The KdV hierarchy and the propagation of solitons on very long distances
H. Leblond
Laboratoire POMA, Universit´e d’Angers, 2Bd Lavoisier 49045 Angers Cedex 1, France Available online 7 March 2005
Abstract
The Korteweg-de Vries (KdV) equation is first derived from a general system of partial differential equations. An analysis of the linearized KdV equation satisfied by the higher order amplitudes shows that the secular-producing terms in this equation are the derivatives of the conserved densities of KdV. Using the multi-time formalism, we prove that the propagation on very long distances is governed by all equations of the KdV hierarchy. We compute the soliton solution of the complete hierarchy, which allows to give a criterion for the existence of the soliton.
© 2004 IMACS. Published by Elsevier B.V. All rights reserved.
Keywords: KdV hierarchy; Higher order KdV; Reductive perturbation
1. Higher orders in multiscale expansions
The Korteweg-de Vries (KdV) equation arises as the first-order approximation in some asymptotic expansion. In many physical contexts, this approximate model is derived from a more general one using the so-called perturbative expansion method, also called multiple scales analysis (or multiscale expansion).
This method was first introduced in 1960[1,2]. It consists in a two-fold asymptotic expansion. On one hand, it uses a very classical perturbative approach since the considered fields are expanded in a power series of the wave amplitude. The characteristic feature the method is that, on the other hand, the propagation variable (time) is also expanded in a power series of the perturbation parameter, yielding a set of different time scales, which are separated since they are treated as independent variables. Increasing the order
E-mail address: herve.leblond@univ-angers.fr (H. Leblond).
0378-4754/$30.00 © 2004 IMACS. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.matcom.2005.01.010
of the perturbation theory can, thus, be done in two different ways. A first possibility is to increase the accuracy of the computation, by using more terms in the expansion of the field relative to the amplitude [3–6]. A second possibility is to increase the order of magnitude of the propagation time. This can be done by increasing the number of time scales considered in the expansion[7–10]. This is the matter of the present paper. Since the reductive perturbation method allows in some sense to convert the information got from one of these problems to the other, they are not equivalent.
Lax discovered that the KdV equation belongs to a family of completely integrable equations, with common properties, which has been called the KdV hierarchy [12]. A natural expectation is that the higher order KdV equations should be related to the higher orders in the multiscale expansions leading to KdV. On this ground has been considered the problem of the asymptotic integrability: if some equation can be reduced to an integrable one at first order in the expansion, is it still integrable when the order is increased? This question has been answered by the negative[4,6]. When the accuracy is increased, the derived higher order equation is in general not that of the hierarchy, and it is not integrable. However, these studies consider the higher orders in the sense that the accuracy is increased for a given propagation time. What happens when the propagation time increases and the accuracy is not modified is another question.
A very remarkable property of the KdV hierarchy is that all equations are compatible. This means that if each equation involve a specific time variable, and if all these time variables are independent, the complete set of equations admits a common solution. This property is very restrictive: it has been checked by explicit computation that the KdV hierarchy is the only set of equations which has both the homogeneity properties of the expansion leading to KdV and this compatibility property, apart from a scaling coefficients for each time variable[10]. The compatibility comes from the fact that all equations of the KdV hierarchy are completely integrable by means of the inverse scattering transform method, with the same spectral problem ([11], p. 96). The resolution of the higher order KdV has been the matter of many works[13,14]. The most commonly considered problem is the resolution of an equation yielded by some linear combination of several equations of the hierarchy, using a single time variable. Such an equation can be obtained when increasing the accuracy in the expansion, without increasing the propagation time.
Here, on the contrary, the time variables involved by the various equations are considered as independent.
We compute this way the one-soliton solution of the whole hierarchy.
However, there is a strong link between the higher order KdV equations and the higher orders of the multiscale expansions leading to KdV. This link appears in the multiple time formalism, i.e. when increasing the propagation time and not the accuracy, as was found first by Kraenkel et al. [10]. The present paper is a survey of this problem.
2. A KdV-type expansion in a general frame
We consider some set of partial differential equations, that can be written as
(∂t+A∂x+E)u=B(u, u), (1)
where the function u of the variables x and t is valued in Rp, A and E are some p×p matrices, and B:Rp×Rp −→Rpis bilinear. We assume that the range Rg(E) of E is in direct sum with its kernel ker(E), which is satisfied if E is skewsymmetric. Physically, it means that the initial system is conservative.
