Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg-de Vries Equation
aa r X i v : . [ m a t h . A P ] N ov Rigorous Justification of the Whitham Modulation Equationsfor the Generalized Korteweg-de Vries Equation
Mathew A. Johnson ∗ Kevin Zumbrun † July 24, 2018
Keywords : Generalized Korteweg-de Vries equation; Periodic waves; Modulationalstability; Whitham theory; WKB.
Abstract
In this paper, we consider the spectral stability of spatially periodic traveling wavesolutions of the generalized Korteweg-de Vries equation to long-wavelength perturba-tions. Specifically, we extend the work of Bronski and Johnson by demonstrating thatthe homogenized system describing the mean behavior of a slow modulation (WKB)approximation of the solution correctly describes the linearized dispersion relation nearzero frequency of the linearized equations about the background periodic wave. Thelatter has been shown by rigorous Evans function techniques to control the spectralstability near the origin, i.e. stability to slow modulations of the underlying solution.In particular, through our derivation of the WKB approximation we generalize themodulation expansion of Whitham for the KdV to a more general class of equationswhich admit periodic waves with nonzero mean. As a consequence, we will show that,assuming a particular non-degeneracy condition, spectral stability near the origin isequivalent with the local well-posedness of the Whitham system.
In the area of nonlinear dispersive waves the question of stability is of fundamental im-portance as it determines those solutions which are most likely to be observed in physicalapplications. In this paper, we consider the stability of the spatially periodic traveling wavesolutions of the generalized Korteweg-de Vries (gKdV) equation(1.1) u t = u xxx + f ( u ) x ∗ Indiana University, Bloomington, IN 47405; [email protected]: Research of M.J. was partially sup-ported by an NSF Postdoctoral Fellowship under NSF grant DMS-0902192. † Indiana University, Bloomington, IN 47405; [email protected]: Research of K.Z. was partiallysupported under NSF grants no. DMS-0300487 and DMS-0801745. INTRODUCTION u is a scalar, x, t ∈ R and f is a suitable nonlinearity. Such equations arise in avariety of applications. For example, the case f ( u ) = u corresponds to the well knownKorteweg-de Vries (KdV) equation which arises as a canonical model of weakly dispersivenonlinear wave propagation [17] [22]. Moreover, the cases f ( u ) = βu , β = ±
1, correspondsto the modified KdV equation which arises as a model for large amplitude internal wavesin a density stratified medium, as well as a model for Fermi–Pasta-Ulam lattices withbistable nonlinearity [1] [4]. In each of these two cases, the corresponding PDE can berealized as a compatibility condition for a particular Lax pair and hence the correspondingCauchy problem can (in principle) be completely solved via the famous inverse scatteringtransform . However, there are a variety of applications in which equations of form (1.1)arise which are not completely integrable and hence the inverse scattering transform cannot be applied. For example, in plasma physics equations of the form (1.1) arise with awide variety of power-law nonlinearities depending on particular physical considerations[16] [18] [19]. Thus, in order to accommodate as many applications as possible the methodsemployed in this paper will not rely on complete integrability of the PDE (1.1). Instead,we will make use of the integrability of the ordinary differential equation governing thetraveling wave profiles: this ODE is always Hamiltonian, regardless of the integrability ofthe corresponding PDE.The stability of such solutions has received much attention as of recently: for example,see [5], [6], [7], [8], [9], [10], [12], [13]. Here, we are interested in the spectral stability tolong wavelength perturbations, i.e. to slow modulations of the underlying wave. There isa well developed (formal) physical theory for dealing with such problems which is knownas Whitham modulation theory [27] [28]. This formal theory proceeds by rescaling thegoverning PDE via the change of variables ( x, t ) ( εx, εt ) then uses a WKB approximationof the solution and looks for a homogenized system which describes the mean behavior ofthe resulting approximation. Heuristically then, one may expect that a necessary conditionfor the stability of such solutions is the hyperbolicity– i.e., local well-posedness– of theresulting first order system of partial differential equations. In order to make this intuitionrigorous, one must study in detail the spectrum of the linearized operator about a periodicsolution in a neighborhood of the origin in the spectral plane, and compare the resultinglow-frequency stability region to that predicted by hyperbolicity of the formal Whithamexpansion.The first part of this analysis was recently conducted by Bronski and Johnson [6]. There,the authors studied the spectral stability of spatially periodic traveling wave solutions to(1.1) to long-wavelength perturbations by using periodic Evans function techniques, i.e.Floquet theory. In particular, an index was derived whose sign, assuming a particular non-degeneracy condition holds, determines the local structure of the spectrum near the originin the spectral plane. This index arises as the discriminant of a polynomial which encodesthe leading order behavior of the Evans function near the origin for such perturbations: thatis, it describes the linearized dispersion relation near zero frequency of the corresponding More precisely, the Cauchy problem for equation (1.1) can be solved via the inverse scattering transformif and only if f is a cubic polynomial. INTRODUCTION f ( u ) = u p +1 , p ∈ N (see [7]).The purpose of this paper is to carry out the second part of the program, connecting theformal Whitham procedure with the rigorous results of [6]. To