in the H¹-subcritical case if the data is in H¹. To improve GWP to scattering, it seems that needs p to be L² super-critical (i.e. p > 1 + 4/d). In this case one can obtain scattering if the data is in L²(|x|² dx) (since one can then use the pseudo-conformal conservation law).
  • In the d ³ 3 cases one can remove the L²(|x|² dx) assumption GiVl1985 (see also Bo1998b) by exploiting Morawetz identities, approximate finite speed of propagation, and strong decay estimates (the decay of t^{-d/2} is integrable). In this case one can even relax the H¹ norm to H^s for some s<1 references:CoKeStTkTa-p7 CoKeStTkTa-p7.
  • For d=1,2 one can also remove the L²(|x|² dx) assumption Na1999c by finding a variant of the Morawetz identity for low dimensions, together with Bourgain's induction on energy argument.

In the L^2-supercritical focussing case one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality

d²_t ò x² |u|² dx ~ H(u);

see e.g. OgTs1991. By scaling this implies that we have instantaneous blowup in H^s for s < s_c in the focusing case. In the defocusing case blowup
is not known, but the H^s norm can still get arbitrarily large arbitrarily quickly for s < s_c [CtCoTa-p2]

Suppose we are in the L^2 subcritical case p < 1 + 2/d, with focusing non-linearity. Then there is a unique positive radial ground state (or soliton) for each energy E. By translation and phase shift one thus obtains a four-dimensional manifold of ground states for each energy. This manifold is H¹-stable Ws1985, Ws1986. Below the H^1 norm, this is not known, but polynomial upper bounds on the instability are in CoKeStTkTa2003b.Multisolitons are also asymptotically stable under smooth decaying perturbations Ya1980, Grf1990, Zi1997, [RoScgSf-p], [RoScgSf-p2], provided that p is betweeen 1+2/d and 1+4/d.

One can go beyond scattering and ask for asymptotic completeness and existence of the wave operators. When p £ 1 + 2/d this is not possible due to the poor decay in time in the non-linear term Bb1984, Gs1977b, Sr1989, however at p = 1+2/d one can obtain modified wave operators for data with suitable regularity, decay, and moment conditions Oz1991, GiOz1993, HaNm1998, ShiTon2004, HaNmShiTon2004. In the regime between the L² and H¹ critical powers the wave operators are well-defined in the energy space LnSr1978, GiVl1985, Na1999c. At the L² critical exponent 1 + 4/d one can define wave operators assuming that we impose an L^p_x,t integrability condition on the solution or some smallness or localization condition on the data GiVl1979, GiVl1985, Bo1998 (see also Ts1985 for the case of finite pseudoconformal charge). Below the L² critical power one can construct wave operators on certain spaces related to the pseudo-conformal charge CaWe1992, GiOz1993, GiOzVl1994, Oz1991; see also GiVl1979, Ts1985. For H^s wave operators were also constructed in Na2001. However in order to construct wave operators in spaces such as L²(|x|² dx) (the space of functions with finite pseudoconformal charge) it is necessary that p is larger than or equal to the rather unusual power

1 + 8 / (sqrt(d² + 12d + 4) + d - 2);

see NaOz2002 for further discussion.

Many of the global results for H^s also hold true for L²(|x|^{2s} dx). Heuristically this follows from the pseudo-conformal transformation, although making this rigorous is sometimes difficult. Sample results are in CaWe1992, GiOzVl1994, Ka1995, NkrOz1997, [NkrOz-p]. See NaOz2002 for further discussion.

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NLS on manifolds and obstacles

The NLS has also been studied on non-flat manifolds. For instance, for smooth two-dimensional compact surfaces one has LWP in H¹ [BuGdTz-p3], while for smooth three-dimensional compact surfaces and p=3 one has LWP in H^s for s>1, together with weak solutions in H¹ [BuGdTz-p3]. In the special case of a sphere one has LWP in H^{d/2 + 1/2} for d³3 and p < 5 [BuGdTz-p3].