We denote byΠ0andQ0, respectively, the projectors onto ker(E) parallel to Rg(E) and onto Rg(E) parallel
to ker(E). This system can describe electromagnetic wave propagation in a ferromagnetic medium, according to the Maxwell and Landau equations[16,17]. The variable u is expanded in a power series of some small parameterεas
u=ε2u2+ε3u3+. . . , (2)
and the slow variables
ξ=ε(x−Vt), τj=ε2j+1t (j1), (3)
where V is a speed to be determined, are introduced. In particular, τ1=τ =ε3t. This multiscale ex- pansion is well-known. It yields the Korteweg-de Vries equation at leading order,ε2, in many physical cases.
At first orders in ε, it is found that the leading term u2 must belong to ker(E) and the velocity V must be an eigenvalue ofΠ0AΠ0[18]. An hypothesis necessary for pursuing the computation is that the characteristic space of the operatorΠ0AΠ0relative to the eigenvalueV has the dimension 1, which is ensured under the assumption that A is completely hyperbolic. Physically, this assumption means that a unique polarization can propagate with the velocityV. If it is not satisfied, interactions between the various waves with same velocity must be taken into account. Then, we calla0 an eigenvector,Π1and Q1 the associated projectors:Π1 projector ona0R, Q1projector on Rg(Π0(A−V)Π0), so thatΠ1+ Q1=Π0. Thenu2=a0ϕ2, where the scalar functionϕ2, the wave amplitude at leading order, has to be determined.
If the conditions
Π1B(a0, a0)=0, (4)
Π1(A−V)Q0E−1Q0(A−V)a0=0, (5)
whereQ0E−1Q0is a partial inverse of E, are both satisfied, the following evolution equation for ϕ2is obtained at orderε5:
∂τϕ2+βϕ2∂ξϕ2+γ∂3ξϕ2=δϕ2
ξ
ϕ22. (6)
The scalar coefficientsβ,γ,δare explicitly computed[18]. If the additional condition
Π1B(a0, Q1(A−V)−1Q1B(a0, a0))=0, (7) whereQ1(A−V)−1Q1is a partial inverse ofΠ0(A−V)Π0, is satisfied,δvanishes and Eq.(6)reduces to the KdV equation
∂τϕ2+βϕ2∂ξϕ2+γ∂3ξϕ2=0. (8)
Hypothesis(5) can be written asd2ω/dk2=0 fork=0, where ω(k) is the dispersion relation of the system. Conditions(4) and (7)are particular expressions of a very general condition called the “trans- parency” one in the rigorous mathematical theory of multiscale expansions by Joly et al.[19]. Condition (4)excludes quadratic self-interaction for the chosen propagation mode, while condition(7) excludes interaction at the same order for different polarizations.
At higher orders, with no further hypotheses, we find that
un=a0ϕn+R[ϕ1, . . . , ϕn−1], (9)
in which the functionalR[ϕ1, . . . , ϕn−1] is explicitly known. The nth order amplitudeϕnsatisfies
∂τ1ϕn+β∂ξ(ϕ2ϕn)+γ∂3ξϕn=n(ϕ2, . . . , ϕn−1). (10) Eq. (10) is the KdV equation (8) linearized about its solution ϕ2, with an additional source term n(ϕ2, . . . , ϕn−1) depending on the previously determined amplitudes. In the general case, the source termn involves not only theξ-derivatives of theϕj, but also many integrations relative to the variable ξ. The propertyQ1B(·,·)≡0 ensures their vanishing.
All conditions required for the above derivation are satisfied by the Maxwell–Landau system which describe the evolution of the magnetic field and magnetization density in an infinite isotropic ferromagnet, when inhomogeneous exchange interaction and damping are neglected[16,20,17].
Further, in this case, homogeneity and symmetry properties allow to prove that the source term n(ϕ2, . . . , ϕn−1) of Eq. (10) vanishes for all odd values of n, therefore, all the ϕn are zero for odd n[17]. This property corresponds to an usual parity property of this kind of expansions (see e.g.[21]), it will be assumed to be satisfied below.
