and variational structure
We study the modulational instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension. We examine how the Jordan block structure of the associated linearized operator bifurcates for small values of the Floquet exponent to derive a criterion governing instability to long wavelengths perturbations in terms of the kinetic and potential energies, the momentum, the mass of the underlying wave, and their derivatives. The dispersion operator of the equation is allowed to be nonlocal, for which Evans function techniques may not be applicable. We illustrate the results by discussing analytically and numerically equations of Korteweg-de Vries type.
We study the stability and instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension, in particular, equations of Korteweg-de Vries (KdV) type
in the theory of wave motion. Here is typically proportional to elapsed time and is usually related to the spatial variable in the primary direction of wave propagation; is real valued, frequently representing the wave profile or a velocity. Throughout we express partial differentiation either by a subscript or using the symbol . Moreover is a Fourier multiplier, defined as and characterizing dispersion in the linear limit, while describes the nonlinearity. In many examples of interest, obeys a power law.
Perhaps the best known among equations of the form (1.1) is the KdV equation
itself, which was put forward in [Bou77] and [KdV95] to model the unidirectional propagation of surface water waves with small amplitudes and long wavelengths in a channel; it has since found relevances in other situations such as Fermi-Pasta-Ulam lattices (see [FPU55], for instance). Observe, however, that (1.1) is nonlocal unless the dispersion symbol is a polynomial of ; examples include the Benjamin-Ono equation (see [Ben70, Ono75], for instance) and the intermediate long wave equation (see [Jos77], for instance), for which and , respectively, while . Another example, proposed by Whitham [Whi74] to argue for breaking of water waves, corresponds to and . Incidentally the quadratic power-law nonlinearity is characteristic of many wave phenomena.
A traveling wave solution of (1.1) takes the form , where and satisfies by quadrature that
for some . In other words, it steadily propagates at a constant speed without changing the configuration. For a broad range of dispersion symbols and nonlinearities, a plethora of periodic traveling waves of (1.1) may be attained from variational arguments, e.g., the mountain pass theorem applied to a suitable functional whose critical point satisfies (1.2). The associated spectral problem
is the subject of investigation here.
As Alan Newell explained in [New85], “if dispersion and nonlinearity act against each other, monochromatic wave trains do not wish to remain monochromatic. The sidebands of the carrier wave can draw on its energy via a resonance mechanism, with the result that the envelop becomes modulated.” Benjamin and Feir [BF67] and Whitham [Whi67] formally argued that Stokes’ periodic waves at the surface of deep water would be unstable, leading to sidebands growth, namely the modulational or Benjamin-Feir instability. Corroborating results arrived nearly simultaneously, albeit independently, by Lighthill [Lig65], Ostrovsky [Ost67], Benney and Newell [BN67], Zakharov [Zak68a, Zak68b], among others; see also [Whi74] and references therein. Modulational instability occurs in numerous physical systems, other than water waves, such as optics (see [Ost67, Zak68a, AL84, THT86, HK95], for instance) and plasmas (see [Has72, MB89], for instance). Furthermore it results in various nonlinear processes such as envelop solitons, dispersive shocks and rogue waves.
Recently a great deal of work has aimed at translating formal modulation theories in [Whi74], for instance, into rigorous mathematical results. It would be impossible to do justice to all advances in the direction, but we may single out a few — [OZ03a, OZ03b, Ser05] for conservation laws with viscosity, [DS09] for nonlinear Schrödinger equations, and [BJ10, JZ10] for equations of KdV type; see also [BM95] for the Benjamin-Feir instability of Stokes waves in water of finite depth.
In particular in [BJ10], a rigorous calculation of long wavelengths perturbations was made for (local) KdV equations with general nonlinearities — henceforth called generalized KdV equations — via Evans function techniques as well as a Bloch wave decomposition. Under certain nondegeneracy conditions, in fact, the spectrum of the associated linearized operator in the vicinity of the origin was shown to take a normal form — either the spectrum consists of the imaginary axis with multiplicity three or it contains three lines through the origin, one in the imaginary axis and two in other directions; the latter implies instability. Furthermore the normal form was determined by an index, which was effectively calculated in terms of conserved quantities of the PDE and their derivatives with respect to constants of integration arising in the traveling wave ODE.
