Equations of Korteweg
de Vries type
Overview
The KdV family of equations are
of the form
u_t + u_{xxx} + P(u)_x = 0
where u(x,t) is a
function of one space and one time variable, and P(u) is some polynomial of
u. One can place various normalizing constants in front of the u_{xxx} and P(u) terms, but they can usually be scaled
out. The function u and the polynomial P are usually assumed to be real.
Historically, these types of equations first arose in the study of 2D
shallow wave propagation, but have since appeared as limiting cases of many
dispersive models. Interestingly, the 2D shallow wave equation can also
give rise to the Boussinesq or 1D NLS-3 equation by making more
limiting assumptions (in particular, weak nonlinearity and slowly varying
amplitude).
The x variable is usually assumed to live on the real line R (so there is
some decay at infinity) or on the torus T (so the
data is periodic). The half-line has also been studied, as has the case
of periodic data with large period. It might be interesting to look at
whether the periodicity assumption can be perturbed (e.g. quasi-periodic data);
it is not clear whether the phenomena we see in the periodic problem are robust
under perturbations, or are number-theoretic artefacts
of perfect periodicity.
When P(u) = c u^{k+1}, then the equation is
referred to as generalized gKdV of order k, or gKdV-k. gKdV-1 is the
original Korteweg de Vries
(KdV) equation, gKdV-2 is the modified KdV (mKdV) equation. KdV and mKdV are quite special,
being the only equations in this family which are completely integrable.
If k is even, the sign of c is important. The c < 0 case is known
as the defocussing case, while c > 0 is the focussing case. When k is odd, the constant c can
always be scaled out, so we do not distinguish focussing
and defocussing in this case.
Drift terms u_x can be added, but they can be
subsumed into the polynomial P(u) or eliminated by a Gallilean transformation [except in the half-line case].
Indeed, one can freely insert or remove any term of the form a'(t) u_x by shifting the x variable by a(t),
which is especially useful for periodic higher-order gKdV
equations (setting a'(t) equal to the mean of P(u(t))).
KdV-type equations on R or T always come with
three conserved quantities:
Mass: \int
u dx
L^2: \int u^2 dx
Hamiltonian: \int u_x^2 - V(u)
dx
where V is a primitive of P. Note that the
Hamiltonian is positive-definite in the defocussing
cases (if u is real); thus the defocussing equations
have a better chance of long-term existence. The mass has no
definite sign and so is only useful in specific cases (e.g. perturbations of a soliton).
In general, the above three quantities are the only conserved quantities
available, but the KdV and mKdV equations come with infinitely
many more such conserved quantities due to their completely integrable
nature.
The critical (or scaling) regularity is
s_c = 1/2 -
2/k.
In particular, KdV, mKdV, and gKdV-3 are subcritical with respect to L^2, gKdV-4 is L^2 critical,
and all the other equations are L^2 supercritical. Generally speaking,
the potential energy term V(u) can be pretty much ignored in the sub-critical
equations, needs to be dealt with carefully in the critical equation, and can
completely dominate the Hamiltonian in the super-critical equations (to the
point that blowup occurs if the equation is not defocussing).
Note that H^1 is always a sub-critical regularity.
The dispersion relation \tau = \xi^3 is always
increasing, which means that singularities always propagate to the left.
In fact, high frequencies propagate leftward at extremely high speeds, which
causes a smoothing effect if there is some decay in the initial data (L^2 will
do). On the other hand, KdV-type equations have
the remarkable property of supporting localized travelling
wave solutions known as solitons, which propagate to
the right. It is known that solutions to the completely integrable equations (i.e. KdV
and mKdV) always resolve to a superposition of solitons as t -> infinity, but it is an interesting open
question as to whether the same phenomenon occurs for the other KdV-type equations.
A KdV-type equation can be viewed as a symplectic flow with the Hamiltonian defined above, and the
symplectic form given by
{u, v} := \int u v_x dx.
Thus H^{-1/2} is the natural Hilbert space in which
to study the symplectic geometry of these
flows. Unfortunately, the gKdV-k equations are
only locally well-posed in H^{-1/2} when k=1.
Airy
estimates
Solutions to the Airy equation and its perturbations are either estimated in
mixed space-time norms L^q_t L^r_x,
L^r_x L^q_t, or in X^{s,b} spaces, defined by
||
u ||_{s,b} = ||
<xi>^s <tau-xi^3>^b \hat{u} ||_2.
Linear space-time estimates in which the space norm is evaluated first are
known as Strichartz estimates,
but these estimates only play a minor role in the theory. A more
important category of linear estimates are the 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 [Bo1993b], although the analogues spaces for
the wave equation appeared earlier [RaRe1982],
[Be1983] in the context of propogation of singularities. See also [Bo1993], [KlMa1993].