3. Secularities and conserved densities
It is well known that the KdV equation(8)is completely integrable, i.e. that the Cauchy problem for it can be solved by use of the inverse scattering transform (IST) method[11,12,22]. For convenience, we write Eqs.(8) and (10)in the normalized form
∂tf −6f∂xf+∂x3f =0, (11)
∂tg−6∂x(fg)+∂3xg=w. (12)
To solve Eq.(12), the use of the IST method is necessary. Indeed, the solution f (proportional toϕ2) of the KdV equation(11)intervenes in it as an essential parameter, and in the general case, it can be only expressed in terms of its inverse transform.
Let us recall some features related to the resolution of the KdV equation (11) through the inverse scattering transform (IST) method. Let us callφkandψkthe Jost functions[11]. For each k real, or purely imaginary zero of the scattering coefficienta(k), we define
Φk =φ2k and Ψk =∂x(ψ2k). (13)
A closure formula for the squared Jost functions function Φk and Ψk [3,11]allows us to compute the solution g of the linearized KdV equation (12) as a sum, g=g1+g2, where g1 is a solution of the homogeneous equation (Eq. (12) with w=0) with the initial datag1(t=0)=g(t=0), and g2 is a solution of the complete Eq.(12)with zero initial data. The source term w can be expanded as
w(x, t)= +∞
−∞ w(k, t)Ψk(x, t)dk+n
l=1
[w(1)l (t)Ψl(x, t)+w(2)l (t)(∂kΨl)(x, t)]. (14) Theng2also expresses as an expansion on the squared Jost functions, as
g2(x, t)= +∞
−∞ Ψk(x, t) t
0
e8ik3tw(k, t)dt
e−8ik3tdk + n
l=1
Ψl(x, t) t
0
e8κ3ltw(1)l (t)dt +24iκ2l t
0
t
0
e8κ3lt w(2)l (t )dtdt
+∂kΨl(x, t) t
0
e8κ3ltw(2)l (t)dt
e−8κ3lt. (15)
Unfortunately, the solution computed this way is not always bounded. As an example, let us assume that att=0,g≡0, and replace the source term w by∂xf. Then, the solution g of Eq.(10)is[3]
g=t∂xf. (16)
A solution like(16)is called “secular”, and the corresponding source term w is called secular-producing.
The secularity phenomenon occurs when a term on the right-hand-side (rhs) is a solution of the homo- geneous equation. The above mentioned closure formula allows us to characterize the secular-producing terms as resonant ones. For a given smooth function w, and for any k with Imk≥0, we define
Mw(k)= lim
t→+∞
1 t
t
0
e8ik3t(Φk, w)dt. (17)
Then w is not secular-producing if and only ifMw(k)=0 for each k[23]. Thus, the secularity phenomenon occurs when a term in the rhs “resonates”, that is, when some component of the inverse scattering transform of the source term w evolves in time in the same way as the corresponding component in the transform of the main term f.
The KdV equation has the very important property of presenting an infinite number of conserved densities. These are algebraically independent functionals Aj of the solution, whose integral over all space is constant in time. They can be written in terms of the squares of the Jost functions[3]as
∂xAj(x, t)=n
l=1
λj,le−8κ3ltΨl(x, t)+ +∞
−∞ λj(k)e−8ik3tΨk(x, t)dk, (18) whereλj,landλj(k) are related to the scattering data. We see that∂xAjis a linear combination and integral of the elementary solutions of the homogeneous linearized KdV equation, thus, it is itself a solution of this equation, and thus, is secular-producing. Reciprocally, if g is a secular-producing term, i.e. a solution of the homogeneous linearized KdV equation, we prove thatU= −∞x g(x, t)dx is a conserved density [23].
4. Multi-time formalism and the KdV hierarchy
The KdV-type asymptotic is valid for times up toT0/ε3, with some finite timeT0. Small corrections to this asymptotic, cumulated on very long times, can indeed modify greatly the behavior of the wave.