motivate our approach, recallthat in [28] Whitham caries out the formal modulation approximation for the cnoidal wavesolutions of the KdV equation, in which everything can be evaluated explicitly in terms ofelliptic integrals. As a first step in his analysis, Whitham introduces a periodic potentialfunction φ with the requirement u = φ x where u is a given solution of the KdV equation.In effect, the introduction of this potential allows for a Lagrangian formulation of the KdVwhich can can analyzed via the modulation theory presented in [28]. However, notice thatby requiring the solution u to be of divergence form one is forcing that the solution havemean zero over one period. This of course presents no problem in the case of the KdVequation since all solutions can be made mean zero by Galilean invariance, but it does posea serious problem when trying to extend Whitham’s method to more general equations ofform (1.1) which admit periodic waves of non-zero mean; for example, the approximationwould not be valid for the well known elliptic function solutions of either the focusing ordefocusing modified KdV equation.In order to consider more general equations of the form (1.1), then, it becomes imperativeto find a way to side step this restriction to mean zero waves. To this end we follow thenovel approach suggested by Serre [26], which is towork with the original variable u andaugment the resulting WKB system with an additional conservation law associated withthe Hamiltonian structure of the gKdV; see also the recent work of Zumbrun and Oh [20]in which this approach was used in the viscous conservation law setting. As we will see,this approach not only works in the KdV case originally considered by Whitham, but alsofor the general Hamiltonian gKdV. In particular, we will see that the linearized dispersionrelation predicted by the linearized Whitham system correctly predicts (assuming particularnon-degeneracy conditions are met) the modulational stability of the given periodic wave.To this end, we compare the linearized dispersion relation coming from the modulationapproximation to that derived by Bronski and Johnson using Evans function techniques.Unlike the works of Serre and Zumbrun and Oh [26, 20], however, we show the equivalenceby an essentially different method using direct computation and identities derived in [6] PERIODIC SOLUTIONS OF THE GKDV
Throughout this paper, we are concerned with the periodic traveling wave solutions of thegKdV equation. To begin then, we recall the basic properties of the periodic traveling wavesolutions of (1.1): for more details, see [6], [7], or [13].Such solutions are stationary solutions of (1.1) in a moving coordinate frame of the form x + ct and whose profiles satisfy the traveling wave ordinary differential equation(2.1) u xxx + f ( u ) x − cu x = 0 . This profile equation is Hamiltonian and hence can be reduced through two integrations tothe nonlinear oscillator equation(2.2) u x E + au + c u − F ( u ) PERIODIC SOLUTIONS OF THE GKDV F ′ = f , F (0) = 0, and a and E are constants of integration. Thus, the existence ofperiodic orbits of (2.1) can verified through simple phase plane analysis: a necessary andsufficient condition is for the effective potential energy V ( u ; a, c ) = F ( u ) − au − c u to havea local minimum. It follows that the periodic solutions of (2.1) form, up to translation , athree parameter family of traveling wave solutions of (1.1) which we can parameterize bythe constants a , E , and c . In particular, on open sets in R = ( a, E, c ) the solution to (2.2)is periodic: the boundary of these open sets corresponds to the equilibrium solutions andsolitary waves (homoclinic/heteroclinic orbits).In addition, we make the assumption that there exist simple roots u ± of the equation E = V ( u ; a, c ) which satisfy u − < u + , and that V ( u ; a, c ) < E for u ∈ ( u − , u + ). As aconsequence, the roots u ± are C functions of the traveling wave parameters ( a, E, c ) and,without loss of generality, we can assume that u (0) = u − . It follows that the period of thecorresponding periodic solution of (2.1) can be expressed by the formula(2.3) T = T ( a, E, c ) = √ Z u + u − du p E − V ( u ; a, c ) = √ I Γ du p E − V ( u ; a, c ) , where integration over Γ represents a complete integration from u − to u + , and then back to u − again: notice however that the branch of the square root must be chosen appropriately ineach direction. Alternatively, you could interpret Γ as a loop (Jordan curve) in the complexplane which encloses a bounded set containing both u − and u + . By a standard procedure,the above integral can be regularized at the square root branch points and hence representsa C function of the traveling wave parameters: see [6] for details.Notice that in general, the gKdV equation admits three conserved quantities which canbe represented as M ( a, E, c ) = Z T u ( x ) dx = √ I Γ u du p E − V ( u ; a, c )(2.4) P ( a, E, c ) = Z T u ( x ) dx = √ I Γ u du p E − V ( u ; a, c )(2.5) H ( a, E, c ) = Z T (cid:18) u x − F ( u ) (cid:19) dx = √ I Γ E − V ( u ; a, c ) − F ( u ) p E − V ( u ; a, c ) du representing the mass, momentum, and Hamiltonian, respectively. As above, these integralscan be regularized at the square root branch points and hence represent C functions ofthe traveling wave parameters. As seen in [6] and related papers, the gradients of theperiod, mass, and momentum plan a large role in the stability of the periodic travelingwave solutions of (1.1). For notational simplicity then, we introduce the Poisson bracket The translation mode is simply inherited by the translation invariance of the governing PDE and canhence be modded out in our theory. In the case of the KdV or mKdV, the complete integrability of the PDE implies the existence of aninfinite number of conservation laws. We will discuss this case more carefully in the following sections.
