B\"acklund Transformation and Quasi-Integrable Deformation of Mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models

0.15 MB·

by  Kumar Abhinav

B¨acklund Transformation and Quasi-Integrable
7 Deformation of Mixed Fermi-Pasta-Ulam and
1
0
Frenkel-Kontorova Models
2

n
a
a b a
J Kumar Abhinav , A Ghose Choudhury and Partha Guha
∗ † ‡
1 a
SN Bose National Centre for Basic Sciences
1

JD Block, Sector III, Salt Lake, Kolkata 700106, India
]
h b
Department of Physics, Surendranath College,
p
-
24/2 Mahatma Gandhi Road, Calcutta 700009, India
h
t
a
m
[

2 Dedicated to the memory of Anjan Kundu
v
9
2
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1 Abstract
0
.
1 In this paper we study a non-linear partial differential equation (PDE), proposed
0
by Kudryashov [arXiv:1611.06813v1[nlin.SI]], using continuum limit approximation of
7
1 mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models. This generalized semi-discrete
: equation can beconsidered as amodelforthedescription of non-linear dislocation waves
v
i in crystal lattice and the corresponding continuous system can be called mixed general-
X
ized potential KdV and sine-Gordon equation. We obtain the B¨acklund transformation
r
a ofthisequationinRiccati formininversemethod. We furtherstudythequasi-integrable
deformation of this model.
PACS: 05.45.Yv, 81.07.De, 63.20.K-,11.10.Lm, 11.27.+d
Keywords: Fermi-Pasta-Ulam equation, Frenkel-Kontorova Models, B¨acklund transforma-
tion, Wadati-Konno formalism, quasi-integrable deformation.
∗E-mail: [email protected]
†E-mail [email protected]
‡E-mail: [email protected]
1
1 Introduction
Recently Kudryashov [1], using the continuous limit approximation, has derived a nonlinear
partial differential equation for the description of dislocations in a crystalline lattice which can
be regarded as a generalization both the Frenkel-Kontorova [2, 3] and the Fermi-Pasta-Ulam
[4, 5, 6] models. This generalized model can be considered to describe nonlinear dislocation
waves in the crystal lattice following,
d2y
i 2 2 2
= (y 2y +y ) k +α(y y )+β(y +y +y y y y y y y )
dt2 i+1 i i1 i+1 i1 i+1 i i1 − i+1 i − i+1 i1 − i i1
2πyh i
i
f sin (1.1)
0
− a
(cid:0) (cid:1)
for all (i = 1,...,N). Here y denotes the displacement of the i-th mass from its original
i
position, t is time, k,α,β,f and a are constant parameters of the system. This equation
0
boils down to Frenkel-Kontorova form for the description of dislocations in the rigid body
when α = 0 and β = 0. In the case of f = 0 and β = 0 the system of equations becomes
the well-known Fermi-Pasta-Ulam model. One must note that the Fermi-Pasta-Ulam model
N and h 0 is transformed to the Korteweg-de Vries (KdV) equation.
→ ∞ →
Kudryashov showed that in the continuum limit the semi-discrete equation takes the fol-
lowing form:
2
u +αu u +3βu u +γu = δsinu. (1.2)
xt x xx x xx xxxx
In the special case when α = 0 and γ = 2β it is shown that one can cast the equation in the
AKNS scheme with the Lax pair given by
iλ q A B
L = − , M = , (1.3)
r iλ C A
(cid:18) (cid:19) (cid:18) − (cid:19)
where q = r = u /2 and the functions A, B and C are
x
− −
δ
−1 2 3
A = cosu(iλ) +βu (iλ)+8β(iλ) ,
−4 x
δ β
−1 3 2
B = sinu(iλ) + βu + u 2βu (iλ)+4βu (iλ) ,
−4 xxx 2 x − xx x
(cid:18) (cid:19)
δ β
−1 3 2
C = sinu(iλ) βu + u 2βu (iλ) 4βu (iλ) .