Linear
Airy estimates
- If u is in X^{0,1/2+} on R,
then
- u is in L^\infty_t L^2_x (energy estimate)
- D_x^{1/4}
u is in L^4_t BMO_x (endpoint Strichartz) [KnPoVe1993]
- D_x
u is in L^\infty_x L^2_t (sharp Kato smoothing
effect) [KnPoVe1993].
Earlier versions of this estimate were obtained in [Ka1979b], [KrFa1983].
- D_x^{-1/4}
u is in L^4_x L^\infty_t (Maximal function) [KnPoVe1993], [KnRu1983]
- D_x^{-3/4-}
u is in L^2_x L^\infty_t (L^2 maximal function)
[KnPoVe1993]
- Remark: Further
estimates are available by Sobolev,
differentiation, Holder, and interpolation. For instance:
- D_x
u is in L^2_{x,t} locally in space [Ka1979b] - use Kato and Holder (can
also be proven directly by integration by parts)
- u is in L^2_{x,t} locally in time - use energy and Holder
- D_x^{3/4-}
u is in L^8_x L^2_t locally in time - interpolate previous with Kato
- D_x^{1/6}
u is in L^6_{x,t} - interpolate energy with
endpoint Strichartz (or Kato with maximal)
- D_x^{1/8} u is in L^8_t L^4_x - interpolate energy with
endpoint Strichartz. (In particular, D_x^{1/8} u is also in
L^4_{x,t}).
- u is in L^8_{x,t}- use previous and Sobolev
in space
- If u is in
X^{0,1/3+}, then u is in L^4_{x,t} [Bo1993b] - interpolate previous with
the trivial identity X^{0,0} = L^2
- If u is in
X^{0,1/4+}, then D_x^{1/2} u is in L^4_x L^2_t
[Bo1993b] - interpolate Kato with
X^{0,0} = L^2
- If u is in X^{0,1/2+} on T,
then
- u
is in L^\infty_t L^2_x (energy estimate).
This is also true in the large period case.
- u
is in L^4_{x,t} locally in time (in fact one
only needs u in X^{0,1/3} for this) [Bo1993b].
- D_x^{-\eps} u is in L^6_{x,t} locally in time. [Bo1993b]. It is conjectured that
this can be improved to L^8_{x,t}.
- Remark: there
is no smoothing on the circle, so one can never gain regularity.
- If u is in X^{0,1/2} on a
circle with large period \lambda, then
- u
is in L^4_{x,t} locally in time, with a bound
of \lambda^{0+}.
- In fact, when u has frequency
N, the constant is like \lambda^{0+} (N^{-1/8}
+ \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line
in L^4 mentioned earlier. [CoKeStTkTa-p2]
Bilinear
Airy estimates
- The key algebraic fact is
\xi_1^3 + \xi_2^3 + \xi_3^3 = 3 \xi_1 \xi_2 \xi_3
(whenever \xi_1 + \xi_2 + \xi_3 = 0)
|| (uv)_x
||_{-3/4+, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}
- The above estimate
fails at the endpoint -3/4. [NaTkTs2001]
- As a corollary of
this estimate we have the -3/8+ estimate [CoStTk1999] on R: If u and v
have no low frequencies ( |\xi| <~ 1 ) then
|| (uv)_x
||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-3/8+, 1/2+}
- The -1/2 estimate [KnPoVe1996] on T: if u,v have mean zero, then for all s >= -1/2
|| (uv)_x
||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}
- The above estimate
fails for s < -1/2. Also, one cannot replace 1/2, -1/2 by 1/2+,
-1/2+. [KnPoVe1996]
- This estimate also
holds in the large period case if one is willing to lose a power of
\lambda^{0+} in the constant. [CoKeStTkTa-p2]
- Remark: In principle,
a complete list of bilinear estimates could be obtained from [Ta-p2].
Trilinear Airy
estimates
- The key algebraic fact is
(various permutations of)
\xi_1^3 + \xi_2^3 + \xi_3^3 + \xi_4^3 = 3
(\xi_1+\xi_4) (\xi_2+\xi_4) (\xi_3+\xi_4)
(whenever \xi_1 + \xi_2 + \xi_3 + \xi_4 = 0)
- The 1/4 estimate [Ta-p2] on R:
|| (uvw)_x
||_{1/4, -1/2+} <~ || u ||_{1/4, 1/2+} || v ||_{1/4, 1/2+} || w
||_{1/4, 1/2+}
The 1/4 is sharp [KnPoVe1996]. We also have
|| uvw
||_{-1/4, -5/12+} <~ || u ||_{-1/4, 7/12+} || v ||_{-1/4, 7/12+} || w
||_{-1/4, 7/12+}
see [Cv-p].
