An index theorem for the stability of periodic traveling waves of KdV type.
There has been a large amount of work aimed at understanding the stability of nonlinear dispersive equations that support solitary wave solutions[20, 7, 3, 6, 19, 29, 28, 30, 35, 36]. Much of this work relies on understanding detailed properties of the spectrum of the operator obtained by linearizing the flow around the solitary wave. These spectral properties, in turn, have important implications for the long-time behavior of solutions to the corresponding partial differential equation[10, 5, 13, 17, 18, 25, 24, 26, 27, 14, 32, 34, 33].
In this paper we consider periodic solutions to equations of Korteweg-Devries type. While the stability theory for periodic waves has received much some attention[1, 8, 2, 9, 15, 16, 21, 12] the theory is much less developed than the analogous theory for solitary wave stability, and appears to be mathematically richer. We prove an index theorem giving an exact count of the number of unstable eigenvalues of the linearized operator in terms of the number of zeros of the derivative of the traveling wave profile together with geometric information about a certain map between the constants of integration of the ordinary differential equation and the conserved quantities of the partial differential equation.
This index can be regarded as a generalization of both the Sturm oscillation theorem and the classical stability theory for solitary wave solutions for equations of Korteweg-de Vries type. In the case of a polynomial nonlinearity this index, together with a related one introduced earlier by Bronski and Johnson, can be expressed in terms of derivatives of period integrals on a Riemann surface. Since these period integrals satisfy a Picard-Fuchs equation these derivatives can be expressed in terms of the integrals themselves, leading to an expression in terms of various moments of the solution. We conclude with some illustrative examples.
1 Introduction and Preliminaries
In the area of nonlinear dispersive waves the question of stability is an important one, as it determines what states one is likely to observe in practice. In this paper we consider periodic traveling wave solutions to equations of KdV type:
where is assumed to be The main goal of this paper is to determine an index expressible in terms of the conserved quantities restricted to the manifold of periodic traveling wave solutions which yields sufficient information for the orbital and spectral stability of the underlying wave. Our results are geometric in nature as they relate to Jacobians of various maps which relate naturally to the structure of the underlying linearized operator, and will be most explicit when is a polynomial. Note that when is a polynomial of degree at most three the above is integrable via the inverse scattering transform, but this is not the case for polynomials of higher degree.
Assuming a traveling wave of the form one is immediately led to the following nonlinear oscillator equation
where is the antiderivative of the nonlinearity Thus the periodic waves depend on three parameters together with a fourth constant of integration corresponding to the translation mode which can be modded out. Thus when we speak of a three parameter family of solutions we will be refering to . On open sets in the solution to the above is periodic. The boundary of this locus of points where the discriminant vanishes, which includes the constant solutions and the solitary waves.
The constants and admit a variational interpretation: defining the Hamiltonian function
so that the Korteweg-DeVries equation can be written
then the traveling waves are critical points of the augmented Hamiltonian functional
and thus represent critical points of the Hamiltonian under the constraint of fixed mass and momentum, with the quantities representing Lagrange multipliers enforcing the constraints of fixed mass and momentum respectively.
We also note the connection with the underlying classical mechanics. The traveling wave ordinary differential equation is (after one integration) Hamiltonian and integrable. The classical action for this ordinary differential equation is
The classical action provides a generating function for the conserved quantities of the traveling waves: specifically the classical action satisfies the following relationships
In this paper, we are interested in both the spectral and orbital (nonlinear) stability of spatially periodic traveling wave solutions of (1). The spectral stability problems has been recently considered [8, 11, 21] in which the authors considered stability to localized perturbations. In this case, the linearized stability takes the form
where is a differential operator with periodic coefficients considered on the real Hilbert space and gives the Hamiltonian structure. This is the standard form for the stability problem for solutions to equations with a Hamiltonian structure, although it must be emphasized that in the KdV case has a non-trivial kernel (spanned by ) which complicates matters somewhat.
