KdV-type equations

Equations of Korteweg-de Vries type

The 'equations of Korteweg-de Vries type are all nonlinear perturbations of the Airy equation. They take the general form

$\partial _{t}u+\partial _{x}^{3}u+\partial _{x}P(u)=0$

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.

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.

Specific equations

Several special cases of KdV-type equations are of interest, including

• The Korteweg-de Vries (KdV) equation (on R, on R^+, or on T), in which $P(u)=3u^{2}$. This equation is completely integrable.
• The modified Korteweg-de Vries (mKdV) equation (on R, on R^+, or on T), in which $P(u)=\pm 2u^{3}$. This equation is also completely integrable.
• The generalized Korteweg-de Vries (gKdV) equation, in which $P(u)=cu^{k+1}$ for some constants $c,k$. The quartic gKdV-4 equation and the quintic (mass-critical) gKdV-5 equation are of special interest. In general, these equations are not completely integrable.

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))).

History

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).

Conservation laws, symmetries, and criticality

KdV-type equations on R or T always come with three conserved quantities:

$Mass:=\int udx,\|u\|_{L_{x}^{2}}^{2}:=\int u^{2}dx,Hamiltonian:=\int u_{x}^{2}-V(u)dx$

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 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

$s_{c}={\frac {1}{2}}-{\frac {2}{k}}.$

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.

Symplectic structure

A KdV-type equation can be viewed as a symplectic flow with the Hamiltonian defined above, and the symplectic form given by

$(u,v):=\int uv_{x}dx$.

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.

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.

Estimates

The perturbation theory for the KdV-type equations rests on a number of linear, bilinear, trilinear, or multilinear estimates for the Airy equation. These estimates involve a number of function space norms, such as the X^s,b spaces. See the page on Airy estimates for more details.

Junk

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 references.html#KnPoVe1993 KnPoVe1993
  • Was shown for s>3/2 in references.html#GiTs1989 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 references.html#KnPoVe1993 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 references.html#CaLo1982 CaLo1982, references.html#Ws1986 Ws1986, references.html#BnSouSr1987 BnSouSr1987 and asymptotically H^1 stable [MtMe-p3], [MtMe-p]

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gKdV_3 on T

• Scaling is s_c = -1/6.
• LWP in H^s for s>=1/2 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3
  • Was shown for s >= 1 in references.html#St1997c St1997c
  • One has analytic ill-posedness for s<1/2 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3 by a modification of the example in references.html#KnPoVe1996 KnPoVe1996.
• GWP in H^s for s>5/6 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3
  • Was shown for s >= 1 in references.html#St1997c St1997c
  • This result may well be improvable by the "damping correction term" method in [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]].
• Remark: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of P(u).

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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]].

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gKdV_4 on T

• Scaling is s_c = 0.
• LWP in H^s for s>=1/2 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3
  • Was shown for s >= 1 in references.html#St1997c St1997c
  • Analytic well-posedness fails for s < 1/2; this is essentially in references.html#KnPoVe1996 KnPoVe1996
• GWP in H^s for s>=1 references.html#St1997c St1997c
  • This is almost certainly improvable by the techniques in references.html#CoKeStTaTk-p3 CoKeStTkTa-p3, probably to s > 6/7. There are some low-frequency issues which may require the techniques in [[references.html#KeTa-p KeTa-p]].
• Remark: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of P(u).

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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.

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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]

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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 references.html#KnPoVe1993 KnPoVe1993
  • Was shown for s>3/2 in references.html#GiTs1989 GiTs1989
  • One has ill-posedness in the supercritical regime references.html#BirKnPoSvVe1996 BirKnPoSvVe1996
  • For small data one has scattering references.html#KnPoVe1993c 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 references.html#BnSouSr1987 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 references.html#St1995 St1995
• On R with any k, gKdV-k is GWP in H^s for s >= 1 references.html#KnPoVe1993 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 references.html#St1997b 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 references.html#KnPoVe-p3 KnPoVe-p3, [KnPoVe-p4].
  • In the completely integrable cases k=1,2 this is in references.html#Zg1992 Zg1992
  • Also, a non-zero solution to gKdV cannot vanish on a rectangle in spacetime references.html#SauSc1987 SauSc1987; see also references.html#Bo1997b Bo1997b.
  • Extensions to higher order gKdV type equations are in references.html#Bo1997b 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... references.html#CtWs1991 CtWs1991
  • A similar result for k > (5+sqrt(73))/4 ~ 3.39... was obtained in references.html#PoVe1990 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 references.html#AbSe1977 AbSe1977
• In the L^2 subcritical case 0 < k < 4, multisoliton solutions are asymptotically H^1-stable [MtMeTsa-p]
  • For a single soliton this is in [MtMe-p3], [MtMe-p], references.html#Miz2001 Miz2001; earlier work is in references.html#Bj1972 Bj1972, references.html#Bn1975 Bn1975, references.html#Ws1986 Ws1986, references.html#PgWs1994 PgWs1994
• A dissipative version of gKdV-k was analyzed in references.html#MlRi2001 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 references.html#St1997b St1997b
• On T with k >= 3, gKdV-k is LWP for s >= 1/2 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3
  • Was shown for s >= 1 in references.html#St1997c St1997c
  • Analytic well-posedness fails for s < 1/2 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3, references.html#KnPoVe1996 KnPoVe1996
  • For arbitrary smooth non-linearities, weak H^1 solutions were constructed in references.html#Bo1993b Bo1993.
• On T with k >= 3, gKdV-k is GWP for s >= 1 except in the focussing case references.html#St1997c St1997c
  • The estimates in references.html#CoKeStTaTk-p3 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 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[references.html#KeTa-p KeTa-p]].

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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]