- The 1/2 estimate [CoKeStTkTa-p3] on T: if u,v,w have mean zero, then
|| (uvw)_x
||_{1/2, -1/2} <~ || u ||_{1/2, 1/2*} || v ||_{1/2, 1/2*} || w
||_{1/2, 1/2*}
The 1/2 is sharp [KnPoVe1996].
- Remark: the trilinear estimate always needs one more derivative of
regularity than the bilinear estimate; this is consistent with the heuristics
from the Miura transform from mKdV to KdV.
Multilinear Airy
estimates
\int u^3 v^2 dx dt <~ || u ||_{1/4+,1/2+}^3 || v ||_{-3/4+,1/2+}^2
\int u^3 v^2 dx dt <~ || u ||_{1/2,1/2*}^3 || v ||_{-1/2,1/2*}^2
In fact, this estimate also holds for large period, but
a loss of lambda^{0+}.
The
KdV equation
The KdV equation is
u_t + u_xxx + u u_x = 0.
It is completely integrable, and has infinitely
many conserved quantities. Indeed, for each non-negative integer k, there
is a conserved quantity which is roughly equivalent to the H^k
norm of u.
The KdV equation has been studied on the line, the circle, and the half-line.
KdV on R
- Scaling is s_c = -3/2.
- LWP in H^s
for s >= -3/4 [CtCoTa-p], using a modified
Miura transform and the mKdV
theory. This is despite the failure of the key bilinear estimate
[NaTkTs2001]
- For s within a
logarithm for s=-3/4 [MurTao-p].
- Was proven for s
> -3/4 [KnPoVe1996].
- Was proven for s
> -5/8 in [KnPoVe1993b].
- Was proven for s
>= 0 in [Bo1993b].
- Was proven for s
> 3/4 in [KnPoVe1993].
- Was proven for s
> 3/2 in [BnSm1975], [Ka1975], [Ka1979], [GiTs1989], [Bu1980], ....
- One has local ill-posedness(in
the sense that the map is not uniformly continuous) for s < -3/4 (in
the complex setting) by soliton examples [KnPoVe-p].
- When the initial
data is a real, rapidly decreasing measure one has a global smooth
solution for t > 0 [Kp1993].
Without the rapidly decreasing hypothesis one can still construct a
global weak solution [Ts1989]
- GWP in H^s
for s > -3/4 (if u is real) [CoKeStTkTa2003].
- Was proven for s
> -3/10 in [CoKeStTkTa2001]
- Was proven for
s>= 0 in [Bo1993b]. Global
weak solutions in L^2 were constructed in [Ka1983], [KrFa1983], and were shown to obey the
expected local smoothing estimate. These weak solutions were shown
to be unique in [Zh1997b]
- Was proven for
s>= 1 in [KnPoVe1993].
- Was proven for
s>= 2 in [BnSm1975], [Ka1975], [Ka1979], ....
- Remark: In
the complex setting GWP fails for large data with Fourier support on the
half-line [Bona/Winther?], [Birnir], ????. This result extends to a wide class of
dispersive PDE.
- By use of the inverse
scattering transform one can show that smooth solutions eventually resolve
into solitons, that two colliding solitons emerge as (slightly phase shifted) solitons, etc.
- Solitons
are orbitally H^1 stable [Bj1972]
- In H^s, 0 <= s < 1, the orbital stability of solitons is at most polynomial (the distance to the
ground state manifold in H^s norm grows like at
most O(t^{1-s+}) in time) [RaySt-p]
- In L^2, orbital
stability has been obtained in [MeVe2003].
The KdV equation can also be
generalized to a 2x2 system
u_t
+ u_xxx + a_3 v_xxx + u u_x + a_1 v v_x + a_2 (uv)_x = 0
b_1 v_t + v_xxx + b_2 a_3 u_xxx + v v_x + b_2 a_2 u u_x + b_2 a_1 (uv)_x + r v_x
where b_1,b_2 are positive
constants and a_1,a_2,a_3,r are real constants. This system was
introduced in [GeaGr1984] to study
strongly interacting pairs of weakly nonlinear long waves, and studied further
in [BnPoSauTm1992]. In [AsCoeWgg1996] it was shown that this
system was also globally well-posed on L^2.
It is an interesting question as to whether these results can be pushed further
to match the KdV theory; the apparent lack of
complete integrability in this system (for generic
choices of parameters b_i, a_i,
r) suggests a possible difficulty.