Here we take matters further and derive a criterion governing spectral instability near the origin of periodic traveling waves for a general class of Hamiltonian systems in one The requirement that the equation is in one spatial dimension merely enters in the discussion of a Pohozaev type identity, which is to avoid inversion of a linearized operator; see Lemma 2.9. spatial dimension. We shall make a few assumptions — mainly the existence of a conserved momentum and a Casimir invariant, interpreted as the mass — but do not otherwise restrict the form of the equation, considerably broadening the scope of applications. Of particular interest are nonlocal equations, for which Evans function techniques Note however that the approach in [GLM07, GLZ08], for instance, realizing the Evans function as a regularized Fredholm determinant may be more generally applicable than the standard ODE based formulation. and other ODE methods (which are instrumental in [BJ10], for instance, in the derivation of index formulae) may not be applicable. Instead we perform a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and replace ODE based arguments by functional analytic ones. Variational properties of the equation will help us to calculate index formulae without recourse to the small amplitude wave limit. Incidentally Lin [Lin08] devised a continuation argument and generalized the stability theory of solitary waves in [GSS87, BSS87, PW92], among others, to a class of nonlinear nonlocal equations.
Our results are most explicit in the case of KdV type equations with fractional dispersion, which we work out in Section 3 and Section 5. In particular we calculate the modulational instability index in terms of the kinetic and potential energies, the momentum and the mass of the underlying wave, together with their derivatives with respect to Lagrange multipliers arising in the traveling wave equation as well as the wave number. In the case of the quadratic power-law nonlinearity we further express the index in terms of the potential energy, the momentum and the mass as functions of Lagrange multipliers associated with conservations of the momentum and the mass. We conduct numerical experiments in Section 4.
2. Abstract framework
We shall derive a sufficient condition of spectral instability to long wavelengths perturbations of periodic traveling waves, for a class of Hamiltonian systems in one spatial dimension, under a few assumptions; they will be stated as they are needed.
Consider a Hamiltonian system of the form
where is a linear skew-symmetric operator, independent of , is a Hamiltonian and denotes variational differentiation.
Throughout we work in the -Sobolev spaces setting. We employ the notation for the -inner product.
Assumption 2.1 (Conservation laws).
Assume that (2.1) possesses in addition to two conserved quantities, denoted and . Assume that , , are smooth in an appropriate function space and invariant under spatial translations. Moreover assume that
We refer to and as the momentum and the mass, respectively. Assumption (P) states that generates spatial translations while assumption (M) implies that is a Casimir invariant of the flow induced by (2.1). Thanks to Noether’s theorem, conservation of the momentum is expected whenever (2.1) is invariant under spatial translations.
satisfies Assumption 2.1, for which ,
Examples include the Benjamin-Bona-Mahony equation (see [BBM72], for instance), for which , and .
A traveling wave solution of (2.1) takes the form , where represents the wave speed, is the spatial translate and satisfies that
Equivalently, arises as a critical point of
for some . That is to say, it satisfies that
Assumption 2.2 (Periodic traveling waves).
The existence of periodic traveling waves of (2.1) usually follows from variational arguments. To illustrate, we shall discuss in Proposition 3.2 minimization problems for a family of KdV equations with fractional dispersion. The symmetry assumption is to break that (2.1) is invariant under spatial translations.
Differentiating (2.4) with respect to and , respectively, we use (P) and (M) to obtain that
Perhaps (2.5a) through (2.5c) are familiar to readers from thermodynamics, where the free energy — in the present setting — serves as a generating function of various quantities of interest. They are in fact found as derivatives of the free energy with respect to Lagrange multipliers for a suitable variational problem. The equality of mixed partial derivatives then leads to relations among their derivatives, known as Kirchhoff’s equations.