·For the cubic equation on two-dimensional surfaces one has LWP in H^s for s > ½ [BuGdTz-p3]

oFor s >= 1 one has GWP Vd1984, OgOz1991 and regularity BrzGa1980

oFor s < 0 uniform ill-posedness can be obtained by adapting the argument in BuGdTz2002 or [CtCoTa-p]

oFor the [#Cubic_NLS_on_RxT sphere], [#Cubic_NLS_on_RxT cylinder], or [#Cubic_NLS_on_T^2 torus] more precise results are known

A key tool here is the development of Strichartz estimates on curved space. For general manifolds one has all the L^q_t L^r_x Strichartz estimates (locally in time), but with a loss of 1/q derivatives, see [BuGdTz-p3]. (This though compares favorably to Sobolev embedding, which would require a loss of 2/q derivatives). When the manifold is flat outside of a compact set and obeys a non-trapping condition, the optimal Strichartz estimates (locally in time) were obtained in [StTt-p].
When instead the manifold is decaying outside of a compact set and obeys a non-trapping condition, the Strichartz estimates (locally in time) with an epsilon loss were obtained by Burq [Bu-p3]; in the special case of L^4 estimates on R^3, and for non-trapping asymptotically conic manifolds, the epsilon was removed in [HslTaWun-p]

Outside of a non-trapping obstacle (with Dirichlet boundary conditions), the known results are as follows.

• If (p-1)(d-2) < 2 then one has GWP in H^1 assuming a coercivity condition (e.g. if the nonlinearity is defocusing) [BuGdTz-p4].
  • Note there is a loss compared with the non-obstacle theory, where one expects the condition to be (p-1)(d-2) < 4.
  • The same is true for the endpoint d=3, p=3 if the energy is sufficiently small [BuGdTz-p4].
  • If d <= 4 then the flow map is Lipschitz [BuGdTz-p4]
  • For d=2, p <= 3 this is in BrzGa1980, Vd1984, OgOz1991
• If p < 1 + 2/d then one has GWP in L^2 [BuGdTz-p4]
  • For d=3 GWP for smooth data is in Jor1961
  • Again, in the non-obstacle theory one would expect p < 1 + 4/d
  • if p < 1 + 1/d then one also has strong uniqueness in the class L^2 [BuGdTz-p4]

On a domain in R^d, with Dirichlet boundary conditions, the results are as follows.

• Local well-posedness in H^s for s > d/2 can be obtained by energy methods.
• In two dimensions when p <=3, global well-posedness in the energy class (assuming energy less than the ground state, in the p=3 focusing case) is in BrzGa1980, Vd1984, OgOz1991, references.html Ca1989.More precise asymptotics of a minimal energy blowup solution in the focusing p=3 case are in [BuGdTz-p], [Ban-p3]
• When p > 1 + 4/d blowup can occur in the focusing case Kav1987

GWP and scattering for defocusing NLS on Schwarzchild manifolds for radial data is in LabSf1999

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NLS with potential

(Thanks to Remi Carles for much help with this section. - Ed.)

One can ask what happens to the NLS when a potential is added, thus

i u_t + D u + l|u|^{p-1} u = V u

where V is real and time-independent. The behavior depends on whether V is positive or negative, and how V grows as |x| -> infinity. In the following results we suppose that V grows like some sort of power of x (this can be made precise with estimates on the derivatives of V, etc.) A particularly important case is that of the quadratic potential V = +- |x|^2; this can be used to model a confining magnetic trap for Bose-Einstein condensation. Most of the mathematical research has gone into the isotropic quadatic potentials, but anisotropic ones (given by quadratic forms other than |x|^2) are also of physical interest.