Throughout this paper will denote the period of the underlying periodic traveling wave. We will let
be the torus of length and , the Hilbert space of square integrable periodic functions. We will frequently restrict to the subspace of mean zero functions. Following the notation of Deconinck and Kapitula we will denote this subspace by
In this paper we give a geometric construction of the spectrum of the linearized operator about a periodic traveling wave solution. This construction depends on the Jacobian determinants of various maps. Throughout this paper we will use the following Poisson bracket style notation for Jacobian determinants
with the analogous notation for larger Jacobian determinants:
We will also define the following eigenvalue counts:
Given the linearized operator acting on we define to be the number of real strictly positive eigenvalues, to be the number of complex-valued eigenvalues with strictly positive real part, and to be the number of purely imaginary (non-zero) eigenvalues with negative Krein signature - in other words eigenvalues such that the corresponding eigenfunctions satisfy
It is worth making a few remarks on this definition. The Krein signature is an important geometric quantity associated with eigenvalue problems having a Hamiltonian structure, and is associated with the sense of transversality of the root of the eigenvalue relation. It is a fundamental result that two eigenvalues of like Krein signature collide they will remain on the axis, while if two eigenvalues of opposite Krein signature collide they will (generically) leave the imaginary axis. It can be shown that a band of spectrum on the imaginary axis with multiplicity one will have eigenvalues of positive Krein signature, while a band of spectrum of multiplicity three will have two eigenvalues of positive Krein signature and one eigenvalue of negative Krein signature. It is not hard to check that outside of a sufficiently large ball in the spectral plane the spectrum lies on the imaginary axis and has multiplicity one: it follows from this that the non-imaginary eigenvalues and imaginary eigenvalues of negative Krein signature (being the roots of an analytic function) must be finite in number. Also notice that by symmetry of the spectrum about the real and imaginary axes, the quantities and are necessarily even, while the quantity has no definite parity. Our immediate goal is to establish the following index theorem, which is the main result of the paper:
Suppose is a periodic solution to (2) and be the period, mass, and momentum of this solution considered as functions of the parameters :
Also let be the number of real, complex and imaginary eigenvalues of negative Krein signature of on as defined above. Finally suppose that none of are zero. Then we have the following equality
The count on the left hand side of (10) clearly gives information concerning the spectral stability with perturbations in . In particular, if the count is zero then one can conclude the underlying periodic wave is spectrally stable to perturbations in . Moreover, if the count is odd one is guaranteed the existence of a real eigenvalue since the quantities and are necessarily even. What is less clear is the relation of this count to the orbital stability of such a solution in . We will discuss this relationship in detail in the next section. The right hand side of (10) relates to geometric information concerning the underlying periodic traveling wave profile, and the equality provides a direct relationship between the conserved quantities of the governing PDE flow restricted to the manifold of periodic traveling wave solutions to the stability of the underlying wave itself.
In the case is polynomial the integrals (7), (8), and (9) are Abelian integrals on a Riemann surface and the above expressions can be greatly simplified. For instance for the case of the Korteweg-de Vries equation the quantities are homogeneous polynomials of degrees one, two and three resepctively in , while for the modifed Korteweg-de Vries equation they are homogeneous polynomials in and . In general for a polynomial nonlinearity they are homogeneous polynomials of degree one, two and three in some finite number of moments of the solution Thus, in the polynomial nonlinearity case Theorem 1 yields sufficient information for the stability of a periodic traveling wave solution in terms of a finite number of moments of the solution itself.
2 Background and Main Results
The study of eigenvalues of operators of the form (6) has a long history. Eigenvalue problems of exactly this form arise in the study of the stability of solitary wave solutions to equations of Korteweg-de Vries type. The basic observation is that, if were positive definite the spectrum of would necessarily be purely imaginary, since this operator is skew-adjoint under the modified inner product , with the standard inner product on . While in the case of nonlinear dispersive waves is never positive definite due to the presence of symmetries one can count the number of eigenvalues off of the imaginary axis in terms of the dimensions of the kernel and the negative definite subspace of
In the case of periodic solutions to the Korteweg-DeVries equation the best results of this type that we are aware of are due to Haragus and Kapitula and Deconinck and Kapitula. Kapitula and Deconinck give the following construction: consider the spectral problem (6) with the linearized operator acting on the real Hilbert space , and let , , and be defined as before. Then one has the count
where denotes the dimension of the negative definite subspace of the appropriate operator acting on , and is a symmetric matrix whose entries are given by
where is any basis for the generalized eigenspace of such that
The importance of this formula is the following: By using the results of , it is known that a sufficient condition for the orbital stability of a periodic traveling wave solution of (1) is given by . Thus, if one is able to prove that
one can immediately conclude orbital stability in . It follows that spectral stability can be upgraded to the orbital stability if there are no purely imaginary eigenvalues of negative Krein signature. Moreover, the count clearly gives information concerning the spectral stability of the underlying periodic wave. In particular, a necessary condition for the spectral stability of such a solution in is for the difference to be even: more will be said on this later.