−4 − xxx 2 x − xx − x
(cid:18) (cid:19)
Equation (1.2) follows from the usual zero-curvature condition, viz
F = L M +[L,M] = 0. (1.4)
tx t x
−
As(1.2)isacontinuoussystemonemayderiveB¨acklundtransformations[7,8]fortheequation
using the method proposed by Konno and Wadati [9, 10] which makes use of the Riccati
equation. In the next section we study this procedure.
2
Real physical systems contain finite number of degrees of freedom, thereby prohibiting
the corresponding field-theoretical models from being integrable in principle, as the latter
requiresinfiniteconserved quantities. However, theyareknowntopossesphysically obtainable
solitonicstates, considerablysimilarinstructuretointegrableones, likesine-Gordon(SG)[11].
Therefore, the study of physical continuous systems can be motivated as slightly deformed
integrable models. Recently [12, 13], the SG model was deformed into an approximate system,
with a finite number of conserved charges. The corresponding connection (curvature) was
almost flat, yielding an anomaly instead of the usual zero-curvature condition. This system
was, therefore, dubbed as quasi-integrable (QI). In a recent paper [14] we obtain the quasi-
integrable deformation of the KdV equation. In the present paper we outline the quasi-
deformation of the new equation proposed by Kudryashov.
1.1 Lagrangian and Hamiltonian
The integrable sector of Eq. 1.2, represented as,
2
u +3βu u +2βu = δsinu, (1.5)
xt x xx xxxx
follows from a Euler-Lagrange form,
∂ ∂ ∂ ∂
L + L + L = L, (1.6)
∂u ∂u ∂u ∂u
(cid:18) t(cid:19)t (cid:18) x(cid:19)x (cid:18) xxx(cid:19)xxx
corresponding the Lagrangian,
1 β
4
= u u + u +βu u δcosu. (1.7)
L 2 x t 4 x x xxx −
Therefore, the system described by Eq. 1.5 has a Lagrangian structure with respect to the
variable u. However, the presence of higher order derivatives therein depicts time-evolution
of u instead. However, the canonical variable is identified to be u, leading to a Hamiltonian
x
through the Legendre transformation,
∂ β
4
:= L δcosu u βu u . (1.8)
H ∂u −L ≡ − 4 x − x xxx
t
2 B¨acklund transformation using the Riccati equation
Classical stable solutions of integrable systems are of paramount interest owing to their Phys-
ical realizability. The standard way to arrive at such a solution is through B¨acklund transfor-
mation [8]. For this purpose, we write the scattering problem as,
∂ ψ iλ q ψ
1 1
= − , (2.1)
∂x ψ2 r iλ ψ2
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
3
∂ ψ A B ψ
1 1
= . (2.2)
∂t ψ2 C A ψ2
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)
The consistency of (2.1) and (2.2) yields the equation (1.4) with the eigenvalues λ being time
independent. By introducing the function
ψ
1
Γ := (2.3)
ψ
2
we obtain from (2.1) and (2.2) the Riccati equations:
∂Γ
2
= 2iλΓ+q rΓ , (2.4)
∂x − −
∂Γ
2
= B +2AΓ CΓ . (2.5)
∂t −
′
To derive a B¨acklund transformation one seeks a transformation Γ Γ which satisfies an
−→
equation identical to (2.4) with q(x,t) replaced by
′
q (x,t) = q(x,t)+f(λ,Γ) (2.6)
for some suitable function f. Then upon eliminating Γ between (2.4) and (2.6) one arrives at
the desired B¨acklund transformation. In our case as r(x,t) = q(x,t) = u /2 we shall take
x
−
1 ∂
′ ′ −1
Γ = and q (x,t) = q(x,t) 2 tan Γ. (2.7)
Γ − ∂x
As,
∂
′ ′ ′
0 = (ΓΓ) = 4iλ+(q +q )(Γ+Γ)
∂x −
′ ′
(where use has been made of (2.4) and its corresponding similar version with q and Γ) we
see that
′
q +q 2Γ ∂
′ −1
= , q q = 2 tan Γ. (2.8)
2iλ 1+Γ2 − ∂x
From the latter equation we notice that as r(x,t) = q(x,t) = u /2,
x
−
′
1 u u
−1 ′
tan Γ = (u u), or Γ = tan − .
