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


Bilinear Airy estimates

\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}


Trilinear Airy estimates

\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 [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 [KnPoVe1996].


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

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^+

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).


KdV on T


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^+


mKdV on T


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].


gKdV_4 on T


gKdV on R^+

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.


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 >= [St1997d]

o       For s > 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.]


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:

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).

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].