In order to reach larger values of the time variable, we have introduced the higher order time variables τj =ε2j+1t,j=1, 2, . . .. The equation giving the evolution of the leading termϕ2with regard toτ3is obtained by imposing that the following termϕ4is bounded, or more exactly sublinear. The evolution of
this term is determined by Eq.(10), which is in this case
∂τ1ϕ4+β∂ξ(ϕ2ϕ4)+γ∂3ξϕ4= −∂τ2ϕ2−γ2∂5ξϕ2+O2. (19) Here, O2 refers here to an expression depending on ϕ2, without linear term, and γ2 is a known real coefficient[24]. The condition to be satisfied is that Eq.(19)does not admit any secular solution. Through an explicit computation in the case whereϕ2is the one-soliton solution of KdV, Kodama and Taniuti[3]
have noticed that the secular-producing terms are the terms linear with regard to the solutionϕ2of lowest order.ϕ4is, thus, nonsecular if the linear terms vanish from the rhs of the Eq.(19). To achieve this, we impose thatϕ2satisfies some partial differential equation, as
∂τ2ϕ2= −γ2∂ξ5ϕ2+O2. (20)
We still need to determine the nonlinear terms of Eq. (20), represented by O2. They are not free but imposed by the compatibility condition between the KdV equation(8)and Eq.(20), which is the Schwartz condition ∂τ1∂τ2ϕ2=∂τ2∂τ1ϕ2. Kraenkel et al.[10] have conjectured and checked on many examples that the only equation that possesses the same homogeneity properties as the rhs member of (19), and that satisfies this condition, is the second equation of what is called the KdV hierarchy.
The KdV hierarchy is the following family of equation[15]:
∂Tnv=∂XLnv (ninteger), (21)
whereLis a recurrence operator, defined by
L= −14∂X2 −v+12 XdX(∂Xv). (22) Forn=1, it is the KdV equation(8). We identify both by settingv=β/(6γ)ϕ2,X =ξ, andT1 =4γτ1. An important property of the hierarchy is that all equations are compatible, i.e. that a solutionv(X, T1, T2, . . .) of the system yielded by all equations of the hierarchy exists. Thus, the Schwartz condition is satisfied at any order. The evolution equation to be satisfied byϕ2is, thus,
−1 16γ2
∂τ2ϕ2 =∂ξL2ϕ2. (23)
This way, the linear terms have been removed from the Eq.(20). Let us show that this procedure removes all secular-producing terms. The recurrence operatorLis related to the conserved densitiesAn through the relation
An = 12Ln−1f, (24)
so that the equations of the KdV hierarchy(21)can be written as
∂Tnf =2∂XAn+1. (25)
The rhs member of Eq.(23)is, thus, proportional to∂XA3, which is exactly the secular-producing term at this order, as shown in section3. The procedure ensures, thus, thatϕ4is sublinear. It is generalized at any order.
Otherwise, the source term of the linearized KdV equation that governs the evolution ofϕ6involvesϕ4, solution of(20). It is, thus, necessary to see whether, when a solution of the linearized KdV equation itself is used in the source term, which part of it is secular-producing, and which part is not. This has been done [23]. Indeed, this solution is given by its expansion on the basis of theΦk, and we have characterized the
fact that a source term is secular-producing or not by some criterion, that involves the coefficients of this expansion and their time-dependency. It remains a last point to be studied: the dependency of the higher order terms with regard to the higher order times. We have shown that it is governed by a linearized KdV hierarchy [18]. Finally, we have been able to justify that the higher order terms are not secular- producing, and to prove that the formal expansion contains bounded terms only. Thus, the equations that give the evolution of the main amplitude with regard to the higher order time variables are that of the KdV hierarchy. There is no contradiction between this statement and the result of[4,6]. Indeed the latter concerns the equations obtained when the evolution of both the main and the higher order amplitudes with regard to the first order time variable are incorporated in a single equation. This equation differs in general from a combination of the equations of the hierarchy.
5. The soliton of the KdV hierarchy
The multi-times formalism yields a set of equations describing the evolution of the main amplitudeϕ2
for each time variableτj, corresponding to a sequence of different time scales. According to the above considerations, these equations are submitted to several constraints: (i) they must be all compatible, (ii) they have the homogeneity properties imposed by the perturbative procedure, (iii) they must cancel the secular-producing terms in the equations satisfied by the higher order amplitudes. All these constraints are satisfied if the equations are that of the KdV hierarchy, apart from a scaling coefficientRnfor each higher order time variableτn, as
∂Tnϕ2=∂ξLnϕ2, with Tj =Rjτj. (26)
Thus, at first orders,R1=4γandR2= −16γ2. Let us now have a look to the solution of the hierarchy.
Notice that many works are devoted to the research of solution of a combination of several equations of the hierarchy for a single time variable[13,14], and very few to the problem of solving simultaneously these equations for a set of independent time variables.