• If V is linear, i.e. V(x) = E.x, then the potential can in fact be eliminated by a change of variables [CarNky-p]
• If V is smooth, positive, and has bounded derivatives up to order 2 (i.e. is quadratic or subquadratic), then the theory is much the same as when there is no potential - one has decay estimates, Strichartz estimates, and the usual local and global well posedness theory (see Fuj1979, Fuj1980, Oh1989)
• When V is exactly a positive quadratic potential V = w^2 |x|^2, then one has blowup for the focusing nonlinearity for negative energy in the L^2 supercritical or critical, H^1 subcritical case Car2002b.
  • In the L^2 critical case one can in fact eliminate this potential by a change of variables Car2002c. One consequence of this change of variables is that one can convert the usual solitary wave solution for NLS into a solution that blows up in finite time (cf. how the pseudoconformal transform is used to achieve a similar effect without the potential).
• When V is exactly a negative quadratic potential, one can prevent blowup even in the focusing case if the potential is sufficiently strong [Car-p]. Indeed, one has a scattering theory in this case [Car-p]
• If V grows faster than quadratic, then there are significant problems due to the failure of smoothness of the fundamental solution; if V is also negative, then even the linear theory fails (for instance, the Hamiltonian need not be essentially self-adjoint on test functions). However for positive superquadratic potentials partial results are still possible YaZgg2001.

Much work has also been done on the semiclassical limit of these equations; see for instance BroJer2000, Ker2002, [CarMil-p], Car2003. For work on standing waves for NLS with quadratic potential, see Fuk2001, Fuk2003, FukOt2003, FukOt2003b.

One component of the theory of NLS with potential is that of Strichartz estimates with potential, which in turn may be derived from dispersive estimates with potential, although it is possible to obtain Strichartz estimates without a dispersive inequality. Both types of estimates can only be expected to hold after first projecting to the absolutely continuous part of the spectrum (although this is not necessary if the potential is positive).

The situation for dispersive estimates (which imply Strichartz), and related estimates such as local L^2 decay, and of L^p boundedness of wave operators (which implies both the dispersive inequality and Strichartz) is as follows. Here we consider potentials which could oscillate; relying mostly on magnitude bounds on V rather than on symbol-type bounds.

• When d=1 one has dispersive estimates whenever <x> V is L^1 [GbScg-p].
  • For potentials such that <x>^{3/2+} V is in L^1, this is essentially in Wed2000; the stronger L^p boundedness of wave operators in this case was established in Wed1999, ArYa2000.
• When d=2, relatively little is known.
  • L^p boundedness of wave operators for potentials decaying like <x>^{-6-}, assuming 0 is not a resonance nor eigenvalue, is in Ya1999, JeYa2002. The method does not quite extend to p=1,infinity and thus does not directly imply the dispersive estimate although it does give Strichartz estimates for 1 < p < infinity.
  • Local L^2 decay and resolvent estimates for low frequencies for polynomially decaying potentials are obtained in JeNc2001
• When d=3 one has dispersive estimates whenever V decays like <x>^{-3-} and 0 is neither a eigenvalue nor resonance [GbScg-p]
  • For potentials which decay like <x>^{-7-} and whose Fourier transform is in L^1, a version of this estimate is in JouSfSo1991
  • A related local L^2 decay estimate was obtained for exponentially decaying potentials in Ra1978; this was refined to polynomially decaying potentials (e.g. <x>^{-3-}) (with additional resolvent estimates at low frequencies) in JeKa1980.
  • L^p boundedness of wave operators was established in Ya1995 for potentials decaying like <x>^{-5-} and for which 0 is neither an eigenvalue nor a resonance.
  • If 0 is a resonance one cannot expect to obtain the optimal decay estimate; at best one can hope for t^{-1/2} (see JeKa1980).
  • Dispersive estimates have also been proven for potentials whose Rollnik and global Kato norms are both smaller than the critical value of 4pi [RoScg-p]. Indeed their arguments partly extend to certain time-dependent potentials (e.g. quasiperiodic potentials), once one also imposes a smallness condition on the L^{3/2} norm of V
  • If the potential is in L^2 and has finite global Kato norm, then one has dispersive estimates for high frequencies at least [RoScg-p].
  • Strichartz estimates have been obtained for potentials decaying like <x>^{-2-} if 0 is neither a zero nor a resonance [RoScg-p]
    1. This has been extended to potentials decaying exactly like |x|^2 and d >= 3 assuming some radial regularity and if the potential is not too negative [BuPlStaTv-p], [BuPlStaTv-p2]; in particular one has Strichartz estimates for potentials V = a/|x|^2, d >= 3, and a > -(n-2)^2/4 (this latter condition is necessary to avoid bound states).
• For d > 3, most of the d=3 results should extend. For instance, the following is known.
  • For potentials which decay like <x>^{-d-4-} and whose Fourier transform is in L^1, dispersive estimates are in JouSfSo1991
  • Local L^2 decay and resolvent estimates for low frequencies for polynomially decaying potentials are obtained in Je1980, Je1984.