The period enters calculations in a slightly different manner from other, periodic traveling wave parameters , , . Although , formally, i.e., with the set of smooth functions as the domain of , nevertheless, is not in general -periodic. Rather exhibits a secular growth linear in . Later we shall take a linear combination of and to develop a Pohozaev type identity.
Here and in the sequel, we may regard , , , evaluated at a periodic traveling wave of (2.1) and restricted to one period, as functions of and , the Lagrange multipliers associated with conservations of the momentum and the mass, respectively, as well as , the period. Therefore we are permitted to differentiate , , with respect to the periodic traveling wave parameters , , . We shall ultimately derive a modulational instability index in terms of , , and their derivatives with respect to , , . Corresponding to translational invariance, plays no significant role in the present development. Hence we may mod it out.
Linearizing (2.1) about a periodic traveling wave in the frame of reference moving at the speed , we arrive at that
Seeking a solution of the form , moreover, we arrive at the spectral problem
We then say that is (spectrally) unstable if the -spectrum of intersects the open, right half plane of . Note that needs not have the same period as , namely a sideband perturbation.
In the case of the (local) generalized KdV equation
the -spectrum of the associated linearized operator was related in [BJ10], for instance, to eigenvalues of the monodromy map (or the periodic Evans function), and its characteristic polynomial led to two stability indices. The first index counts modulo two the total number of positive eigenvalues in the periodic functions setting and it extends that in [GSS87, BSS87, PW92], among others, governing stability of solitary waves. The second index, on the other hand, furnishes a sufficient condition of instability to long wavelengths perturbations and it justifies the formal modulation theory in [Whi74], for instance. The present purpose is to extend the second index to a general class of Hamiltonian systems allowing nonlocal dispersion, for which Evans function techniques and other ODE methods may not be applicable. Instead we rely upon a Bloch wave decomposition of the related spectral problem. Lin [Lin08] devised a continuation argument and extended the first index to solitary waves for a class of nonlinear nonlocal equations; see [HJ12], for instance, for an adaptation in the periodic wave setting.
denotes the Floquet exponent. Accordingly (2.8) leads to the one-parameter family of spectral problems
suggesting us to study -spectra of . Notice that the spectrum of consists merely of discrete eigenvalues. Furthermore
One does not expect to be able to compute the spectrum of for an arbitrary , however, except in few special cases, e.g., completely integrable systems (see [BD09], for instance). Instead we are going to restrict the attention to the Floquet exponent and the eigenvalue both small. Physically this amounts to long wavelengths perturbations or slow modulations of the underlying wave. We shall first study the spectrum of the unmodulated operator at the origin; zero is an eigenvalue of , thanks to the latter equations in (2.5a) and (2.5c). We shall then examine how the spectrum near the origin of the modulated operator bifurcates from that of for small.
We promptly discuss “nondegeneracy” assumptions for a periodic traveling wave of (2.1), under which the generalized -null space of supports a Jordan block structure.
Assumption 2.5 (Nondegeneracies).
is not constant;
In what follows, we employ the notation
and write .
Assumption (N1) states that the underlying, periodic traveling wave is nondegenerate. It is not a serious assumption since if the profile of the underlying wave is constant then its stability proof is easy. In the case of the quadratic power-law nonlinearity, for which the related, time dependent equation obeys Galilean invariance, this amounts to understanding the stability of the zero state.
Assumption (N2) states the nondegeneracy of the linearized operator associated with the traveling wave equation; that is to say, the kernel is spanned merely by spatial translations. It proves a spectral condition, which plays a central role in the stability of traveling waves (see [Wei87, Lin08], among others) and the blowup (see [KMR11], for instance) for the related, time dependent equation, and it necessitates a proof. Actually one may concoct a polynomial nonlinearity, say, , for which the kernel of at the underlying, periodic traveling wave is two dimensional at isolated points.