In another paper Bronski and Johnson  considered the analogous spectral stability problem to localized perturbations from the point of view of Whitham modulation theory. Bronski and Johnson gave a normal form calculation for the spectral problem in a neighborhood of the origin in the spectral plane, which amounts to studying the spectral stability of a periodic traveling wave solution of (1) to long-wavelength perturbations: so called modulational instability. It was found that the presence of such an instability could be detected by computing various Jacobians of maps from the conserved quantities of the gKdV flow to the parameter space used to parameterize the periodic traveling waves. By deriving an asymptotic expansion of the periodic Evans function
in a neighborhood of , where is the monodromy matrix associated with third order ODE (6) and is the three-by-three identity matrix, it was found that the spectrum of the operator in a neighborhood of the origin is determined by the modulational instability index
In particular, it was found that if then the spectrum locally consists of a symmetric interval on the imaginary axis with multiplicity three, while if the spectrum locally consists of a symmetric interval of the imaginary axis with multiplicity one, along with two branches which, to leading order, bifurcate from the origin along straight lines with non-zero slope. Such information is important if one wishes to consider spectral or nonlinear stability to perturbations whose fundamental period is an integer multiple of that of the underlying wave.
A similar geometric construction was later found useful by Johnson  to prove that such a solution with fundamental period is orbitally stable in if the quantities , , and are all positive. Notice that for this sign pattern Theorem 1 also implies orbital stability in , and hence the index theorem can be regarded as an extension of the theorem of Johnson. Both the calculations of Bronski and Johnson and of Johnson required a detailed understanding of the structure of the kernel and generalized kernels of the linear operators and with periodic boundary conditions. These kernels were constructed by taking infinitesimal variations in each of the defining parameters , , and as well as the translation mode: this construction will be reviewed in the Proposition 1. This information on the structure of the null-spaces, together with some additional work, will allow us to prove our main result.
The operators we consider in this paper are non-self-adjoint and the null-spaces typically have a non-trivial Jordan structure. We adopt the following notation:
Given an operator acting on for some , we define the generalized kernel as follows
Thus is the usual kernel, , and so on.
We begin by stating a preliminary lemma regarding the Jordan structure of
Given any , one generically has , , and for . In particular, we have the following result:
If then If then
If and do not simultaneously vanish then
If and simultaneously vanish then
Since the defining ordinary differential equation is third order the kernel cannot be more than three dimensional.
The generalized kernels are one dimensional unless and vanish simultaneously.
Assuming the subsequent generalized kernels () are empty as long as
This follows from the observation that the derivatives of the wave profile with respect to the parameters satisfy the following equations
reflecting the fact that the constants arise as Lagrange multipliers to enforce the mass and momentum constraints. In the above equality denotes the formal operator without consideration for boundary conditions. In order to find elements of the kernel one must impose periodic boundary conditions. It is not hard to see that is periodic while derivatives with respect to the quantities are not periodic - since the period depends on “secular” terms (in the sense of multiple scale perturbation theory) arise: in particular one sees that the change across a period is proportional to derivatives of the period:
with similar expressions for the change in the across a period. Thus the quantity
is periodic and satisfies Similarly the quantity
is by construction periodic and satisfies and thus Note that while is essentially uniquely determined is only determined up to an element of the kernel. Here we have chosen to make have mean zero since this is the convention required in the work of Deconinck and Kapitula. More will be said on this choice later.
The rest of the calculation follows in straightforward way from calculations of this sort. For instance the existence of a second element of the generalized kernel is equivalent to the solvability of
By the Fredholm alternative and thus the above is solvable if only if
The rest of the claims follow similarly. In the case that the genericity conditions do not hold we do not attempt to compute the Jordan form, but we do remark that the algebraic multiplicity of the zero eigenvalue must jump from three to at least five, and is necessarily odd. ∎
In essence the above proposition shows that the elements of the kernel of are given by elements of the tangent space to the (two-dimensional) manifold of solutions of fixed period at fixed wavespeed, while the element of the first generalized kernel is given by a vector in the tangent space to the (three-dimensional) manifold of solutions of fixed period with no restrictions on wavespeed. As one might expect all of the geometric information on independence in the above proposition can be expressed in terms of various Jacobians. The next fact we note is that the signs of certain of these quantities conveys geometric information about the various operators.