As mentioned above, all equation of the KdV hierarchy are compatible together, in the sense that for a given initial data, a functionϕ2(ξ, T1, T2, T3, . . .) satisfying equation(26)for each value of n can be found. This solution can be found using the inverse scattering transform (IST) method, at least in principle. Indeed, all equations of the hierarchy are completely integrable by means of the IST method.
Furthermore, they can all be described in the IST formalism using the same spectral problem ([11], p.
96), which ensures their compatibility. The scattering data (R+(k), D+,j, kj) (see [11], p. 141 sq., for the precise definition of these quantities) are defined in the same way for all equations, only their time evolution differ for each time variableTn. These time evolution is given by ([11], p. 149)
R+(k, Tn)=R+(k,0)eΩn(k)Tn, (27)
D+,j(Tn)=D+,j(0)eΩn,jTn, (28)
kj(Tn)=kj(0). (29)
The index n refers to the nth equation of the hierarchy. The evolution factors are Ωn,j =Ωn(kj), and Ωn(k)= −iωj(2k), whereωj(k) is the dispersion relation of the nth equation of the hierarchy linearized.
Its seen from relation (29) that the discrete spectrum (kj) is constant with regard to any of the time variablesTn. Therefore, the number of solitons and their characteristics are not modified by the higher
order time evolution. The evolution of the spectral data with regard to all the higher order time variables can then be written as a single exponential factor for each spectral component,
R+(k, T1, T2, . . .)=R+(k,0,0, . . .) exp ∞
n=1
Ωn(k)Tn
. (30)
For a value of the spectral parameterk=kj =iκjbelonging to the discrete spectrum, using the expression (21) and (22)of the equations of the hierarchy, and the definition(3)of the time variableTj, we get the expression of the complete time evolution factor, as
∞ n=1
Ωn,jTn =Ωjt, with Ωj = −2 ∞ n=1
(−1)n(εκj)2n+1Rn. (31)
Obviously, formula (31) is valid only if the power series converges. Notice that the coefficients of the latter are the time scaling coefficientsRn. We find this way or by direct computation the expression of the one-soliton solution of the complete hierarchy, as[24]
v=2κ21sech2κ1
ξ+∞
n=1
(−κ12)nTn
. (32)
ReplacingξandTnby their expressions we get ε2ϕ2 = 12γ
β p2sech2p(x−Vt), (33)
where
V=V −∞
n=1
(−1)np2nRn. (34)
We have setp=εκ1. The soliton speed is, thus, given by a power series of the soliton parameter p, whose coefficients are essentially the time scaling coefficients (Rn)n1. Obviously if this series diverges so does the whole perturbative scheme. Reciprocally, the convergence of the series defining the velocity should favour that of the perturbative scheme, although the latter is by no means proven. The power series which definesVconverges when
p < pM =lim inf
n−→∞ εn, where εn = |Rn|−1/2n. (35)
In the case where the system (1) is the Maxwell–Landau model which describes the evolution of an electromagnetic wave in an infinite isotropic lossless saturated ferromagnetic medium[16], the scaling time coefficients Rj have been computed explicitly [17,24] as a function of the physical parameters, which are in this case the ratioαof the external magnetic field to the magnetization saturation, and the angleθ between the external magnetic field and the propagation direction. The five firstεn are plotted onFig. 1, against the angleθ. Insofar as the extrapolation of the few computed terms is valid, the figure gives the upper boundpMof the soliton parameter.
Fig. 1. Logarithmic plot of the five firstεn, built from the time scaling coefficientsRn, against the angleθbetween the propagation direction and the exterior field.
6. Conclusion
We derived the KdV equation from a general system of partial differential equations, under some conditions which are specified. The higher order amplitudes in the perturbative scheme obey linearized KdV equations, which are solved by means of the IST method. This allows to show that the secular- producing terms in these equations are the derivatives of the conserved densities of KdV. Using the multi-time formalism, we prove that the higher order equations in the expansion are that of the KdV hierarchy, if we mean by “higher order”, that not the accuracy, but the propagation distance is increased.
We compute the soliton solution of the complete hierarchy, and express it in terms of the variables of the initial system: its velocity expresses as a power series of the soliton parameter, whose coefficients are scaling coefficients for the higher order time variables. This yields a criterion for the existence of the soliton, which does not depend on the perturbation parameter.
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