KdV-type equations

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Equations of Korteweg de Vries type

Overview


The KdV family of equations are of the form

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 [schrodinger.html#Cubic NLS on R 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:

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 KdV] and [#mkdv mKdV] equations come with infinitely many more such conserved quantities due to their completely integrable nature.

The critical (or scaling) regularity is

In particular, [#kdv KdV], [#mkdv 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

.

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

Linear space-time estimates in which the space norm is evaluated first are known as [#kdv_linear 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 [#kdv_bilinear bilinear estimates], although more recently [#KdV_multilinear multilinear estimates have begun to appear]. These spaces and estimates first appear in the context of the Schrodinger equation in references.html#Bo1993b Bo1993b, although the analogues spaces for the wave equation appeared earlier references.html#RaRe1982 RaRe1982, references.html#Be1983 Be1983 in the context of propogation of singularities. See also references.html#Bo1993 Bo1993, references.html#KlMa1993 KlMa1993.

Linear Airy estimates
  • If on R, then
  • 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) references.html#Bo1993b Bo1993b.
    • D_x^{-\eps} u is in L^6_{x,t} locally in time. references.html#Bo1993b 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. references.html#CoKeStTaTk-p2 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+}

|| (uv)_x ||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-3/8+, 1/2+}
|| (uv)_x ||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}
  • Remark: In principle, a complete list of bilinear estimates could be obtained from [[references.html#Ta-p2 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)
|| (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 references.html#KnPoVe1996 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].

|| (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 references.html#KnPoVe1996 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
  • We have the quintilinear estimate on R: [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]
\int u^3 v^2 dx dt <~ || u ||_{1/4+,1/2+}^3 || v ||_{-3/4+,1/2+}^2
  • The analogue for this on T is: [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2], [references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]
\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 [#kdv_on_R line], the [#kdv_on_T circle], and the [#KdV_on_R+ half-line].

KdV on R

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 references.html#GeaGr1984 GeaGr1984 to study strongly interacting pairs of weakly nonlinear long waves, and studied further in references.html#BnPoSauTm1992 BnPoSauTm1992. In references.html#AsCoeWgg1996 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 [[references.html#BnSuZh-p BnSuZh-p]] (assuming that there is a drift term).
    • Was proven for data in sufficiently weighted H^1 spaces in references.html#Fa1983 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 [[references.html#BnSuZh-p BnSuZh-p]]
    • Was proven for smooth data in references.html#BnWi1983 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 references.html#KnPoVe1996 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 references.html#Bo1993b Bo1993b.
    • Analytic ill posedness at s<-1/2, even in the real case references.html#Bo1997 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], references.html#Bo1999 Bo1999.
  • C^0 GWP in H^s for s >= -1, in the real case [KpTp-p].
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 [#mKdV_on_R line], [#mKdV_on_T circle], and [#gKdV_on_R+ half-line].

The Miura transformation v = u_x + u^2 transforms a solution of defocussing mKdV to a solution of [#kdv 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 references.html#KnPoVe1993 KnPoVe1993
    • Was shown for s>3/2 in references.html#GiTs1989 GiTs1989
    • This is sharp in the focussing case [[references.html#KnPoVe-p KnPoVe-p]], in the sense that the solution map is no longer uniformly continuous for s < 1/4.
    • 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 references.html#Ka1983 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 references.html#GiTs1989 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 references.html#CoKeStTaTk-p2 CoKeStTkTa-p2, via the KdV theory and the Miura transform, for both the focussing and defocussing cases.
    • Was proven for s>3/5 in references.html#FoLiPo1999 FoLiPo1999
    • Is implicit for s >= 1 from references.html#KnPoVe1993 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 GWP in L^2 in the defocusing case [KpTp-p2]
    • Analytic GWP in H^s for s >= 1/2 [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]], via the KdV theory and the Miura transform, for both the focussing and defocussing cases.
    • Was proven for s >= 1 in references.html#KnPoVe1993 KnPoVe1993, references.html#Bo1993b Bo1993b.
    • One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) references.html#Bo1995c 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^+
gKdV_3 on T
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 references.html#KnPoVe1993 KnPoVe1993
    • Was shown for s>3/2 in references.html#GiTs1989 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 references.html#KnPoVe1993 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], references.html#MtMe2001 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 references.html#KnPoVe1993 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 references.html#MtMe2001 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 [[references.html#MtMe-p MtMe-p]].
gKdV_4 on T
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 references.html#KnPoVe1993 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_on_R+ KdV], [#mKdV_on_R mKdV], [#gKdV_3_on_R gKdV-3] , and [#gKdV_4_on_R 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 [schrodinger.html#Cubic_NLS_on_R cubic NLS equation] , the [schrodinger.html#dnls-3_on_R derivative cubic NLS equation], [#mKdV_on_R complex mKdV], and a cubic nonlinear Airy equation.This equation is a general model for propogation of pulses in an optical fiber references.html#Kod1985 Kod1985, references.html#HasKod1987 HasKod1987

·When c=delta=epsilon = 0, scaling is s=-1.When c=gamma=0, scaling is –1/2.

·LWP is known when s >= ¼ references.html#St1997d St1997d

oFor s > ¾ this is in references.html#Lau1997 Lau1997, references.html#Lau2001 Lau2001

oThe s>=1/4 result is also known when c is a time-dependent function [Cv2002], [CvLi2003]

oFor s < -1/4 and delta or epsilon non-zero, the solution map is not C^3 [CvLi-p]

oWhen delta = epsilon = 0 LWP is known for s > -1/4 references.html#Cv2004 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.]

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 references.html#KnPoVe1994 KnPoVe1994, and independently by Cai (ref?); see also references.html#CrKpSr1992 CrKpSr1992.The case j=2 was studied by Choi (ref?).The non-scalar diagonal case was treated in references.html#KnSt1997 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 KdV]; when a=0 this is the Benjamin-Ono equation (BO) [[references.html#Bj1967 Bj1967]], references.html#On1975 On1975, which models one-dimensional internal waves in deep water. Both of these equations are completely integrable (see e.g. references.html#AbFs1983 AbFs1983, references.html#CoiWic1990 CoiWic1990), though the intermediate cases 0 < a < 1 are not.

When a=0, scaling is s = -1/2, and the following results are known:

When 0 < a < 1, scaling is s = -1/2 - a, and the following results are known:

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 references.html#KnPoVe1994b 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 references.html#KnPoVe1994b KnPoVe1994b.
  • For k>4, LWP for small data in H^s, s >=3/4 was obtained in references.html#KnPoVe1994b 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 references.html#Li1999 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-posednessin H^3 and above references.html#GuoTan1992 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 references.html#OttSud1982 OttSud1982.

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