Let be the dimension of the negative definite subspace of as an operator on with periodic boundary conditions. Then
As noted above and thus zero is a band-edge of . From the Sturm oscillation theorem it is clear that either (if zero is an upper band-edge) or if zero is a lower band-edge. The Floquet discriminant has positive slope at an upper band-edge and negative slope at a lower band-edge (and vanishes at a double point), and thus serves to distinguish the two cases. It can be shown (see appendix) that the sign of is equal to the sign of and thus the result follows. Note that the sign of the slope of the Floquet discriminant has an interpretation as the Krein signature of the eigenvalues in the band. ∎
This lemma implies that the vanishing of signals an eigenvalue of passing through the origin and a change in the dimension of . The next proposition gives a similar interpretation for .
Let denote the space of mean-zero, periodic functions, and denote the dimension of the negative-definite subspace of the restriction of the operator to this subspace. Assume that and never vanish simultaneously. Then we have the equality
Since we are restricting to a codimension one subspace the Courant minimax principle immediately implies that we have either or .
Next note that when then the function has mean zero and lies in . Further one has that
Therefore the vanishing of signals a change in the dimension of . A local perturbation analysis shows that near a zero of we have that there is an eigenvalue of which is given by
Thus as long as vanishes for some parameter value the above count is correct. The case that is non-vanishing will be handled later. ∎
The last two results show that geometric quantities associated to the classical mechanics of the traveling waves contain information about changes in the nature of the spectrum of the linearized problem. Specifically:
Vanishing of signals a change in the dimension of , the negative definite subspace of
Vanishing of signals a change in the , the negative definite subspace of .
Vanishing of signals a change in the length of the Jordan chain of .
Finally, to conclude the proof of Theorem 1, we must calculate . This is the content of the following lemma.
Under the assumptions of Theorem 1, one has that with
Thus, is either or depending if is negative or positive, respectively.
Under the assumptions of Theorem 1, we know that and
from Proposition 1. It follows that the matrix is a real number in this case, with value equal to
To complete the proof of proposition 1 note that the results of Bronski and Johnson imply the following identity:
Since and are even they do not change the count modulo two. In the case is non-vanishing the count is determined to within one, and is thus the count is exact if one knows the parity. Applying the result of Bronski and Johnson thus determines the count. ∎
It should be noted that the quantity computed in Lemma 1 also arose naturally in  when considering orbital stability of periodic traveling wave solutions of (1) to perturbations with the same periodic structure. There, the negativity of was necessary in order to prove the quadratic form induced by acting on was positive definite on an appropriate subspace. As the methods therein are based on classical energy functional calculations, such a requirement was necessary to classify the periodic traveling wave as a local minimizer of the Hamiltonian subject to the momentum and mass constraints.
The proof of Theorem 1 is now complete. Notice that Theorem 1 gives a sufficient requirement for a spatially periodic traveling wave of (1) to be orbitally stable in for any and any sufficiently smooth nonlinearity . In the next two sections, we analyze Theorem 1 in the case of a power-nonlinearity by using complex analytic methods to reduce the expression for the Jacobians involved in (10) in terms of moments of the underlying wave itself. This has the obvious advantage of being more susceptible to numerical experiments as one no longer has to numerically differentiate approximate solutions with respect to the parameters , , and . We will also discuss the computation of for power-law nonlinearities. In particular, we will prove a new theorem in the case of the focusing and defocusing MKdV which relates the modulational stability of a spatially periodic traveling wave to the number of distinct families of periodic solutions existing for the given parameter values.
3 Polynomial Nonlinearities and the Picard-Fuchs System
One major simplification of this theory occurs when the nonlinearity is polynomial. In this case the fundamental quantities are given by Abelian integrals of the first, second or third kind on a Riemann surface. While we cannot give a detailed exposition of this theory here the basics are very straightforward. Suppose that is a polynomial of degree . If the polynomial is of degree or then the quantity is an Abelian differential on a Riemann surface of genus . If we define the moment of the solution as follows:
then one obviously has
for any loop in the correct homotopy class. For our purposes we are interested in branch cuts on the real axis though none of what will be said in this section assumes this. In the context of the stability problem one only needs since can all be expressed in terms of these five quantities, but the theory requires that one consider all such moments. The main observation is that the above integrals are again Abelian integrals and thus can be expressed in terms of
In practice the simplest way to do this is to use the identity
for . This gives a linear system of equations in unknowns :
The matrix which arises in the above linear systems is the Sylvester matrix of and . It is a standard result of commutative algebra that the Sylvester matrix of and is singular if and only if the polynomials and have a common root. In our case and having a common root is equivalent to having a root of higher multiplicity. In the case where has a multiple root the a pair of branch points degenerate to a pole and the genus of the surface decreases by one. We will later work an example where this occurs.
