In mathematics, the Korteweg–De Vries (KdV) hierarchy is an infinite sequence of mutually compatible nonlinear evolution equations containing the Korteweg–de Vries equation as its first nontrivial member. It is one of the central examples in the theory of integrable systems and soliton equations, because it combines several characteristic features of integrability: a Lax formulation, infinitely many commuting flows and conserved quantities, Hamiltonian and bi-Hamiltonian structures, and exact solution methods such as the inverse scattering transform and finite-gap integration.
It is most commonly formulated as a family of Lax equations for the one-dimensional Schrödinger operator
or, equivalently up to sign conventions, . The hierarchy consists of commuting flows in auxiliary variables , each of which preserves the spectral data of . In periodic and quasiperiodic settings, this spectral interpretation leads to the theory of finite-gap and algebro-geometric solutions, while more broadly the hierarchy serves as a prototype for many later integrable hierarchies and reductions such as the modified KdV and KP hierarchies.
Definition Let
The KdV hierarchy may be defined by requiring that evolve according to the family of Lax equations
where each is an odd-order differential operator determined by . A compact way to construct these operators uses formal pseudodifferential operators:
where denotes the differential part of a formal pseudodifferential operator. This formulation implies that the flows commute:
so one may regard the hierarchy as a compatible overdetermined system for depending on infinitely many times.
Equivalent descriptions can be given in terms of commuting Hamiltonian vector fields or the Lenard–Magri recursion scheme. In all of these formulations, the distinguishing feature of the hierarchy is that it produces infinitely many compatible flows, all associated with the same Schrödinger operator.
First flows Although the Lax operator is written here as , it is customary to display the nonlinear equations in an equivalent convention obtained by replacing by and rescaling the time variables. In that convention, the first few equations of the hierarchy are
The -flow is the translation flow. The -flow is the Korteweg–de Vries equation itself, up to the conventional replacement . Higher members are called the higher KdV equations. Different sign conventions for and different normalizations of the times produce equivalent formulas with altered coefficients or signs.
Computing the first flows The first few equations of the hierarchy can be derived explicitly from the pseudodifferential definition. Let
and define
where denotes the differential part of a formal pseudodifferential operator. The KdV hierarchy is then
To compute the first flows one begins with the formal square root
whose coefficients are determined by the requirement
The basic multiplication rule is
valid for all integers , where the generalized binomial coefficients are used for negative . In particular,
Substituting the series for into and comparing coefficients of powers of gives the coefficients recursively. The coefficient of is
so
The coefficient of is
so
The coefficient of is
so
The coefficient of is
so
Thus
The t0-flow Since
the first Lax equation is
Now
so comparing the zeroth-order terms gives
This is the translation flow.
The t1-flow Next,
Using the expansion above and keeping only nonnegative powers of gives
To compute the corresponding evolution equation, expand the commutator:
Using
and
one finds that the and terms cancel and
Since has zeroth-order part , this yields
Writing and , and then renaming as , gives the more familiar form
The t2-flow Similarly,
A straightforward calculation gives
Substituting this into
and again comparing zeroth-order terms gives
Writing and , and then renaming as , gives
Remarks on conventions The coefficients above depend on the sign convention for the Schrödinger operator and on the normalization of the time variables. Many references use instead of , or rescale the variables . These changes alter some signs and numerical factors but not the underlying hierarchy.
This computation shows explicitly how the first members of the hierarchy arise from the formal half-powers of : one expands , extracts the differential parts of the odd powers, and then reads off the induced evolution equation for from the commutator .
Hamiltonian structure The KdV hierarchy is also a Hamiltonian hierarchy. The KdV equation admits infinitely many conserved functionals, and choosing these successively as Hamiltonians produces the higher flows. In the standard Poisson formulation, the hierarchy can be written schematically as
where are conserved quantities in involution. This Hamiltonian point of view is closely related to the Gelfand–Dikii formalism and to the bi-Hamiltonian structure of KdV.
Hamiltonians and Lenard–Magri recursion In the sign and time normalization of the displayed equation , one introduces two compatible Poisson operators
If are Hamiltonians whose variational derivatives satisfy the Lenard–Magri recursion
then the KdV flows are
This is the bi-Hamiltonian formulation of the hierarchy.
The Hamiltonians may also be constructed directly from the Lax operator. In the formal pseudodifferential approach, one first considers the odd half-powers of
and takes their residues, that is, the coefficients of . Up to conventional numerical factors and the addition of total derivatives, the functionals
are conserved quantities of the hierarchy. Applying the potential sign change and time rescalings described above gives Hamiltonians for the displayed convention. The corresponding residue densities are the Gelfand–Dikii polynomials associated with .
With this normalization one may take, for example,
Then
the translation flow, while
which is the Korteweg–de Vries equation itself. Higher Hamiltonians generate the higher KdV equations in the same way.