EVANS FUNCTION CALCULATIONS { f, g } x,y = det (cid:18) ∂ ( f, g ) ∂ ( x, y ) (cid:19) for two-by-two Jacobians, and { f, g, h } x,y,z for analogous three-by-three Jacobians.It should be pointed out that in [6] it was shown when E = 0 gradients in the periodcan be interchanged for gradients of the conserved quantities via the relation(2.6) E ∇ a,E,c T + a ∇ a,E,c M + c ∇ a,E,c + ∇ a,E,c H = 0where ∇ a,E,c = h ∂ a , ∂ E , ∂ c i . Thus, although all results of this paper will be expressed interms of Jacobians involving the quantities T , M , and P , it is possible to re-express themcompletely in terms of conserved quantities of the gKdV flow. Such an interpretation seemspossibly more natural and desired from a physical point of view. Now, suppose u = u ( · ; a, E, c ) represents a particular periodic traveling wave solution of(1.1). As we are interested in studying the spectral stability of this solution to localizedperturbations, we must study the linearized equation about u , namely ∂ x L [ u ] v = − v t where v ∈ L ( R ) and the operator L [ u ] = − ∂ x − f ′ ( u ) + c is a self adjoint periodic Hilloperator. Taking the Laplace transform in time yields the linearized spectral problem(3.1) ∂ x L [ u ] v = µv where µ represents the Laplace frequency. We refer to the background solution u as beingspectrally stable if the L spectrum of the linear operator ∂ x L [ u ] is confined to the imaginaryaxis: notice the spectrum is symmetric about the real and imaginary axis.The spectral analysis in L p ( R ), 1 ≤ p < ∞ , of differential operators with periodiccoefficients is known as Floquet theory. In this theory, one rewrites the linearized spectralproblem (3.1) as a first order system of the form(3.2) Y x = H ( x, µ ) Y and lets Φ( x, µ ) be a matrix solution satisfying the initial condition Φ(0 , µ ) = I for all µ ∈ C , where I is the standard 3 × M ( µ ), orthe period map, is then defined to be M ( µ ) := Φ( T, µ ) . Notice that given any vector solution Y of the above first order system, the monodromyoperator acts via the formula M ( µ ) Y ( x, µ ) = Y ( x + T, µ ) EVANS FUNCTION CALCULATIONS x ∈ R and µ ∈ C . In particular, it follows that the L spectrum of the operator ∂ x L [ u ] is purely continuous and µ ∈ spec( ∂ x L [ u ]) if and only ifdet( M ( µ ) − e iκ I ) = 0for some κ ∈ R . Such an equality is equivalent with the existence of a nontrivial solution v of (1.1) such that(3.3) v ( x + mT ) = e imκ v ( x )for all x ∈ R and m ∈ Z , which is equivalent with (3.1) admitting a nontrivial uniformlybounded solution.Following Gardner (see [9] and [10]), we define the periodic Evans function for ourproblem to be(3.4) D ( µ, κ ) = det (cid:0) M ( µ ) − e iκ I (cid:1) , ( µ, κ ) ∈ C × R . The complex constant e iκ is called the Floquet multiplier and the constant κ (which is notuniquely defined) is called the Floquet exponent . In particular, by the above discussionwe see µ ∈ spec( ∂ x L [ u ]) if and only if D ( µ, κ ) = 0 for some κ ∈ R . Moreover, since thecoefficient matrix H ( µ, x ) in (3.2) depends analytically on µ , it follows that D is analyticin the variables µ and κ and is real whenever µ is real.From a computational viewpoint, however, it is more convenient to define the periodicEvans function by choosing a basis { Φ j ( · , µ ) } j =1 of the kernel of the linear operator ∂ x L [ u ] − µ and noticing from (3.4) that(3.5) D ( µ, κ ) = (cid:16) det(Φ j (0 , µ ) (cid:12)(cid:12) j =1 ) (cid:17) − det (cid:16) Φ j ( T, µ ) − e iκ Φ j (0 , µ ) (cid:12)(cid:12) j =1 (cid:17) This view is particularly convenient in our case: as a consequence of the integrability ofthe traveling wave ODE (2.1) one can easily verify that the set of functions { u x , u E , u a } islinearly independent and satisfy L [ u ] u x = 0 , L [ u ] u E = 0 , and L [ u ] u a = − . The first of these is a reflection of the translation invariance of (1.1). Thus, by takingvariations of the underlying periodic solution u ( · ; a, E, c ) of (2.1) in the traveling waveparameters generates an explicit basis for the (formal) kernel of ∂ x L [ u ]. In particular,one can easily verify that u x and a linear combination of u a and u E constitute bonafide T -periodic elements of the kernel of ∂ x L [ u ], and hence D ( µ,
0) = O ( | µ | ) for | µ | ≪ ∂ x L u c = − u x and hence an appropriate linearcombination of u c , u a , and u E lies in the first T -periodic Jordan chain in the translationdirection and hence D ( µ,
0) = O ( | µ | ) for | µ | ≪
1. Moreover, one can verify that if the By (3.3), the Floquet exponent encodes a class of admissible perturbations. Here, we are considering the formal operator ∂ x L [ u ] with out any reference to boundary conditions. EVANS FUNCTION CALCULATIONS { T, M, P } a,E,c is non-zero, then linear combinations of u x , u a , u E , and u c generatethe entire generalized null space of ∂ x L [ u ] and hence the Evans function should indeed beof order O ( | µ | ) near the origin.Using perturbation theory one can then analyze the way these solutions bifurcate fromthe ( µ, κ ) = (0 ,
0) state by using (3.5) with Φ ( x, ( x, ( x,
0) correspondingto u x , u a , and u E respectively. Moreover, it follows that the quantity ∂∂µ Φ ( x, µ ) (cid:12)(cid:12)(cid:12) µ =0 corresponds to a linear combination of u c , u a , and u E , and that the first order variations inthe u a and u E directions can be computed via variation of parameters. The details of thisasymptotic calculation were carried out in Lemma 2 and Theorem 3 of [6], which we nowrecall as the main result of this section. Theorem 1.