KdV on R^+
- The KdV
Cauchy-boundary problem on the half-line is
u_t
+ u_{xxx} + u_x + u u_x = 0; u(x,0) = u_0(x); u(0,t) = h(t)
The sign of u_{xxx}
is important (it makes the influence of the boundary x=0 mostly negligible),
the sign of u u_x is not. The drift term u_x appears naturally from the derivation of KdV from fluid mechanics. (On R, this drift term can
be eliminated by a Gallilean transform, but this is
not available on the half-line).
- Because one is restricted to
the half-line, it becomes a little tricky to use the Fourier
transform. One approach is to use the Fourier-Laplace
transform instead.
- Some compatibility conditions
between u_0 and h are needed. The higher the regularity, the more
compatibility conditions are needed. If the initial data u_0 is in H^s, then by scaling heuristics the natural space for
h is in H^{(s+1)/3}. (Remember that time
has dimensions length^3).
- LWP is known for initial
data in H^s and boundary data in H^{(s+1)/3} for s >= 0 [CoKe-p],
assuming compatibility. The drift term may be omitted because of the
time localization.
- For s > 3/4 this
was proven in [BnSuZh-p]
(assuming that there is a drift term).
- Was proven for data
in sufficiently weighted H^1 spaces in [Fa1983].
- From the real line
theory one might expect to lower this to -3/4, but there appear to be
technical difficulties with this.
- GWP is known for initial
data in L^2 and boundary data in H^{7/12},
assuming compatibility.
- for initial data in
H^1 and boundary data in H^{5/6}_loc this was proven in [BnSuZh-p]
- Was proven for smooth
data in [BnWi1983]
KdV on T
- Scaling is s_c = -3/2.
- C^0 LWP in H^s for s >= -1, assuming u is real [KpTp-p]
- C^0 LWP in H^s for s >= -5/8 follows (at least in principle)
from work on the mKdV equation by [Takaoka and Tsutsumi?]
- Analytic LWP in H^s for s >= -1/2, in the complex case [KnPoVe1996]. In addition to
the usual bilinear estimate, one needs a linear estimate to keep the
solution in H^s for t>0.
- Analytic LWP was
proven for s >= 0 in [Bo1993b].
- Analytic ill posedness at s<-1/2, even in the real case [Bo1997]
- This has been
refined to failure of uniform continuity at s<-1/2 [CtCoTa-p]
- Remark: s=-1/2 is the
symplectic regularity, and so the machinery of
infinite-dimensional symplectic geometry
applies once one has a continuous flow, although there are some
technicalities involving approximating KdV by a
suitable symplectic finite-dimensional
flow. In particular one has symplectic
non-squeezing [CoKeStTkTa-p9], [Bo1999].
- C^0 GWP in H^s for s >= -1, in the real case [KpTp-p].
- Analytic GWP in H^s in the real case for s >= -1/2 [CoKeStTkTa-p2]; see also [CoKeStTkTa-p3].
- A short proof for the
s > -3/10 case is in [CoKeStTkTa-p2a]
- Was proven for s
>= 0 in [Bo1993b].
- GWP for real initial
data which are measures of small norm [Bo1997]
The small norm restriction is presumably
technical.
- Remark:
measures have the same scaling as H^{-1/2}, but
neither space includes the other. (Measures are in H^{-1/2-\eps} though).
- One has GWP for real
random data whose Fourier coefficients decay like 1/|k| (times a Gaussian
random variable) [Bo1995c].
Indeed one has an invariant measure.
- Solitons
are asymptotically H^1 stable [MtMe-p3], [MtMe-p]. Indeed, the solution decouples into a
finite sum of solitons plus dispersive
radiation [EckShr1988]
The
modified KdV equation
The (defocussing) mKdV
equation is
u_t + u_xxx = 6 u^2 u_x.
It is completely integrable, and has infinitely
many conserved quantities. Indeed, for each non-negative integer k, there
is a conserved quantity which is roughly equivalent to the H^k
norm of u. This equation has been studied on the line,
circle, and half-line.
The Miura transformation v = u_x + u^2
transforms a solution of defocussing mKdV to a solution of KdV
v_t + v_xxx = 6 v v_x.
Thus one expects the LWP and GWP theory for mKdV
to be one derivative higher than that for KdV.
The focussing mKdV
u_t
+ u_xxx = - 6 u^2 u_x
is very similar, except that the Miura transform is
now v = u_x + i u^2.
This transforms focussing mKdV
to complex-valued KdV, which is a slightly
less tractable equation. (However, the transformed solution v is still
real in the highest order term, so in principle the real-valued theory carries
over to this case).
