The Short Pulse Hierarchy

0.13 MB·English

by  Jose Carlos Brunelli

The Short Pulse Hierarchy
J. C. Brunelli
Departamento de F´ısica, CFM
Universidade Federal de Santa Catarina
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0 Campus Universit´ario, Trindade, C.P. 476
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CEP 88040-900
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Florian´opolis, SC, Brazil
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Abstract

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. We studyanewhierarchy of equations containing theShortPulseequation, whichdescribesthe
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i evolution of very short pulses in nonlinear media, and the Elastic Beam equation, which describes
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nonlinear transverse oscillations of elastic beams under tension. We show that the hierarchy of
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equations is integrable. We obtain the two compatible Hamiltonian structures. We construct an
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v infinite series of both local and nonlocal conserved charges. A Lax description is presented for both
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systems. For the Elastic Beam equations we also obtain a nonstandard Lax representation.
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1 Introduction:
The cubic nonlinear Schro¨dinger equation is used in the description of the propagation of pulses
in nonlinear media such as optical fibers. Recently, technology progress for creating very short
pulses was achieved, however, the description of the evolution of those pulses lies beyond the
usualapproximations leadingto thenonlinear Schro¨dinger equation. Various approaches have been
proposed to replace the nonlinear Schro¨dinger equation in these conditions. In Ref. [1] Scha¨fer and
Wayneproposedanalternative modeltoapproximatetheevolutionofveryshortpulsesinnonlinear
media. They derived the short pulse (SP) equation
1
u = u+ u3 . (1)
xt 6 xx
(cid:0) (cid:1)
Chung et al [2] showed numerically that the SP equation provides a better approximation to the
solution of Maxwell’s equations than the nonlinear Schro¨dinger equation as the pulse length gets
short. Also, Sakovich and Sakovich [3] have studied the integrability of (1).
In this paper we will study the integrability of the nonlocal representation of the SP equation
(1)
1
u = (∂−1u)+ u2u , (2)
t x
2
as well the hierarchy of equations associated with it. The equation in this form is more feasible for
a Hamiltonian description and can be obtained integrating (1) with respect to x. In what follows
we will refer to the equation (2) simply as the SP equation.
The local nonlinear equation
u
xx
u = , (3)
t
(cid:20)(1+u2)3/2(cid:21)
x x
will also appear in our hierarchy of equations. It can be embedded in the Wadati-Konno-Ichikawa
(WKI) system [4] and its x derivative can be shown to describe nonlinear transverse oscillation of
elastic beams under tension [5]. Therefore, we will simply call (3) the equation for elastic beams
(EB).
Our paper is organized as follows. In Sec. 2, we shown that our system is integrable through its
bi-Hamiltonian nature. We give the two Hamiltonian structures associated with the Hamiltonian
description of equations (2) and (3). The method of prolongation is used to prove the Jacobi
identity as well the compatibility of the Hamiltonian structures. In Sec. 3, we construct the
recursion operator and its inverse to obtain the first local and nonlocal charges recursively. We
also obtain the first local and nonlocal equations of the hierarchy recursively, which includes the
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EB and SP equations, respectively. In Se.c 4 we obtain the Lax representation for the system. For
the EB equation, and for other local equations in the hierarchy, we also give a nonstandard Lax
representation. In Sec. 5, we summarize our results and present our conclusions.
TheresultsinthispaperaremuchliketheonesobtainedfortheHarryDymandHunter-Saxton1
hierarchy of equations [6], deformed Harry Dym and Hunter-Saxton hierarchy of equations [7] and
for the Non-local Gas hierarchy of equations [8]. In these works we have as a main characteristic
a hierarchy of integrable equations with positive and negative flows. Also, throughout this paper
the calculations involving pseudo-differential operators were performedor checked by the computer
algebra program PSEUDO [9].
2 Bi-Hamiltonian Structure:
Following [7] let us introduce
u
F2 ≡ 1+u2 , A ≡ x , (4)
x F
which satisfy the following useful properties
F2(1−A2)= 1 ,
F
x
= AA ,
F3 x
u
xx
= A . (5)
F3 x
Using (4) the SP equation (2) assumes the forms
1
F = u2F ,
t
(cid:18)2 (cid:19)
x
u 1
A = + u2A , (6)
t x
F 2
and, similarly, the EB equation (3) can be written in one of the forms
u = A ,
t xx
F = AA ,
t xxx
A
xxx
A = . (7)
t F3
1In References [6] and [7] the equation obtained by Hunterand Saxton in SIAM J. Appl. Math. 51, 1498 (1991)
waserroneouslynamed,bythepresentauthor,Hunter-Zheng. Wesincerelyapologizetheauthorsaboutthismistake.
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LetusstressthatthebasicfieldisuandthatAandF arejustplaceholdersusedtomakeexpressions
more compact. Also, it is interesting to observe that the EB equation when written in the form
(7) has the same structural form of the deformed Harry Dym equation studied in [7], however, the
definitions in (4) are different.
From (6) and (7) it is straightforward to note that
H = − dxF , (8)
0
Z
is conserved under both the SP and EB flows.