In the case of generalized KdV equations (see (2.9)), the nondegeneracy of the linearization at a periodic traveling wave was characterized in [BJ10], for instance, through the wave amplitude as a function of the period. Furthermore it was verified in [Kwo89], among others, at ground states in all dimensions. By a ground state, incidentally, we mean a traveling wave solution that is positive and radial and which vanishes asymptotically. The proofs rely upon shooting arguments and the Sturm-Liouville theory for ODEs, which may not be applicable to nonlocal equations. Nevertheless, Frank and Lenzmann [FL12] recently obtained (N2) of Assumption 2.5 at ground states for a family of nonlinear nonlocal equations. We shall adapt it in Proposition 3.5 in the periodic wave setting.
Assumption (N3) implies that the mapping is of and locally invertible, namely the nondegeneracy of the constraint set for the periodic, traveling wave equation. We shall achieve it in Lemma 3.10 in the case of KdV equations with fractional dispersion near the solitary wave limit. Note in passing that may vanish along a curve of co-dimension one. We shall in fact indicate that a change in the sign of signals an eigenvalue of crossing from the left half plane of to the right through the origin.
Lemma 2.6 (Jordan block structure).
for an integer.
form a basis and a dual basis, respectively:
Here and elsewhere, the dagger means adjoint. Moreover
Note from the latter equations in (2.5a) and (2.5c) that . Note from Assumption 2.2 that are even functions and are odd. Since by (N1) of Assumption 2.5, we infer from the former equations in (2.5a) and (2.5c) that and are linearly independent. Since is at most two dimensional by (N2) of Assumption 2.5, furthermore, . A duality argument then leads to that and, therefore, we deduce from (N3) of Assumption 2.5 that and form a basis of .
To proceed, notice that by the latter equation in (2.5b), but by (N1) of Assumption 2.5. If then , which in light of (N3) of Assumption 2.5 admits no solutions other than by the Fredholm alternative. A dual statement follows mutatis mutandis, noting that is orthogonal to and thereby is in . Similarly must be empty since admits no solutions by the Fredholm alternative. ∎
In case , the Jordan block structure for the periodic, generalized null space of is necessarily larger than (J1)-(J3). Since if , indeed, either there must be an element in linearly independent of and , or since ’s would then lie in , there must be elements in and linearly independent of . Here and elsewhere, the superscript means the orthogonal complement.
In the case of the generalized KdV equation (see (2.9)), whose traveling wave equation reduces by quadrature to that
for some , stability indices were effectively calculated in [BJ10], for instance, in terms of , , as functions of , , (although the results may be restated in terms of , , ). When dealing with abstract Hamiltonian systems, however, the second constant of integration may not be available and we opt to choose as a periodic traveling wave parameter instead. Accordingly we shall express the modulational instability index in terms of , , as functions of , , . The present approach seems to lead to advantages that , , are -periodic, as opposed to in [BJ10], and they form a basis of the periodic, generalized null space of , in connection to variational properties of the equation. Formulae do become cumbersome. But in the absence of extra features, this seems the only way forward.
2.2. Jordan block perturbation and modulational instability
Let be a periodic traveling wave of (2.1). Under Assumption 2.1, Assumption 2.2 and Assumption 2.5 we shall examine the -spectrum of , defined in (2.10), in the vicinity of the origin for small, where a Baker-Campbell-Hausdorff expansion reveals that
Notice that , , are well-defined in even though is not. In the case of the generalized KdV equation (see (2.9)),
Recall from Lemma 2.6 that zero is a generalized -eigenvalue of , with algebraic multiplicity three and geometric multiplicity two (with a basis and a dual basis , , of the generalized -null space). For small, therefore, three eigenvalues of will branch off from the origin. We make an effort to understand when an eigenvalue of leaves the imaginary axis.
where are near , and . Note that is related to the Floquet exponent via .