Assume that { T, M, P } a,E,c is non-zero. Then in a neighborhood of ( µ, κ ) =(0 , the periodic Evans function admits the following asymptotic expansion: D ( µ, κ ) = − µ { T, M, P } a,E,c + iκµ { T, P } E,c + 2 { M, P } a,E ) + iκ + O ( | µ | + κ ) . Notice that from Theorem 1, spectral stability in a neighborhood of the origin is equiv-alent with the cubic polynomial(3.6) P ( y ) = − y + y { T, P } E,c + 2 { M, P } a,E ) − { T, M, P } a,E,c having three real solutions, and hence modulational stability can be inferred from the dis-criminant of the polynomial P . Indeed, if this discriminant is positive then the polynomial P has three real roots and hence, using the Hamiltonian structure of the linearized operator ∂ x L [ u ], the spectrum in a neighborhood of the origin consists of the a triple covering of theimaginary axis. If the discriminant is negative, however, then P must have a pair of complexroots y , with non-zero imaginary part and hence the spectrum in a neighborhood of theorigin consists of the imaginary axis with multiplicity one, along with two curves which aretangent to lines through the origin with angle arg( iy , ) to the imaginary axis. In particular,the underlying periodic traveling wave is modulationally unstable in the latter case. Thisis the main result of [6] concerning the stability of periodic traveling wave solutions of (1.1)to long wavelength perturbations.It should be pointed out that by standard abelian integral calculations, the Jacobiansarising in Theorem 1 were shown in [7] to be explicitly computable in terms of the travelingwave parameters and moments of the underlying wave over a period in the case of a powerlaw nonlinearity f ( u ) = u p +1 , p ∈ N . For example, in the case of the KdV equation with f ( u ) = u it was shown that the discriminant of P takes the formdisc( P ( · )) = ( α , T + α , T M + α , T M + α , M ) disc( E − V ( · ; a, c )) EVANS FUNCTION CALCULATIONS α j, − j represent nonlinear combinations of the traveling wave param-eters a , E , and c . In particular, it follows that the underlying periodic traveling wave ismodulationally stable provided the equation E − V ( y ; a, c ) = E + ay + c y − y = 0has three real solutions, which is equivalent with the constants ( a, E, c ) corresponding toa periodic orbit of the traveling wave equation (2.1). Since every periodic traveling wavesolution of the KdV can be represented in terms of the Jacobbi elliptic function cn( x ; γ ),it follows that the cnoidal wave solutions of the KdV are always spectrally stable to long-wavelength perturbations as predicted by Whitham [28]. A similar representation holdsin the case of the modified KdV equations with f ( u ) = ± u and hence a given periodicsolution u ( · ; a, E, c ) of the modified KdV is modulationally stable provided the polynomialequation E + ay + c y − y = 0has four real solutions. While the same procedure can be implemented for other power lawnonlinearities f ( u ) = u p +1 with p ∈ N , one must resort to numerical studies to compute thedesired moments of the underlying periodic solution.Finally, it should also be noticed that taking κ = 0 in the low frequency expansion ofthe Evans function provided in Theorem 1 suggests that the stability indexsgn ( D ( µ, D (Λ , < µ ≪ ≪ Λ < ∞ is determined precisely by the Jacobian { T, M, P } a,E,c .As a result, the underlying periodic traveling wave solution is spectrally unstable to co-periodic perturbations if { T, M, P } a,E,c <
0, and (see [6]) is spectrally stable to such per-turbations provided that { T, M, P } a,E,c and T E >