4 − 4
(cid:18) (cid:19)
On the other hand the first part of (2.8) with Γ as given above yields,
′ ′
q +q 2Γ u u
= = sin − ,
2iλ 1+Γ2 2
(cid:18) (cid:19)
whence,
′
u u
′
u +u = 4iλsin − . (2.9)
x x 2
(cid:18) (cid:19)
To find the time-part we use the form of Γ found above in (2.5) to obtain after simplification
′ ′
u u u u
′
u u = 2(C B)+4Asin − 2(C +B)cos − . (2.10)
t − t − 2 − 2
(cid:18) (cid:19) (cid:18) (cid:19)
Equations (2.9) and (2.10) constitute the desired B¨acklund transformation.
4
One-Soliton Solution: In order to construct a one-soliton solution of (1.2) from the
B¨acklund transformation given by (2.9) and (2.10) we set λ = iµ, µ R and note that
′ ∈
as u = 0 is a trivial solution of (1.2) we have
u u δ
3
u = 4µsin , u = 4Asin with A = 8βµ . (2.11)
x t
2 − 2 4µ −
It now follows that
δt
u = 4arctan(eθ), where θ = 2µx +16βµ3t
± − 2µ
which matches the solution given in [1] when η = 2µ, δ δ and β β.
→ − → −
3 Derivation of the conservation laws
Given the Lax pair one can quite easily derive an infinite number of conserved quantities for
equation (1.2). Firstly it follows from the zero-curvature condition that
A +rB qC = 0 (3.1)
x
−
q B 2(iλ)B 2qA = 0 (3.2)
t x
− − −
r C +2(iλ)C +2rA = 0. (3.3)
t x
−
Using (2.5) and the above set of equations one easily derives the following equation
∂ ∂
(rΓ) = ( A+CΓ), (3.4)
∂t ∂x −
which has the general form of an conservation law for the conserved densities and the flows.
In fact from (2.4) we have
2
r[(rΓ)/r] = 2(iλ)(rΓ)+qr (rΓ) , (3.5)
x
− −
where the suffix represents the usual partial differentiation with respect to x. We may expand
(rΓ) in inverse powers of (iλ) as
∞
rΓ = f (iλ)−n, (3.6)
n
n=1
X
which when inserted into (3.5) yields the following recurrence relation for the conserved den-
sities, namely
n−1
f
s
2fs+1 = r qrδs,0 + fqfs−q, s = 0,1,2, , (3.7)
− r − ···
(cid:18) (cid:19)x q=1
X
5
therebyconfirmingintegrabilityofthesystem byyielding infinitenumber ofconserved charges.