The Miura transformation can be generalized. If v and w solve the
system
v_t
+ v_xxx = 6(v^2 + w) v_x
w_t + w_xxx = 6(v^2 + w) w_x
Then u = v^2 + v_x + w is a
solution of KdV. In particular, if a and b are
constants and v solves
v_t
+ v_xxx = 6(a^2 v^2 + bv) v_x
then u = a^2 v^2 + av_x + bv solves KdV (this is the Gardener transform).
mKdV on R and R^+
- Scaling is s_c = -1/2.
- LWP in H^s
for s >= 1/4 [KnPoVe1993]
- Was shown for
s>3/2 in [GiTs1989]
- This is sharp in the
focussing case [KnPoVe-p],
in the sense that the solution map is no longer uniformly continuous for
s < 1/4.
- This has been
extended to the defocussing case in [CtCoTa-p], by a high-frequency approximation of mKdV by NLS.
(This high frequency approximation has also been utilized in [Sch1998]).
- Below 1/4 the
solution map was known to not be C^3 in [Bo1993b], [Bo1997].
- The same result has
also been established for the half-line [CoKe-p],
assuming boundary data is in H^{(s+1)/3} of
course.
- Global weak
solutions in L^2 were constructed in [Ka1983].
Thus in L^2 one has global existence but no uniform continuity.
- Uniqueness is also
known when the initial data lies in the weighted space <x>^{3/8}
u_0 in L^2 [GiTs1989]
- LWP has also been
demonstrated when <xi>^s hat(u_0) lies in L^{r’} for 4/3 <
r <= 2 and s >= ½ - 1/2r [Gr-p4]
- GWP in H^s
for s > 1/4 [CoKeStTkTa-p2],
via the KdV theory and the Miura transform, for
both the focussing and defocussing
cases.
- Was proven for
s>3/5 in [FoLiPo1999]
- Is implicit for s
>= 1 from [KnPoVe1993]
- On the half-line GWP
is known when s >= 1 and the boundary data is in H^{11/12}, assuming
compatibility and small L^2 norm [CoKe-p]
- GWP for smooth data
can also be achieved from inverse scattering methods [BdmFsShp-p];
the same approach also works on an interval [BdmShp-p].
- Solitions
are asymptotically H^1 stable [MtMe-p3], [MtMe-p]
mKdV on T
- Scaling is s_c = -1/2.
- C^0 LWP in L^2 in the
defocusing case [KpTp-p2]
- C^0 LWP in H^s for s > 3/8 [Takaoka and Tsutsumi?] Note one has to gauge away a nonlinear
resonance term before one can apply iteration methods.
- Analytic LWP in H^s for s >= 1/2, in both focusing and defocusing
cases [KnPoVe1993], [Bo1993b].
- This is sharp in the
sense of analytic well-posedness [KnPoVe1996] or uniform well-posedness [CtCoTa-p]
- C^0 GWP in L^2 in the
defocusing case [KpTp-p2]
- Analytic GWP in H^s for s >= 1/2 [CoKeStTkTa-p2], via the KdV theory and the Miura transform, for both the focussing and defocussing
cases.
- Was proven for s
>= 1 in [KnPoVe1993], [Bo1993b].
- One has GWP for
random data whose Fourier coefficients decay like 1/|k| (times a Gaussian
random variable) [Bo1995c].
Indeed one has an invariant measure. Note that such data barely
fails to be in H^{1/2}, however one can modify
the local well-posedness theory to go below
H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the
Fourier coefficients (which is indeed the case almost surely).
gKdV_3 on R and R^+
- Scaling is s_c = -1/6.
- LWP in H^s
for s > -1/6 [Gr-p3]
- Was shown for
s>=1/12 [KnPoVe1993]
- Was shown for
s>3/2 in [GiTs1989]
- The result s >=
1/12 has also been established for the half-line [CoKe-p],
assuming boundary data is in H^{(s+1)/3} of
course..
- GWP in H^s
for s >= 0 [Gr-p3]
- For s>=1 this is
in [KnPoVe1993]
- Presumably one can
use either the Fourier truncation method or the "I-method" to
go below L^2. Even though the equation is not completely integrable, the one-dimensional nature of the
equation suggests that "correction term" techniques will also
be quite effective.
- On the half-line GWP
is known when s >= 1 and the boundary data is in H^{5/4}, assuming
compatibility and small L^2 norm [CoKe-p]
- Solitons
are H^1-stable [CaLo1982], [Ws1986], [BnSouSr1987] and asymptotically H^1
stable [MtMe-p3], [MtMe-p]
gKdV_3 on T
- Scaling is s_c = -1/6.