0. In particular, the low-frequency ex-pansion in Theorem 1 provides no information for the stability to co-periodic perturbationsin the case where the Jacobian { T, M, P } a,E,c is zero, i.e. when the map( a, E, c ) ∈ R → ( T ( a, E, c ) , M ( a, E, c ) , P ( a, E, c ))is a local diffeomorphism. Thus, by (2.6), when E = 0 the nonvanishing of { T, M, P } a,E,c is equivalent with the statement that the conserved quantities of the PDE flow defined by(1.1) provide good local coordinates for nearby periodic traveling waves. This nondegen-eracy condition has been seen in the stability analysis of periodic gKdV waves in severalother contexts: for example, it appears in the nonlinear (orbital) stability analysis of suchsolutions [7] [13] to periodic perturbations, as well in the instability analysis to transverseperturbations in higher dimensional models [15]. In particular, using the elliptic integralmethods [7] it has been seen that the quantity { T, M, P } a,E,c is generically nonzero for awide variety of nonlinearities, including the most physically relevant examples of the KdV It was shown in [6] by elementary asymptotic ODE theory that D (Λ , < > SLOW MODULATION APPROXIMATION
We now complement the rigorous results of the previous section with a formal Whithamtheory calculation. In particular, we want to show that the linearized dispersion relation P ( · ) in (3.6) can be derived through a slow modulation (WKB) approximation, and hencethat the formal homogenization procedures suggested by Whitham [27] [28] and Serre [26]correctly describe the stability of the periodic traveling wave solutions of (1.1) to longwavelength perturbations. To this end, recall from Section 2 that the gKdV admits a fourparameter family T of periodic traveling wave solutions for some triple ( a, E, c ). We canthus form the quotient space P := T / R under the relation( u R v ) ⇐⇒ ( ∃ ξ ∈ R ; v = u ( · − ξ )) . and we have the class functions T = T ( ˙ u ) , a = A ( ˙ u ) , E = E ( ˙ u ) , c = C ( ˙ u ) . Similarly, since the conserved quantities are translation invariant it follows that M and P can be interpreted as class functions on P themselves.Let ( a , E , c ) correspond to a particular non-constant periodic solution u of (2.1) in P , that is ˙ u ( x ) = ¯ v ( x ; a , E , c )for some v ∈ P , where a = A ( ˙ u ), E = E ( ˙ u ), and c = C ( ˙ u ). In particular, from thediscussion in Section 2 it follows that since u is nonconstant the projection u ˙ u T 7→ P is locally a fibration (where the fibers are circles), and hence P is locally of dimension three.Now, consider equation (1.1) in the moving coordinate frame x + c t :(4.1) u t = u xxx + f ( u ) x − c u x . A periodic solution of the corresponding traveling wave ODE is a stationary solution of(4.1), i.e. is a solution with wave speed s = 0 in this moving coordinate frame. Letting ε >
0, we now rescale (4.1) by the change of variables ( x, t ) ( εx, εt ) so that the rescaledequation takes the form(4.2) u t = ε u xxx + f ( u ) x − c u x , SLOW MODULATION APPROXIMATION x and t now refer to the slow variables εx and εt .Following Whitham then [27] [28], we consider a solution with a WKB type approxima-tion of the form(4.3) u ε ( x, t ) = u (cid:18) x, t, φ ( x, t ) ε (cid:19) + εu (cid:18) x, t, φ ( x, t ) ε (cid:19) + O ( ε )where y u ( x, t, y ) is an unknown 1-periodic function. It follows that the local periodof oscillation of the function u is ε/∂ x φ , where we assume the unknown phase a priori satisfies ∂ x φ = 0. We now plug (4.3) into (4.2) and collect like powers of ε . The lowestpower of ε present is ε − which leads to the equation φ t ∂ y u = ( φ x ∂ y ) u + ( φ x ∂ y ) f ( u ) − c ( φ x ∂ y ) u . By defining(4.4) s = − φ t φ x , and ω = φ x , it follows that the O ( ε − ) equation is precisely the traveling wave ODE (2.1) in the variable ωy with wavespeed c − s ; In a similar way as above we may define class functions S and Ωon P , respectively, associated to the quantities s and ω . Moreover, we may choose u ( x, t, · )to be spatially periodic and satisfy the nonlinear oscillator equation(4.5) (cid:0) u y (cid:1) E + au + c − s (cid:0) u (cid:1) − F ( u )where a = a ( x, t ), E = E ( x, t ), and s = s ( x, t ) are independent of y . That is, we identifyfor each ( x, t ) ∈ R the projection of the function y u ( x, t, y ) into P as being a