The first few these conserved densities, at the lowest order, are given by,
1
f = qr, (3.8)
1
2
1
f = rq , (3.9)
2 −22 x
1 1
2
f = rq (qr) , (3.10)
3 23 xx − 23
1
2
f = (q (q r) ) 2qrq . (3.11)
4 −24 xxx − x − x
(cid:0) (cid:1)
Consequently (3.4) becomes
∞
∂
fn,t(iλ)−n = A−1(iλ)−1 A1(iλ) A3(iλ)3
∂x − − −
Xn=1 h
∞
f
+ C−1(iλ)−1 +C0 +C1(iλ)+C2(iλ)2 n (iλ)−n) . (3.12)
r
(cid:0) (cid:1)Xn=1(cid:18) (cid:19) i
where the coefficients of the various powers of (iλ) are as follows:
δ
2
A = cosu, A = βu , A = 8β,
1 −4 1 x 3
δ β
3
B−1 = C−1 = −4 sinu, B0 = −C0 = βuxxx + 2ux,
B = C = 2βu , B = C = 4βu
1 1 xx 2 2 x
− −
It can be verified that for all positive powers of (iλ) this relation are identically satisfied, while
for various negative powers of (iλ) we have the following relations:
∂ 1 ∂ δ β β 3β
−1 2 2 4
(iλ) : u = cosu+ u u u + u , (3.13)
∂t −8 x ∂x 4 2 x xxx − 4 xx 16 x
(cid:18) (cid:19) (cid:18) (cid:19)
which leads to Eq. 1.5 itself. The next order continuity equation has the form,
∂ ∂
−2 2 2
(iλ) : [ (u ) ] = u 2δsinu+5βu u +4βu , (3.14)
∂t − x x ∂x x − xx x xxxx
(cid:2) (cid:0) (cid:1)(cid:3)
and so on.
6
4 QI Deformation
The most trivial way to QI-deform the present system is by deforming the periodic sinu
function, like that in Ref. [12, 13] or any other way. Incorporating this into the time-
component M of the Lax pair as usual, only the first terms of (1/λ) in B and C changes. As
O
only C appears in the continuity Eq. 3.4, and further, as its (1/λ) part does not contribute
O
while obtaining Eq. 3.13 from Eq. 3.12, the (1/λ) conserved quantity remains conserved
O
even after the deformation. However, the higher order equations coming from Eq. 3.12 gets
effected and hence they do not support conserved charges in general. Thus, quasi-integrability
is achieved by definition. Moreover, the deformation need not to be small or finite, as far as
Eq. 3.13 is concerned.
As an attempt for non-trivial QI deformation of the present system, we opt to deform the
highest powered term in u in the temporal component of the Lax pair: u3 u3+ǫ. This
x x → x
exclusively changes the C contribution in Eq. 3.12, leading to
0
∂ 1 ∂ δ β β β
u2 = cosu+ u u u2 + u4(2uǫ +1) , (4.1)
∂t −8 x ∂x 4 2 x xxx − 4 xx 16 x x
(cid:18) (cid:19) (cid:20) (cid:21)
thatfallsbacktoEq. 3.13forǫ 0. Herealso, evenfinitevaluesofǫwillworkforconservation
→
of the (1/λ) charge. In fact, as the RHS of Eq. 3.4 is a total derivative, as long as a
O
deformation does not induce boundary non-zeros, the system will remain integrable.
Thus, only sensible and simplest way to make the system QI for sure is to deform sinu in B
and C in a way that it leads to non-zero boundary value, as the (1/λ) charge remains unaf-
O
fected, and thus conserved still. This requirement is naturally satisfied by a shift deformation
of the type,
δ r
(B, C)−1 = sinu (B, C)−1 +D−1; D−1 := gm(u), (4.2)
(cid:18) −4 (cid:19) → m=1 fm−1
X
where g (u) are functions of u and its derivatives, which is finite at the boundary. Then, for
m
m = n+1, nrepresenting the summation index inEq. 3.12, there will be a non-vanishing term
on the RHS of Eq. 3.12. For m < n+1, however, g will be multiplied with positive powers
m
of u and its derivatives, leading to conservation again. For m > n + 1, negative powers of
x
u and its derivatives will come into play, leading to infinities (non-conserved charges again).
x
One can very well truncate the vale of m to avoid such infinities. All in all, quasi-integrability
will be obtained as desired.
One can very well attribute this deformation to the level of the Lagrangian (Eq. 1.7) or
the Hamiltonian (Eq. 1.8). As discussed in subsection 1.1, although the system dynamics
contains higher derivatives, the canonical variable is still u, making δsinu equivalent to the
‘potential’ of the system. Therefore, the above deformation of Eq. 4.2, being QI, in principle
7
corresponds to that of the sine-Gordon system [12, 13]. However, the present deformation
needs to be of shift nature (subsec. 1.1), unlike the power modification of the previous cases.