For a given polynomial it is rather straightforward to work these out, particularly with the aid of computer algebra systems. In this paper we did many of the more laborious calculations with Mathematica. Some examples are presented in the next section.
4.1 The Korteweg -de Vries Equation (KdV-1)
The Korteweg-Devries (KdV) equation
is, of course, completely integrable. The spectrum of the linearized flow can in principle be understood by the machinery of the inverse scattering transform, in particular by the construction via Baker-Akheizer functions detailed in the text of Belokolos, Bobenko, Enolskii, Its and Matveev). Nevertheless this problem provides a good test for our methods, which we believe to be considerably simpler and easier to calculate than the algebro-geometric approach.
Assuming a traveling wave the ordinary differential equation integrates up to
where denotes the effective potential for the Hamiltonian system. A fundamental quantity is the discriminant of the polynomial , which is given by
Notice the KdV equation has periodic solutions if and only if is positive. Moreover, by scaling (and possibly a map ) the wave speed can be assumed to be .
In this case, the Picard-Fuchs system is the following set of five linear equations:
These quantities are all positive. The positivity of follows from the result of Schaaf mentioned earlier. The non-negativity of is clear: in principle the cubic polynomial in the numerator could vanish but numerics shows that it does not in the region where The positivity of is clear. Finally is positive from Jensen’s inequality since
Recall from  that the Jacobian arises naturally as an orientation index for the gKdV linearized spectral problem for a sufficiently smooth nonlinearity. Indeed, one has that is sufficient to imply the existence of a non-zero real periodic eigenvalue of the linearized operator , i.e. an unstable real eigenvalue in . Moreover, from  it follows that if , then such an eigenvalue can not exist if is positive: however, no such claim can be made in the case where .
Theorem 1 now implies the following index result: if one considers the linearized operator acting on for then has roots in and the number of real eigenvalues, complex eigenvalues, and imaginary eigenvalues of negative Krein signature satisfy
In particular when , so one is considering stability to perturbations of the same period, the only eigenvalues lie on the imaginary axis and have positive Krein signature thus proving orbital stability of such solutions in . Furthermore, considered as an operator on the spectrum in a neighborhood of the origin in the spectral domain consists of the imaginary axis with multiplicity three, thus implying modulational stability of the periodic traveling wave solutions of the KdV equation.
4.2 Example: Modified Korteweg- de Vries (KdV-2)
The MKdV equation
arises as a model for wave propagation in plasmas and as a model for the propagation of interfacial waves in a stratified medium. It is also integrable and the same caveats apply as for the KdV regarding the algebro-geometric construction of the spectrum of the linearized operator. The MKdV is invariant under the scaling and thus the wavespeed can be scaled to be The most physically and mathematically interesting case is the focusing MKdV (the plus sign above) with rightmoving waves where can be scaled to . If we scale the focusing MKdV equation such that the parameter space contains the familiar swallow-tail fold: see Figure 1.
For the focusing MKdV, the swallowtail curve is defined implicitly by the equation
or by the polynomial parametric representation
The soliton solution corresponds to the origin Along the upper (dashed) branch () there are two solutions: a constant solution and a solitary waves homoclinic to some (non-zero) constant value. Along the lower (dotted) branch there are again two solutions: a constant solution and a non-constant solution. Along the remaining portions of the curve there is only a constant solution. The Riemann surface associated to the traveling wave solutions is of genus one (a torus)
except along the swallowtail curve where the discriminant vanishes. In the case of vanishing discriminant the torus “pinches off” and degenerates to a cylinder. In this case all of the elliptic integrals can be evaluated in terms of elementary functions. Unlike the KdV case, where the only periodic solutions on the curve of vanishing discriminant are constant, the MKdV admits non-trivial periodic solutions on the swallowtail curve.
The modulational instability index turns out, in this case, to be particularly simple. After solving the Picard-Fuchs system
one finds the following expressions for the various Jacobians:
as before and