For finite rank perturbations of the Laplacian, where each rank one perturbation is generated by a bump function and the bump functions are sufficiently far apart in physical space, decay and Schrodinger estimates were obtained in NieSf2003.The bounds obtained grow polynomially in the number of rank one perturbations.

Local smoothing estimates seem to be more robust than dispersive estimates, holding in a wider range of situations.For instance, in R^d, any potential in L^p for p >= d/2, as well as inverse square potentials 1/|x|^2, and linear combinations of these, have local smoothing RuVe1994.Note one does not need to project away the bound states for such estimates (as the bound states tend to already be rather smooth).However, for p < d/2, one can have breakdown of local smoothing (and also dispersive and Strichartz estimates) [Duy-p]

For time-dependent potentials, very little is known.If the potential is small and quasiperiodic in time (or more generally, has a highly concentrated Fourier transform in time) then dispersive and Strichartz estimates were obtained in [RoScg-p]; the smallness is used to rule out bound states, among other things.In the important case of the charge transfer model (the time-dependent potential that arises in the stability analysis of multisolitons) see Ya1980, Grf1990, Zi1997, [RoScgSf-p], [RoScgSf-p2], where energy, dispersive, and Strichartz estimates are obtained, with application to the asymptotic stability of multisolitons.

The nonlinear interactions between the bound states of a Schrodinger operator with potential (which have no dispersion or decay properties in time) and the absolutely continuous portion of the spectrum (where one expects dispersion and Strichartz estimates) is not well understood.A preliminary result in this direction is in [GusNaTsa-p], which shows in R^3 that if there is only one bound state, and Strichartz estimates hold in the remaining portion of the spectrum, and the non-linearity does not have too high or too low a power (say 4/3 <= p <= 4, or a Hartree-type nonlinearity) then every small H^1 solution will asymptotically decouple into a dispersive part evolving like the linear flow (with potential), plus a nonlinear bound state, with the energy and phase of this bound state eventually stabilizing at a constant.In [SfWs-p] the interaction of a ground state and an excited state is studied, with the generic behavior consisting of collapse to the ground state plus radiation.

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Unique continuation

A question arising by analogy from the theory of unique continuation in elliptic equations, and also in control theory, is the following: if u is a solution to a nonlinear Schrodinger equation, and u(t_0) and u(t_1) is specified on a domain D at two different times t_0, t_1, does this uniquely specify the solution everywhere at all other intermediate times?

• For the 1D cubic NLS, with D equal to a half-line, and u assumed to vanish on D, this is in Zg1997.
• For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in Bo1997b.
• For D the complement of a convex cone, and arbitrary NLS of polynomial growth with a bounded potential term, this is in KnPoVe2003
• For D in a half-plane, and allowing potentials in various Lebesgue spaces, this is in [IonKn-p]
• A local unique continuation theorem (asserting that a non-zero solution cannot vanish on an open set) is in reference:Isk1993 Isk1993

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Derivative non-linear Schrodinger

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

u_t - i D u = f(u, u, Du, 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. We often assume the natural gauge invariance condition

f(exp(i q) u, exp(-i q) u, exp(i q) Du, exp(-i q) Du) = exp(i q) f(u, u, Du, 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 u Du is preferable to u Du (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 L²; 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' ³ 0.