4.1 Loop-algebraic treatment
From Eq. 1.3 the Lax pair can be re-expressed as,
L = iλσ3 +qσ+ +rσ− and M = Aσ3 +Bσ+ +Cσ−, (4.3)
−
where the Pauli matrices σ3,± satisfy SU(2) algebra:
[σ3,σ±] = 2σ± and [σ+,σ−] = σ3, (4.4)
±
allowing for the construction of the SU(2) loop algebra [14],
bn,Fm = 2Fm+n, [Fn,Fm] = λbm+n; where ,
1,2 2,1 1 2
1 1
(cid:2)bn = λnσ(cid:3)3, F1n = λn(λσ+ σ−) and F2n = λn(λσ+ +σ−). (4.5)
√2 − √2
Such algebraic structure enables sl(2) gauge rotation of the Lax pair, elegantly demonstrating
the quasi-integrability of the system [12].
We now consider the QI deformation of Eq. 4.2, modifying the curvature defined through
the zero-curvature condition in Eq. 1.4; leading to,
1 1
Ftx F¯tx = E¯(u)σ+ + E¯(u)σ+ + σx, σx = σ+ +σ−, (4.6)
→ −2 2 X
where E¯(u) is the Euler function yielding the Quasi-modified equation while equated to zero:
2
uxt +3βuxuxx +2βuxxxx = δsinu 4D−1, (4.7)
−
and is the anomaly term given as,
X
= iD−1,x. (4.8)
X
Gauge transformation: Based on the standard gauge-fixing of the QI systems [12, 13, 14],
we undertake a gauge transformation with respect to the operator,
∞
g = exp J , where, J = anFn +anFn. (4.9)
n n 1 1 2 2
n=−∞ !
X
The coefficients an are to be chosen such that the new spatial Lax pair component L¯ =
1,2
gLg−1+g g−1 contains only bns:
x
8
L¯ βnbn, (4.10)
≡ L
n
X
making it diagonal in the SU(2) basis. From the BCH formula:
1 1
eXYe−X = Y +[X,Y]+ [X,[X,Y]]+ [X,[X,[X,Y]]]+ ,
2! 3! ···
the gauge-transformed spatial component takes the form,
1 1
L¯ = J +L+[J ,L]+ [J ,[J ,L]]+ [J ,[J ,[J ,L]]]+ , (4.11)
n,x n m n l m n
2! 3! ···
with summations understood over all semi-positive integers. Few of the lowest order commu-
tators are,
q r
[J ,L] = 2iλ(anFn +anFn)+ (an an)bn + (an +an)bn+1,
n 1 2 2 1 1 2 1 2
√2 − √2
1 q
[J ,[J ,L]] = iλ(aman aman)bm+n+1 (an an) amFm+n +amFm+n
m n 1 1 2 2 1 2 1 2 2 1
2! − − √2 −
r (an +an) amFm+n+1 +amFm+n+1 (cid:0), (cid:1)
1 2 1 2 2 1
−√2
(cid:0) (cid:1)
1 2
[J ,[J ,[J ,L]]] = i λ(aman aman) alFl+m+n+1 +alFl+m+n+1
l m n 1 1 2 2 1 2 2 1
3! − 3 −
q
(an an) alam(cid:0) alam bl+m+n+1 (cid:1)
1 2 1 1 2 2
−3√2 − −
r (an +an)(cid:0)alam alam(cid:1)bl+m+n+2,
1 2 1 1 2 2
−3√2 −
. (cid:0) (cid:1)
.