- LWP in H^s
for s>=1/2 [CoKeStTkTa-p3]
- GWP in H^s
for s>5/6 [CoKeStTkTa-p3]
- Was shown for s
>= 1 in [St1997c]
- This result may well
be improvable by the "damping correction term" method in [CoKeStTkTa-p2].
- Remark: For this equation
it is convenient to make a "gauge transformation'' to subtract off
the mean of P(u).
gKdV_4 on R and R^+
(Thanks to Felipe Linares for help
with the references here - Ed.) A good
survey for the results here is in [Tz-p2].
- Scaling is s_c = 0 (i.e. L^2-critical).
- LWP in H^s
for s >= 0 [KnPoVe1993]
- Was shown for
s>3/2 in [GiTs1989]
- The same result s
>= 0 has also been established for the half-line [CoKe-p],
assuming boundary data is in H^{(s+1)/3} of
course..
- GWP in H^s
for s > 3/4 in both the focusing and defocusing cases, though one must
of course have smaller L^2 mass than the ground state in the focusing case
[FoLiPo-p].
- For s >= 1 and
the defocusing case this is in [KnPoVe1993]
- Blowup has recently
been shown for the focussing case for data
close to a ground state with negative energy [Me-p]. In such
a case the blowup profile must approach the ground state (modulo scalings and translations), see [MtMe-p4], [MtMe2001]. Also, the blow up
rate in H^1 must be strictly faster than t^{-1/3}
[MtMe-p4], which is the rate suggested by scaling.
- Explicit
self-similar blow-up solutions have been constructed [BnWe-p]
but these are not in L^2.
- GWP for small L^2
data in either case [KnPoVe1993].
In the focussing case we have GWP whenever the
L^2 norm is strictly smaller than that of the ground state Q (thanks to
Weinstein's sharp Gagliardo-Nirenberg
inequality). It seems like a reasonable (but difficult) conjecture
to have GWP for large L^2 data in the defocusing case.
- On the half-line GWP
is known when s >= 1 and the boundary data is in H^{11/12}, assuming
compatibility and small L^2 norm [CoKe-p]
- Solitons
are H^1-unstable [MtMe2001].
However, small H^1 perturbations of a soliton
must asymptotically converge weakly to some rescaled soliton
shape provided that the H^1 norm stays comparable to 1 [MtMe-p].
gKdV_4 on T
- Scaling is s_c = 0.
- LWP in H^s
for s>=1/2 [CoKeStTkTa-p3]
- Was shown for s
>= 1 in [St1997c]
- Analytic well-posedness fails for s < 1/2; this is essentially
in [KnPoVe1996]
- GWP in H^s
for s>=1 [St1997c]
- This is almost
certainly improvable by the techniques in [CoKeStTkTa-p3], probably to s
> 6/7. There are some low-frequency issues which may require the
techniques in [KeTa-p].
- Remark: For this
equation it is convenient to make a "gauge transformation'' to
subtract off the mean of P(u).
gKdV on R^+
- The gKdV
Cauchy-boundary problem on the half-line is
u_t
+ u_{xxx} + u_x + u^k u_x = 0; u(x,0) =
u_0(x); u(0,t) = h(t)
The sign of u_{xxx}
is important (it makes the influence of the boundary x=0 mostly negligible),
the sign of u u_x is not. The drift term u_x is convenient for technical reasons; it is not known
whether it is truly necessary.
- LWP is known for initial
data in H^s and boundary data in H^{(s+1)/3} when s > 3/4 [CoKn-p].
- The techniques are
based on [KnPoVe1993] and a
replacement of the IVBP with a forced IVP.
- This has been
improved to s >= s_c = 1/2 - 2/k when k >
4 [CoKe-p].
- For KdV, mKdV, gKdV-3 , and gKdV-4 see the corresponding sections on this
page.
Nonlinear
Schrodinger-Airy equation
The equation
u_t + i c u_xx + u_xxx
= i gamma |u|^2 u + delta |u|^2 u_x
+ epsilon u^2 u_x
on R is a combination of the cubic NLS equation , the derivative cubic NLS equation, complex mKdV, and a cubic
nonlinear Airy equation. This equation
is a general model for propogation of pulses in an
optical fiber [Kod1985], [HasKod1987]
·
When c=delta=epsilon = 0, scaling is s=-1. When c=gamma=0, scaling is –1/2.
·
LWP is known when s >= ¼ [St1997d]
o For
s > ¾ this is in [Lau1997], [Lau2001]
o The
s>=1/4 result is also known when c is a time-dependent function [Cv2002],
[CvLi2003]
o For
s < -1/4 and delta or epsilon non-zero, the solution map is not C^3 [CvLi-p]
o When
delta = epsilon = 0 LWP is known for s > -1/4 [Cv2004]
§
For s < -1/4 the solution map is not C^3 [CvLi-p]
Miscellaneous
gKdV results
[Thanks to Nikolaos Tzirakis for some corrections - Ed.]