Introducing the Clebsch potential
u= φ , (9)
x
the equation (2) can be written as
1
φ = (∂−1φ)+ φ3 . (10)
t 6 x
This equation (10) can be obtained from a variational principle, δ dtdxL, with the Lagrangian
density R
1 1 1
L = φ φ − φ4 + φ2 . (11)
2 t x 24 x 2
Thisis afirstorderLagrangian density and, consequently, theHamiltonian structurecan bereadily
read out, or we can use, for example, Dirac’s theory of constraints [10] to obtain the Hamiltonian
and the Hamiltonian operator associated with (11). The Lagrangian is degenerate and the primary
constraint is obtained to be
1
Φ = π− φ , (12)
x
2
where π =∂L/∂φ is the canonical momentum. The total Hamiltonian can be written as
t
1 1 1
H = dx(πφ −L+λΦ)= dx φ4 − φ2+λ π− φ , (13)
T Z t Z (cid:20)24 x 2 (cid:18) 2 x(cid:19)(cid:21)
where λ is a Lagrange multiplier field. Using the canonical Poisson bracket relation
{φ(x),π(y)} = δ(x−y), (14)
with all others vanishing, it follows that the requirement of the primary constraint to be stationary
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under time evolution,
{Φ(x),H } = 0 ,
T
determines the Lagrange multiplier field λ in (13) and the system has no further constraints.
Using the canonical Poisson bracket relations (14), we can now calculate
1 1
K(x,y) ≡ {Φ(x),Φ(y)} = ∂ δ(y−x)− ∂ δ(x−y). (15)
y x
2 2
This shows that the constraint (12) is second class and that the Dirac bracket between the basic
variables has the form
{φ(x),φ(y)} = {φ(x),φ(y)}− dzdz′{φ(x),Φ(z)}J(z,z′){Φ(z′),φ(y)} = J(x,y) ,
D
Z
where J is the inverse of the Poisson bracket of the constraint (15),
dzK(x,z)J(z,y) = δ(x−y).
Z
This last relation determines
∂ J(x,y) = −δ(x−y),
x
or
J(x,y) = Dδ(x−y),
where
D = −∂−1 , (16)
and can be thought of as the alternating step function in the coordinate space. We can now set
the constraint (12) strongly to zero in (13) to obtain
1 1
H = dx φ4 − φ2 . (17)
T Z (cid:18)24 x 2 (cid:19)
Using (9) and the transformation properties of Hamiltonian operators [11], we get
D = ∂(D)(∂)∗ = ∂ , (18)
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and the SP equation (2) can be written in the Hamiltonian form as
δH
2
u = D ,
t 1
δu
with H given by
2
1 1
H = dx u4− (∂−1u)2 , (19)
2
Z (cid:20)24 2 (cid:21)
which can be easily checked to be conserved by both the SP and EB equations.
We will show that the SP and EB equations belong to the same hierarchy of equations, at this
point we note that
δH
0
u = D ,
t 1
δu
with H given by (8), yields the EB equation (3).
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It is easy to show that the charges
1
H = dxFA2 , (20)
−1 2 Z x
1
H = dxu2 (21)
1
2 Z
arealsoconserved byboththeSPandEBequations. Therefore,theSPequation (2)can bewritten
in the Hamiltonian form as
δH
1
u = D ,
t 2
δu
and the EB equation (3) as
δH
−1
u = D ,
t 2
δu
where we have defined
D = ∂−1+u ∂−1u = F2−u ∂−1u ∂−1 . (22)
2 x x x xx
(cid:0) (cid:1)
The skew symmetry of this Hamiltonian structures is manifest. The proof of the Jacobi identity
for this structure as well its compatibility with (18) can be shown through the standard method of
prolongation [12] which we describe briefly.
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We can construct the two bivectors associated with the two structures as
1 1
Θ = dx {θ∧D θ}= dxθ∧θ ,
D1 2 Z 1 2Z x
1 1
Θ = dx {θ∧D θ}= dx θ∧(∂−1θ)+u θ∧(∂−1u θ) .
D2 2 Z 2 2Z x x
(cid:8) (cid:9)
Using the prolongation relations,
pr~v (u) = θ ,
D1θ x
pr~v (u) = (∂−1θ)+u (∂−1u θ),
D2θ x x
pr~v (u ) = (pr~v (u)) ,
D2θ x D2θ x
(23)
it is straightforward to show that the prolongation of the bivector Θ vanishes,
D2
pr~v (Θ )= 0 ,
D2θ D2
implying that D satisfies Jacobi identity. Using (23), it also follows that
2
pr~v (Θ )+pr~v (Θ ) = 0 ,
D1θ D2 D2θ D1
showing that D and D are compatible. Namely, not only are D ,D genuine Hamiltonian struc-
1 2 1 2
tures, any arbitrary linear combination of them is as well. As a result, the dynamical equations (2)
and (3) are bi-Hamiltonian with the same compatible Hamiltonian structures and, consequently,
are integrable [12, 13].