periodictraveling wave solution of the gKdV in of form˙ u ( x, t, y ) = ¯ u ( y ; A ( ˙ u ) , E ( ˙ u ) , c − S ( ˙ u ))for some function ¯ u ∈ P , where we have used the notation from Section 2. In particu-lar, notice that the parameters ( A ( ˙ u ) , E ( ˙ u ) , S ( ˙ u )) defining ˙ u in P depend on the slowvariables x and t introduced above.Continuing, we find that the O ( ε ) equation reads ∂ t u = ∂ x f ( u ) − c∂ x u + ∂ x (cid:0) φ x ∂ y u (cid:1) + ∂ y ( · · · )Averaging over a single period in y and rescaling we find(4.6) ∂ t (cid:0) M ( ˙ u )Ω( ˙ u ) (cid:1) − ∂ x G ( ˙ u ) = 0where, using T = T ( ˙ u ) for convenience, G ( ˙ u ) = 1 T Z T (cid:0) f ( ˙ u ) − c ˙ u + ∂ y ˙ u (cid:1) dy. SLOW MODULATION APPROXIMATION u in the variable y , it follows that G ( ˙ u ) = 1 T Z T (cid:0) A ( ˙ u ) − S ( ˙ u ) ˙ u (cid:1) dy = ( A − SM Ω) ( ˙ u ) . Moreover, the Schwarz identity φ xt = φ tx provides us with the additional conservation law(4.7) ∂ t Ω( ˙ u ) + ∂ x (cid:0) S ( ˙ u )Ω( ˙ u ) (cid:1) = 0 . Now, as noted above, the manifold P has dimension three and hence equations (4.6) and(4.7) do not form a closed system for the unknown function ˙ u . In [28] this problem wasremedied in the case of the KdV equation by restricting to mean zero periodic waves, whichforces the equation count to be correct. However, as noted in the introduction, this require-ment can not be implemented in the general case considered here since the gKdV in generaladmits periodic waves of non-zero mean. In order to close the system then, we follow theapproach of Serre [26] and augment the above equations with an additional conservation lawarising from the Hamiltonian structure of (1.1). The choice of this extra conservation lawseems somewhat arbitrary where there is more than one to choose; for example, in the caseof the modified KdV where complete integrability implies the existence of infinitely manyconservation laws. To identify a useful conserved quantity in our calculations, notice fromthe Section 3 that the conserved quantity P must play a significant role in the modulationalstability of the underlying periodic solution. With this as motivation, we notice from (4.1)that (cid:18) u (cid:19) t = u ( u xxx + f ( u ) x − c u x )= (cid:18) uf ( u ) + uu xx − F ( u ) − c u − ( u y ) (cid:19) x (4.8)Substituting (4.3) into (4.8) as before and collecting like powers of ε , we find that the O ( ε − )equation is the traveling wave ODE equation for u , as before, multiplied by the profile u ,and averaging the O ( ε ) equation over a single period in y yields the conservation law(4.9) ∂ t (cid:0) P ( ˙ u )Ω( ˙ u ) (cid:1) − ∂ x Q ( ˙ u ) = 0where(4.10) Q ( ˙ u ) = 2 T Z T (cid:18) ˙ u f ( ˙ u ) + ˙ u ˙ u yy − F ( ˙ u ) − (cid:0) ˙ u y (cid:1) − c u ) (cid:19) dy. Again, using (4.5) one finds that u f ( u ) − F ( u ) = c − s u ) + (cid:0) u y (cid:1) − E − u u yy SLOW MODULATION APPROXIMATION Q ( ˙ u ) = − T Z T (cid:18) S ( ˙ u )2 (cid:0) ˙ u (cid:1) + E ( ˙ u ) (cid:19) dy = − ( SP Ω + 2 E ) ( ˙ u ) . (4.11)The homogenized system (4.6), (4.7), and (4.9) is a closed system of three conservationlaws in three unknowns, since ˙ u belongs to P . In particular, (4.6)-(4.9) describe the meanbehavior of the WKB approximation (4.3). This system is precisely the Whitham systemwe were seeking, and it can be written in closed form as(4.12) ∂ t ( M Ω , P Ω , Ω) ( ˙ u ) − ∂ x ( A − SM Ω , − SP Ω − E, − S Ω) ( ˙ u ) = 0 . We now wish to make a connection between the system (4.12) and the modulational stabilityof the original solution u . To this end, we make the assumption that the matrix(4.13) ∂ ( ˙ u ) ∂ ( a, E, s ) (cid:12)(cid:12) ( a,E,s )=( a ,E , is invertible, which implies that nearby periodic waves in P can be coordinatized by thetraveling wave parameters ( a, E, c ) near ( a , E , c ). In particular, we can parameterize theWhitham system in terms of the variables ( a, E, s ) near ( a , E ,