- On R with k > 4, gKdV-k is LWP down to scaling: s >= s_c = 1/2 - 2/k [KnPoVe1993]
- Was shown for
s>3/2 in [GiTs1989]
- One has ill-posedness in the supercritical regime [BirKnPoSvVe1996]
- For small data one
has scattering [KnPoVe1993c]. Note that one cannot have scattering in
L^2 except in the critical case k=4 because one can scale solitons to be arbitrarily small in the non-critical
cases.
- Solitons
are H^1-unstable [BnSouSr1987]
- If one considers an
arbitrary smooth non-linearity (not necessarily a power) then one has LWP
for small data in H^s, s > 1/2 [St1995]
- On R with any k, gKdV-k is GWP in H^s for s
>= 1 [KnPoVe1993], though for
k >= 4 one needs the L^2 norm to be small; global weak solutions were
constructed much earlier, with the same smallness assumption when k >=
4. This should be improvable below H^1 for all k.
- On R with any k, gKdV-k has the H^s norm
growing like t^{(s-1)+} in time for any integer s >= 1 [St1997b]
- On R with any non-linearity,
a non-zero solution to gKdV cannot be supported
on the half-line R^+ (or R^-) for two different times [KnPoVe-p3], [KnPoVe-p4].
- In the completely integrable cases k=1,2 this is in [Zg1992]
- Also, a non-zero
solution to gKdV cannot vanish on a rectangle
in spacetime [SauSc1987]; see also [Bo1997b].
- Extensions to higher
order gKdV type equations are in [Bo1997b], [KnPoVe-p5].
- On R with non-integer k, one
has decay of O(t^{-1/3}) in L^\infty for small decaying data if k > (19 - sqrt(57))/4 ~ 2.8625... [CtWs1991]
- A similar result for
k > (5+sqrt(73))/4 ~ 3.39... was obtained in [PoVe1990].
- When k=2 solutions
decay like O(t^{-1/3}), and when k=1 solutions decay generically like
O(t^{-2/3}) but like O( (t/log t)^{-2/3}) for exceptional data [AbSe1977]
- In the L^2 subcritical case 0 < k < 4, multisoliton
solutions are asymptotically H^1-stable [MtMeTsa-p]
- A dissipative version of gKdV-k was analyzed in [MlRi2001]
- On T with any k, gKdV-k has the H^s norm
growing like t^{2(s-1)+} in time for any integer s >= 1 [St1997b]
- On T with k >= 3, gKdV-k is LWP for s >= 1/2 [CoKeStTkTa-p3]
- Was shown for s
>= 1 in [St1997c]
- Analytic well-posedness fails for s < 1/2 [CoKeStTkTa-p3], [KnPoVe1996]
- For arbitrary smooth
non-linearities, weak H^1 solutions were
constructed in [Bo1993].
- On T with k >= 3, gKdV-k is GWP for s >= 1 except in the focussing case [St1997c]
- The estimates in [CoKeStTkTa-p3] suggest that this
is improvable to 13/14 - 2/7k, but this has only been proven in the
sub-critical case k=3 [CoKeStTkTa-p3].
In the critical and super-critical cases there are some low-frequency
issues which may require the techniques in [KeTa-p].
The
KdV Hierarchy
The KdV equation
V_t + V_xxx = 6 V_x
can be rewritten in the Lax Pair form
L_t = [L,
P]
where L is the second-order operator
L = -D^2 + V
(D = d/dx) and P is the third-order antiselfadjoint operator
P = 4D^3 + 3(DV + VD).
(note that P consists of the zeroth
order and higher terms of the formal power series expansion of 4i L^{3/2}).
One can replace P with other fractional powers of L. For instance, the
zeroth order and higher terms of 4i L^{5/2} are
P = 4D^5 + 5(D^3 V + V D^3) - 5/4 (D V_xx + V_xx D) + 15/4 (D V^2 +
V^2 D)
and the Lax pair equation becomes
V_t
+ u_xxxxx = (5 V_x^2 + 10 V V_xx
+ 10 V^3)_x
with Hamiltonian
H(V)
= \int V_xx^2 - 5 V^2 V_xx
- 5 V^4.
These flows all commute with each other, and their
Hamiltonians are conserved by all the flows simultaneously.
