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 $u(t,x)$ 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 cases k=1,2 are KdV and mKdV respectively. The quartic gKdV-3 equation and the quintic (mass-critical) gKdV-4 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 Galilean 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 $P(u(t))$).

The Korteweg-de Vries equation is also a member of the KdV hierarchy. One can also couple the KdV equation to other equations, creating for instance the nonlinear Schrodinger-Airy system.

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 cubic NLS 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 and mKdV equations come with infinitely many more such conserved quantities due to their completely integrable nature.

The critical (or scaling) regularity is

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

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 $t\rightarrow \infty$, 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.