. (4.12)
The gauge-fixing condition of vanishing coefficients for Fn s leads to the order-by-order con-
1,2
sistency relations:
r q q
0 0 0 0 0 0
F : a = + a a a 2iλa ,
O 1 1,x √2 √2 1 − 2 2 − 2 − √2λ
(cid:0) 0(cid:1) 0 r q(cid:0) 0 (cid:1)0 0 0 q
F : a = + a a a 2iλa ,
O 2 2,x −√2 √2 1 − 2 1 − 1 − √2λ
(cid:0) 1(cid:1) 1 r 0 0(cid:0) 0 q(cid:1) 0 0 1 1 1 0 1
F : a = a +a a + a a a + a a a 2iλa + ,
O 1 1,x √2 1 2 2 √2 1 − 2 2 1 − 2 2 − 2 ···
(cid:0) 1(cid:1) 1 r (cid:0) 0 0(cid:1) 0 q (cid:2)(cid:0) 0 0(cid:1) 1 (cid:0) 1 1(cid:1) 0(cid:3) 1
F : a = a +a a + a a a + a a a 2iλa + ,
O 2 2,x √2 1 2 1 √2 1 − 2 1 1 − 2 1 − 1 ···
(cid:0) (cid:1) . (cid:0) (cid:1) (cid:2)(cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
.
. (4.13)
9
leaving behind the coefficients for bns as,
q
0 0 0
β = iλ+ a a ,
L − √2 1 − 2
q (cid:0) r (cid:1) q
1 1 1 0 0 0 0 0 0 0 0
β = a a + a +a + iλ a a a a a a ,
L √2 1 − 2 √2 1 2 − 3√2 1 − 2 1 1 − 2 2
(cid:20) (cid:21)
q (cid:0) (cid:1) r (cid:0) (cid:1) (cid:0) (cid:1) q(cid:0) (cid:1)
2 2 2 1 1 0 1 0 1 0 0 0 1 0 1
β = a a + a +a +2iλ a a a a 2 a a a a a a
L √2 1 − 2 √2 1 2 1 1 − 2 2 − 3√2 1 − 2 1 1 − 2 2
(cid:0) (cid:1) (cid:0) (cid:1) r (cid:0) (cid:1) h (cid:0) (cid:1)(cid:0) (cid:1)
1 1 0 0 0 0 0 0 0 0 0 0
+ a a a a a a a +a a a a a +
1 2 1 1 2 2 1 2 1 1 2 2
− − − 3√2 − ···
i
. (cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
.
. (4.14)
Therefore, the transformed spatial Lax component L¯ of Eq. 4.10 is completely determined
[12, 13, 14]. The gauge-transformed temporal Lax component,
M¯ = gMg−1 +g g−1 = [βn bn +ϕnFn +ϕnFn], (4.15)
t M 1 1 2 2
n
X
can also be evaluated similarly. Some of the lowest order commutators are,
B C
[J ,M] = 2A(anFn +anFn)+ (an an)bn + (an +an)bn+1,
n 1 2 2 1 1 2 1 2
− √2 − √2
1 B
[J ,[J ,M]] = A(aman aman)bm+n+1 (an an) amFm+n +amFm+n
m n 1 1 2 2 1 2 1 2 2 1
2! − − − √2 −
C (cid:0) (cid:1)
(an +an) amFm+n+1 +amFm+n+1 ,
1 2 1 2 2 1
−√2
1 2 (cid:0) (cid:1)
[J ,[J ,[J ,M]]] = A(aman aman) alFl+m+n+1 +alFl+m+n+1
l m n 1 1 2 2 1 2 2 1
3! 3 −
B (an an) al(cid:0)am alam bl+m+n+1 (cid:1)
1 2 1 1 2 2
−3√2 − −
C (cid:0) (cid:1)
(an +an) alam alam bl+m+n+2,
1 2 1 1 2 2
−3√2 −
. (cid:0) (cid:1)
.
. (4.16)
that leads to the lowest order coefficients,
10