0) and hence(4.14) ∂ t ( M ω, P ω, ω ) ( a, E, s ) − ∂ x ( a − sM ω, − sP ω − E, − sω ) ( a, E, s ) = 0 . where now we have dismissed the use of the class functions on P previously defined andwork directly with the functions M , P , and T = ω − defined on R defined in section 2.We now linearize (4.14) about the constant solution ( a, E, s ) = ( a , E ,
0) correspondingto the original solution u chosen above. To begin, notice from the integral representations(2.3)-(2.5) that ∂ s h M, P, ω i = − ∂ c h M, P, ω i , and hence assuming the quantitydet (cid:18) ∂ ( M ω, P ω, ω ) ∂ ( a, E, s ) (cid:12)(cid:12) ( a,E,s )=( a ,E , (cid:19) = 1 T { T, M, P } a,E,c ( a , E , c )is nonzero, the corresponding linearized system is of evolutionary type, i.e.the system (4.12)is of evolutionary type provided that the conserved quantities of the flow defined by (1.1)provide good local coordinates for the nearby periodic traveling waves. Recall from Section2 that this property is generically true for several physically relevant nonlinearities, and is amajor technical assumption in much of the current work on the stability of such solutions:see again [6], [7], [13], and [15]. Moreover the Cauchy problem for the linearized system islocally well posed provided that it is hyperbolic, which is equivalent with the characteristicequation(4.15) det (cid:18) µ ∂ ( M ω, P ω, ω ) ∂ ( a, E, s ) − iκT ∂ ( a − sM ω, − sP ω − E, − sω ) ∂ ( a, E, s ) (cid:19) ( a , E ,
0) = 0
SLOW MODULATION APPROXIMATION iκµT , i.e. the matrix ∂ ( M ω, P ω, ω ) ∂ ( a − sM ω, − sP ω − E, − sω ) ( a , E , e P ( µ, κ ; a, E, s ) = det (cid:18) µ ∂ ( M ω, P ω, ω ) ∂ ( a, E, s ) − iκT ∂ ( a − sM ω, − sP ω − E, − sω ) ∂ ( a, E, s ) (cid:19) at the point ( a , E , e P ( µ, κ ; a , E ,
0) = µ det M a ω M E ω − M c ωP a ω P E ω − P c ω − T a ω − T E ω T c ω − iκµT − − ω = 2 µ T − P (cid:18) − iκµ (cid:19) + (2 M a − P E − T c ) (cid:18) iκµ (cid:19) ! where P ( · ) is the cubic polynomial given in (3.6) encoding the modulational stability of theperiodic traveling wave solution u ( · ; a , E , c ). At first sight, this result is disturbing as itsuggests that the formal WKB/homogenization calculation does not accurately describe themodulational stability of such solutions. However, this apparent discrepancy can be easilyexplained using the integral representations of the period, mass, and momentum given in(2.3)-(2.5). Indeed, notice from this representation that M a = P E = 2 T c = − √ I Γ u du ( E − V ( u ; a, c )) / and hence we have the relation P E + 2 T c − M a = 0. In particular, it follows that(4.17) e P ( µ, κ ; a , E ,
0) = − µ T P (cid:18) − iκµ (cid:19) . Notice that the root of the unexpected µ
7→ − µ conversion in (4.17) stems from the factthat in the Evans function calculation described in Section 3 we considered left movingwaves (waves constant in the moving coordinate frame x + ct ), while in homogenizationcalculations of the above form it is customary to use right moving waves. Indeed, from ourdefinition of s in (4.4) it follows that the quantity φ ( x − st ) is constant, i.e. φ correspondsto a right moving plane wave. Since changing right moving waves to left moving waves isequivalent to time reversal, which in turn arises as a µ
7→ − µ at the level of the linearizedequations, this added negative sign seems necessary in the framework presented here. Inparticular, we have proven the following theorem. SLOW MODULATION APPROXIMATION Theorem 2.
Assume that { T, M, P } a,E,c is non-zero and that the matrix in (4.13) is non-singular. Then there exists a nonzero constant Γ such that the periodic Evans functionadmits the following asymptotic expansion in a neighborhood of ( µ, κ ) = (0 , : D ( µ, κ ) = Γ e P ( − µ, κ ; a , E ,
0) + O ( | µ | + κ ) . That is, the linearized dispersion relation arising from the formal WKP approximationcorrectly describes the true linearized dispersion relation of the spectrum of the linearizationabout the underlying periodic traveling wave at the origin.
Corollary 1.