The KdV hierarchy are
examples of higher order water wave models; a general formulation is
u_t + partial_x^{2j+1} u = P(u, u_x, ..., partial_x^{2j} u)
where u is real-valued and P is a polynomial with
no constant or linear terms; thus KdV and gKdV correspond to j=1, and the higher order equations in
the hierarchy correspond to j=2,3,etc.
LWP for these equations in high regularity Sobolev
spaces is in [KnPoVe1994], and
independently by Cai (ref?);
see also [CrKpSr1992]. The case j=2 was studied by Choi (ref?). The
non-scalar diagonal case was treated in [KnSt1997];
the periodic case was studied in [Bo-p3].
Note in the periodic case it is possible to have ill-posedness
for every regularity, for instance u_t + u_xxx = u^2 u_x^2 is ill-posed in every H^s
[Bo-p3]
Benjamin-Ono
equation
[Thanks to Nikolay Tzvetkov and Felipe Linares for
help with this section - Ed]
The generalized Benjamin-Ono equation BO_a
is the scalar equation
u_t
+ D_x^{1+a} u_x + uu_x = 0
where D_x
= sqrt{-Delta} is the positive differentiation
operator. When a=1 this is KdV;
when a=0 this is the Benjamin-Ono equation (BO) [Bj1967], [On1975],
which models one-dimensional internal waves in deep water. Both of these
equations are completely integrable (see e.g. [AbFs1983], [CoiWic1990]), though the intermediate
cases 0 < a < 1 are not.
When a=0, scaling is s = -1/2, and the following results are known:
- LWP in H^s
for s >= 1 [Ta-p]
- For s >= 9/8 this
is in [KnKoe-p]
- For s >= 5/4 this
is in [KocTz-p]
- For s >= 3/2 this
is in [Po1991]
- For s > 3/2 this
is in [Io1986]
- For s > 3 this is
in [Sau1979]
- For no value of s is
the solution map uniformly continuous [KocTz-p2]
- For s < -1/2
this is in [BiLi-p]
- Global weak solutions exist
for L^2 data [Sau1979], [GiVl1989b], [GiVl1991], [Tom1990]
- Global well-posedness in H^s for s >=
1 [Ta-p]
- For s >= 3/2 this
is in [Po1991]
- For smooth solutions
this is in [Sau1979]
When 0 < a < 1, scaling is s = -1/2 - a, and the following results are known:
- LWP in H^s
is known for s > 9/8 – 3a/8 [KnKoe-p]
- GWP is known when s >=
(a+1)/2 when a > 4/5, from the conservation of the Hamiltonian [KnPoVe1994b]
- The LWP results are obtained
by energy methods; it is known that pure iteration methods cannot work [MlSauTz2001]
- However, this can be
salvaged by combining the H^s norm || f ||_{H^s} with a weighted Sobolev
space, namely || xf ||_{H^{s - 2s_*}}, where
s_* = (a+1)/2 is the energy regularity.[CoKnSt-p4]
One can replace the quadratic non-linearity uu_x by higher powers u^{k-1} u_x, in analogy with KdV and gKdV, giving rise to the gBO-k
equations (let us take a=0 for sake of discussion). The scaling exponent is 1/2 - 1/(k-1).
- For k=3, one has GWP for
large data in H^1 [KnKoe-p] and LWP for small
data in H^s, s > ½ [MlRi-p]
- For small data in H^s, s>1, LWP was obtained in [KnPoVe1994b]
- With the addition of
a small viscosity term, GWP can also be obtained in H^1 by complete integrability methods in [FsLu2000], with asymptotics under the additional assumption that the
initial data is in L^1.
- For s < ½, the
solution map is not C^3 [MlRi-p]
- For k=4, LWP for small data
in H^s, s > 5/6 was obtained in [KnPoVe1994b].
- For k>4, LWP for small
data in H^s, s >=3/4 was obtained in [KnPoVe1994b].
- For any k >= 3 and s
< 1/2 - 1/k the solution map is not uniformly continuous [BiLi-p]
The
KdV-Benjamin Ono (KdV-BO)
equation is formed by combining the linear parts of the KdV
and Benjamin-Ono equations together. It
is globally well-posed in L^2 [Li1999],
and locally well-posed in H^{-3/4+} [KozOgTns] (see also [HuoGuo-p]
where H^{-1/8+} is obtained). Similarly
one can generalize the non-linearity to be k-linear, generating for instance
the modified KdV-BO equation, which is locally
well-posed in H^{1/4+} [HuoGuo-p]. For general gKdV-gBO
equations one has local well-posedness in H^3 and above [GuoTan1992]. One can also add damping terms Hu_x to the equation; this arises as a model for ion-acoustic
waves of finite amplitude with linear Landau damping [OttSud1982].