Eigenvalues of (2.18) in the neighborhood of the origin in general bifurcate merely continuously in the perturbation parameter , but not in the manner. Rather they admit Puiseaux series in fractional powers of . Requiring that
however, eigenvalues do depend upon the perturbation parameter in the manner; a proof based upon the Fredholm alternative may be found in [BJ10, Theorem 4]. In applications in Section 3 and Section 5 we shall in fact demonstrate that
where and . Hence we may posit that
In the case of generalized KdV equations (see (2.9)), incidentally, eigenvalue bifurcation is analytic.
Note from the Fredholm alternative that is solvable if and the solution is defined up to an element in . Below we write the solution as , with the understanding that . As a reminder,
whence for some .
At the order of , correspondingly, we find that
which by virtue of the Fredholm alternative is solvable if
Therefore, The kernel of is spanned by two elements while for small but non-zero supports three eigenfunctions at the origin, which in the limit as tend to the same limit. Numerical experiments bear this out. and
Since is determined merely up to an element in one must add to it. Any component in the direction, however, may be absorbed to . Hence, without loss of generality, we set . This amounts to fixing normalization in the perturbation theory for symmetric operators.
Continuing, at the order of , we find that
which by the Fredholm alternative is solvable if and . Substituting (2.23) we arrive at that
Observe that (2.24) is a quadratic eigenvalue problem, which may be transformed into a linear matrix pencil of the form , after the change of variables and , so long as . Consequently (2.24) is equivalent to that
is the effective dispersion matrix.
Theorem 2.7 (Normal form).
Furthermore a complex eigenvalue of implies modulational instability, and the discriminant of its characteristic polynomial, or equivalently , leads to a modulational instability index. A straightforward calculation reveals that
We summarize the conclusion.
Corollary 2.8 (Modulational instability index).
Under Assumption 2.1, Assumption 2.2, Assumption 2.5 and (2.19) a periodic traveling wave of (2.1) is unstable to long wavelengths perturbations if admits a complex root, or equivalently, if its discriminant
We therefore obtain the modulational instability index in terms of various inner products between basis and dual basis elements of the generalized null space of , together with , , . We may further express the index in terms of , , and their derivatives with respect to , , with the help of variational properties of the equation.
We remark that while is made up of terms up to second order in various inner products, its characteristic polynomial is homogeneous. In fact, is linear in inner products, is quadratic and is cubic.
2.3. Pohozaev identity techniques
One may calculate various inner products in (2.25a)-(2.25e) and (2.25f)-(2.25j) using definitions in (2.12a)-(2.12c) and (2.17), except and in (2.25c) and (2.25h), respectively. The goal of this subsection is to develop Pohozaev type identities, which assist us in calculating them without recourse to inversion of .
Lemma 2.9 (Periodic Pohozaev-type identity).
formally in the non-periodic functions setting. Unfortunately we must modify it in the periodic functions setting. For one thing, although is well-defined in the periodic setting, and individually are not. Another, related, is that is not -periodic, but it develops a jump in the derivative over one period:
On the other hand, this is what causes to fail to lie in the periodic kernel of ; see Remark 2.4. Indeed , acting on smooth functions, although is not -periodic. Rather
We then observe that the jump in the derivative of offsets that in , so that makes a -periodic function. Therefore (2.32) follows. Note in passing that the vector field corresponds to simultaneous rescaling of the spatial variable and the period, maintaining periodicity.
Here we tacitly assume the continuity of and across the period so that lies in the domain of the Hamiltonian. We shall establish in Proposition 3.2 that a periodic traveling wave of a KdV equation with fractional dispersion is in fact smooth if it arises as a local constrained minimizer.
The apparent lack of an identity like (2.32) is the main obstruction in extending the present development to higher dimensions.
Concluding the subsection we discuss another Pohozaev identity, relating inner products involving to derivatives with respect to the wave number.
If is smooth and then
where denotes the wave number.
After integration by parts,
it follows that
3. Application: KdV equations with fractional dispersion
We shall illustrate the results in Section 2 by discussing the KdV equation with fractional dispersion and the quadratic power-law nonlinearity
where and is defined via the Fourier transform as . In the range , alternatively,