Under the assumptions of Theorem 2, a necessary condition for the spectralstability of the periodic traveling wave solution u ( · ) = u ( · ; a , E , c ) is that the spectrumof the three-by-three matrix A ( a, E, s ) := (cid:18) ∂ ( a − sM ω, − sP ω − E, − sω ) ∂ ( a, E, s ) (cid:19) − (cid:18) ∂ ( M ω, P ω, ω ) ∂ ( a, E, s ) (cid:19) = ∂ ( M ω, P ω, ω ) ∂ ( a − sM ω, − sP ω − E, − sω ) be real at ( a , E , : equivalently, that the Whitham system (4.14) be hyperbolic at ( a , E , .Proof. First, a straight forward calculation shows that the matrix ∂ ( a + sM ω, sP ω − E, − sω ) ∂ ( a, E, s ) (cid:12)(cid:12)(cid:12) ( a,E,s )=( a ,E , is invertible provided { T, M, P } a,E,c is non-zero at ( a, E, c ) = ( a , E , c ) and the matrixin (4.13) is nonsingular at this special point. Moreover, if the matrix A ( a , E ,
0) had anonzero eigenvalue η , then it follows from Theorem 2 that there is a branch of spectrumbifurcating from the origin admitting and asymptotic expansion µ = iηκT + O ( κ )for | κ | ≪
1. Thus, if η has non-zero imaginary part one immediately has a modulationalinstability of the underlying periodic traveling wave solution.It follows that we have established that a necessary condition for the spectral stabilityof a periodic traveling wave solution of (1.1) is that the Whitham system (4.6), (4.7),and (4.9) be hyperbolic at the corresponding solution. Moreover, from the analysis in [6],if the matrix A ( a , E ,
0) has only simple real eigenvalues, it follows that the underlyingperiodic wave is modulationally stable. Indeed, the Hamiltonian structure of the linearizedoperator ∂ x L [ u ] implies the spectrum is symmetric about the imaginary and real axes, andhence if the matrix A ( a , E ,
0) has real distinct eigenvalues α , α , and α , then thereare three branches µ j ( κ ) which bifurcate from the origin with leading order expansions µ j ( κ ) = iα j κT + O ( κ ). If one did not have spectral stability near the origin, then two of the DISCUSSION AND OPEN PROBLEMS µ and µ , would satisfy µ ( κ ) = − µ ( κ ). It follows by equating like powersof κ that α = α , contradicting the simplicity of the spectrum of A ( a , E , In this paper, we have rigorously verified that the formal homogenization procedure intro-duced by Whitham to study the modulational stability of a periodic traveling wave profileof the gKdV equation does indeed describe the spectral stability near the origin. This inparticular applies to the KdV case for which the WKB expansion is carried out in Chapter14 of Whitham [28] by Lagrangian methods making use of integrability to carry out com-putations in terms of elliptic integrals. (The rigorous connection between the Lagrangianformulation and the type of WKB expansion carried out here is an important result of [28].)Recall from Section 4 that there is at first sight a “missing equation” in the WKB equa-tion for (KdV) and (gKdV). This is remedied in [28] by introducing a “potential” consistingof the anti-derivative of the solution, thus viewing the equation as a one higher order PDEfor which the equation count is correct. Here, we follow instead the approach suggested bySerre [26] of augmenting the system by an additional conservation law associated with theintegral of motion coming from Hamiltonian structure. This approach works not only in theintegrable KdV and mKdV cases, but also for the general Hamiltonian (gKdV). The deriva-tion of the Whitham equations seems also slightly simpler from this point of view. Indeed,using this approach of Serre together with identities developed by Bronski and Johnson[6], it seems remarkable that we are able to obtain the Whitham system (4.12) explicitlyin terms of moments of the underlying periodic wave and the traveling wave parameters.Because of this, the verification that the hyperbolicity of the Whitham system correctly de-scribes the spectral stability near the origin proved to be much simpler and straight forwardthan cases previously considered by Oh and Zumbrun [20] and Serre.This issue of a missing equation comes up when there are more periodic solutions thanwould generically expected [26], as arises when there is an integral of motion for the travelingwave profile ODE. Serre states on p. 262 of [26] that “a supplementary integral has acounterpart at the profile level”, that is, such ODE integrals of motion often come fromconserved quantities The “supplementary integral” he refers to is exactly an additionalconservation law, which he proposes to use as above to fill the same missing equation itcauses. That is, this same basic approach, by the argument of Serre, should work in a widerange of situations.In situations such as the KdV, where there may be still further conservation laws,there is flexibility in deriving the Whitham system, but all such derived systems mustbe equivalent. This would appear to correspond to existence for the Whitham systemof additional conservation laws, or hyperbolic “entropies”, another interesting possibilitypointed out by Serre.We here just briefly investigate further the statement of Serre that supplementary PDE
EFERENCES η ( u ) of the unknown and no derivatives, such that η ( u ) = ∂ x ( ... ), then this is indeed always true. (Note: the quantity R η ( u ( x )) dx is thenconserved in time.) Proof.
Proof: we then have additional traveling wave relation − sη ( u )+ q ( u )+ C = L ( u, u x , ... ) , where C is a constand of integration and L is an expression of the same order as thetraveling-wave ODE. But, combined with existing equations, we can then eliminate thehighest derivative to get an algebraic relation between some lower derivatives, i.e., andintegral of motion of the traveling-wave profile ODE.2. In the Hamiltonian case, both properties can be inherited from an overarching La-grangian structure, as in the multi-symplectic form system studied by many authors; see forexample [2, 3]. This may tie in an interesting way to the variational approach of Witham.We cite as open problems the rigorous verification of the Witham approximation atthe level of behavior, either in the small ε limit as in [11], or in large-time behavior as in[23, 24, 25, 21]. Moreover, it is expected that a similar homogenization procedure can beused to justify the Whitham equations for the generalized Benjamin-Bona-Mahony equation,where the low frequency asymptotics were recently computed in a similar way as to thegKdV case: see [14]. Acknowledgement.
Many thanks to Jared Bronski for providing several useful insightsinto the connection of this work to that presented in [28] by Whitham. Also, we thank thereviewer for a careful and detailed check of the formulas and derivations presented in sectionfour.
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