3 The Short Pulse Hierarchy:
When a system is bi-Hamiltonian, we can naturally define a hierarchy of commuting flows through
the relation
δH δH
n+1 n
u = K [u] = D =D , n = ...,−2,−1,0,1,2,... . (24)
tn n 1 δu 2 δu
For n = 1 and n = −1 we get the SP and EB equations, respectively. For n = 0 we have
δH
1
K = D = (∂u) = u ,
0 1 x
δu
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and
δH
K = D 0 = F2−u ∂−1u ∂−1A
0 2 x xx x
δu
(cid:0)(cid:0) (cid:1) (cid:1)
= F2A−u ∂−1u A = Fu −u F = 0 , (25)
x xx x x
(cid:0) (cid:1)
where we have used (5), i.e., F2A = Fu and u A = F . Therefore, we would be lead to
x xx x
conclude that H is a Casimir of D . We can resolve this apparent contradiction being careful
0 2
while performing calculations with the antiderivative ∂−1. We can use the following representation
+∞
(∂−1f)≡ (∂−1f)(x) = dyǫ(x−y)f(y),
x
Z
−∞
where
1/2 for x >y ,
ǫ(x−y)=
(cid:26) −1/2 for x <y .
Then, it can be shown that
1
∂−1f = f − (f(+∞)+f(−∞)) , (26)
x x 2
(cid:0) (cid:1)
Now, if we assume u(n) → 0 as |x|→ ∞ (which yields A| = 0 and F| = 1) then it follows
±∞ ±∞
(∂−1A ) = A and (∂−1F )= F −1 . (27)
x x
Using this last result in the naive calculation (25) we obtain the desired term u . This sort of
x
missing or “ghost” terms given rise to apparent contradictions in nonlocal theories were already
observed in the literature (see [14] and references therein).
Let us introduce the recursion operator following from the two Hamiltonian structures as
R = D D−1 . (28)
2 1
Then, it follows from (24) that
δH δH
n+1 = R† n , n = 0,1,2,... , (29)
δu δu
where
R† = ∂−2+∂−1u ∂−1u = ∂−2 F2+u ∂−1u (30)
x x xx x
(cid:0) (cid:1)
istheadjointofR. Theconservedchargesforthehierarchycan,ofcourse,bedeterminedrecursively
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from (29) and give the infinite set of (nonlocal) conserved Hamiltonians
H = − dxF ,
0
Z
1
H = dxu2 ,
1
2 Z
1 1
H = dx u4− (∂−1u)2 ,
2
Z (cid:20)24 2 (cid:21)
1 1 1 1
H = dx u6+ (∂−2u)2+ (∂−2u3)u− (∂−1u)2u2 ,
3
Z (cid:20)720 2 6 4 (cid:21)
.
.
. . (31)
Thecorrespondingflows(thefirstfew,sincetheequationsbecomeextremelynonlocalasweproceed
further in the recursion) have the forms
u = u ,
t0 x
1
u = (∂−1u)+ u2u ,
t1 2 x
(32)
1 1
u = (∂−3u)+ (∂−1u3)+u ∂−1 u ∂−2u + u4u ,
t2 6 x x 24 x
. (cid:0) (cid:0) (cid:0) (cid:1)(cid:1)(cid:1)
.
. .
For negative values of n, the gradients of the Hamiltonians will satisfy the recursion [9]
δH δH
n = (R†)−1 n+1 , n = −1,−2,... . (33)
δu δu
Writing the recursion operator (28) in the form
R = F2−u ∂−1u ∂−2 , (34)
x xx
(cid:0) (cid:1)
and using the identities (5) the inverse can be easily checked to be
1 1 u
R−1 = ∂2 +A∂−1A = ∂2 ∂F∂−1 xx . (35)
(cid:18)F2 x(cid:19) u F3
xx
Note that (35) can be recognized as the recursion operator obtained in [3] for the SP equation
in the form (1) using cyclic basis techniques. The corresponding conserved charges can now be
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recursively constructed from (33) and have the forms
1
H = dxFA2 ,
−1 2 Z x
1 A2
H = dx FA4 −4 xx ,
−2 8 Z (cid:18) x F (cid:19)
1 AA3 A2A2 A2
H = dx FA6 +8 xx −12 x xx +8 xxx ,
−3 16 Z (cid:18) x F F F3 (cid:19)
.
.
. . (36)
The corresponding flows (the first few) have the forms
u = A ,
t−1 xx
A 1
u = xx + A2A ,
t−2 (cid:18) F2 2 x (cid:19)
xx
(37)
A 1A2A A A A 3A2 A 3
u = xxxx + x xx −2 xxx x − xx −A A2A2+ A4A ,
t3 (cid:18) F4 2 F2 F2 2 F2 xx x 8 x (cid:19)
xx
.
.
..
4 The Lax Representation:
Conserved charges for our systems can be determined in principle recursively from (29) and (33).
However, toconstructtheconservedchargesdirectlywelookforaLaxrepresentationforthesystem
of SP and EB equations.
It is well known [15, 16] that for a bi-Hamiltonian system of evolution equations, u = K [u],
tn n
a natural Lax description
∂M
= [B,M] ,
∂t
n
is easily obtained where, we can identify
M ≡ R ,
B ≡ K′ .
n
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