In this formulation the conserved quantities are in involution with respect to both Poisson brackets, and the compatibility of and implies that the Hamiltonian vector fields commute. Thus the Hamiltonian, recursion, and Lax formulations are equivalent descriptions of the same hierarchy.
Spectral interpretation A Lax equation implies that the operator evolves isospectrally. Thus the spectral data of the auxiliary Schrödinger problem are preserved along each KdV flow. In the case of periodic potentials, this may be expressed in terms of Floquet or Bloch spectral data. Let be the translation operator . For a 1-periodic potential , one studies pairs for which there exists a nonzero solution of
The resulting Bloch–Floquet spectrum is invariant under the KdV flows. This periodic spectral problem is one of the main starting points for finite-gap integration.
Tau-function For many purposes, a solution of the KdV hierarchy is encoded by a tau-function , defined up to multiplication by the exponential of an arbitrary linear function of the times. In the normalization used in the matrix-resolvent construction below,
More generally, higher logarithmic derivatives of give the correlation functions or -point functions of the hierarchy. This provides a compact way to organize the commuting KdV flows and their bilinear identities, and it is a standard language for exact solutions of integrable hierarchies.
Recent work of Dubrovin, Yang, and Zagier expresses the generating series of the logarithmic derivatives of the tau-function in terms of the basic matrix resolvent of the Schrödinger operator, and equivalently in terms of an associated wave function and dual wave function.
A distinguished example is the Witten–Kontsevich tau-function, whose logarithm is the generating function for intersection numbers of -classes on the Deligne–Mumford moduli spaces . Edward Witten conjectured that this generating function is a tau-function of the KdV hierarchy satisfying the string equation, and Maxim Kontsevich proved the conjecture using the matrix Airy function. This result established a major connection between the KdV hierarchy, two-dimensional gravity, and intersection theory on moduli spaces of curves.
Matrix resolvent and wave-function approach Another formulation of the KdV hierarchy uses a matrix resolvent associated with the Schrödinger operator. In a common normalization one writes the scalar Lax operator as and introduces the corresponding first-order matrix operator
The basic matrix resolvent is the unique formal -valued Laurent series satisfying
Here lower order refers to the standard principal grading, in which and the upper- and lower-triangular generators of have degrees and , respectively. The upper-right entry has a recursive expansion whose coefficients determine the commuting KdV derivations.
In this approach the tau-function is encoded by generating series built from . Dubrovin, Yang, and Zagier showed that for any solution of the hierarchy the logarithmic derivatives of the tau-function can be written in terms of the basic matrix resolvent, and then gave equivalent formulas in terms of a wave function and its dual . These satisfy the auxiliary linear problems for the hierarchy, and their product recovers the resolvent entry:
From them one forms the kernel
and the generating series of the -point correlation functions of is obtained as a cyclic sum of products of these kernels. An equivalent formula uses a gauge-invariant kernel built only from .
This does not replace the usual Lax or pseudodifferential definitions of the hierarchy, but it gives another effective way to compute its tau-function data. In particular, after specializing to , a pair of wave functions determined by the initial data yields all logarithmic derivatives of of order at least two, and hence determines up to its exponential-linear ambiguity. This makes the method useful for explicit solutions such as the Witten–Kontsevich, generalized Brézin–Gross–Witten, and Lamé-type solutions discussed by Dubrovin, Yang, and Zagier.
Writing the basic matrix resolvent as
the equation gives
and hence the scalar third-order equation
The normalization is equivalent to
Expanding
one obtains the recursive differential relation
together with an algebraic recursion that determines each uniquely from the previous coefficients. The first terms are
These coefficients generate the hierarchy recursively: after the same choice of time normalizations used elsewhere in this article, they reproduce the translation flow, the KdV equation, and the higher KdV equations. In this form, the differential recursion is the Lenard–Magri recursion written in matrix-resolvent language.
Soliton solutions Among the best-known exact solutions of the KdV hierarchy are its soliton solutions. In a common normalization they are written in terms of a tau-function by
For the Korteweg–de Vries equation itself,
the one-soliton solution is
where and are constants. It is a localized travelling wave of permanent shape, and its speed is proportional to its amplitude.
More generally, the -soliton solutions may be written as
where
For the KdV equation, one may take
with constants . The resulting solutions describe nonlinear superpositions of solitary waves. After interaction, the solitons re-emerge with the same amplitudes and speeds, differing only by phase shifts; this elastic scattering behavior is one of the characteristic features of integrable systems.
The same tau-function form extends to the full KdV hierarchy. One lets each phase depend linearly on all hierarchy times
with coefficients proportional to odd powers of . Thus a single choice of soliton parameters determines a compatible solution of all commuting KdV flows. In this sense, the hierarchy soliton solutions are the simultaneous soliton solutions of the entire family of equations, not just of the first KdV flow.