• If m ³ [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 H³ 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 L²_{t,x,loc}.
    • If f is cubic or better one can take m=0KnPoVe1993c.
    • 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 u, then one can do much better. For instance [Gr-p], when f = (Du)^k one can obtain LWP when s > s_c = d/2 + 1 - 1/(k-1), s³1, and k ³ 2 is an integer; similarly when f = D(u^k) one has LWP when s > s_c = d/2 - 1/(k-1), s ³0, and k ³ 2 is an integer. In particular one has GWP in L² when d=1 and f = i(u²)_x and GWP in H¹ when d=1 and f = i({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 u Du + u Du De2002.

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Quasilinear NLS (QNLS)

These are general equations of the form

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

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).

Specific Schrodinger Equations

Monomial semilinear Schrodinger equations are indexed by the degree of the nonlinearity and the spatial domain. A taxonomy of these and other specific Schrodinger equations appears on the specific equations page.

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.

Cubic NLS

The cubic nonlinear Schrodinger equation is of the form

$i\partial _{t}u+\Delta u=\pm |u|^{2}u$

Quartic NLS

A nonlinear Schrodinger equation with nonlinearity of degree 4 is a quartic nonlinear Schrodinger equation.

Quintic NLS

Equations of the form

$i\partial _{t}u+\Delta u=\pm |u|^{4}u$

are quintic nonlinear Schrodinger equations.

Septic NLS

Equations of the form

$i\partial _{t}u+\Delta u=\pm |u|^{6}u$

are septic nonlinear Schrodinger equations.

$L^{2}$-critical NLS

The nonlinear Schrodinger equation

$i\partial _{t}u+\Delta u=\pm |u|^{\frac {4}{d}}u$

posed for $x\in R^{d}$ is scaling invariant in $L_{x}^{2}$. This family of nonlinear Schrodinger equations is therefore called the $L^{2}$ critical nonlinear Schrodinger equation.

Higher order NLS

One can study higher-order NLS equations in which the Laplacian is replaced by a higher power.One class of such examples comes from the infinite hierarchy of commuting flows arising from the completely integrable [Schrodinger:cubic NLS| cubic NLS] on $R$ .Another is the [kdv:Schrodinger_Airy nonlinear Schrodinger-Airy equation].

Schrodinger maps

A geometric Schodinger equation that has been intensively studied is the Schrodinger map equation.

Cubic DNLS on $R$

The Schrodinger:cubic DNLS on R deriviative cubic nonlinear Schrodinger equation has nonlinearity of the form $i\partial _{x}(|u|^{2}u).$

Hartree Equation

The Hartree equation has a nonlocal nonlinearity given by convolution.

Maxwell-Schrodinger system

A Schrodinger-wave system closely related to the [wave:Maxwell-Klein-Gordon|Maxwell-Klein-Gordon equation]] is the Schrodinger:Maxwell-Schrodinger system.

Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms $L_{t}^{q}L_{x}^{r}$ or $L_{x}^{r}L_{t}^{q}$, or in $X^{s,b}$ spaces, defined by

$||u||_{X^{s,b}}=||u||_{s,b}:=||\langle \xi \rangle ^{s}\langle \tau -|\xi |^{2}\rangle ^{b}{\hat {u}}||_{L_{\tau ,\xi }^{2}}.$

Note that these spaces are not invariant under conjugation.

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates. They are useful for NLS without derivatives, but are much less useful for derivative non-linearities. Other linear estimates include smoothing estimates and maximal function estimates. The X^{s,b} spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces and estimates first appear in the context of the Schrodinger equation in [Bo1993], although the analogous spaces for the wave equation appeared earlier [RaRe1982], [Be1983] in the context of propogation of singularities. See also [Bo1993b], [KlMa1993].

See Schrodinger estimates

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