Relativistic Burgers and Nonlinear Schrödinger Equations

0.14 MB·English

by  Oktay K. Pashaev

Relativistic Burgers and Nonlinear Schr¨odinger
Equations
9
0
0
Oktay K. PASHAEV
2

n
a Department of Mathematics, Izmir Institute of Technology
J
Urla-Izmir, 35430, Turkey

0
1
January 13, 2009

]
h
p
Abstract
-
h
Relativistic complex Burgers-Schr¨odinger and Nonliner Schr¨odinger
t
a equations are constructed. In the non-relativistic limit they reduce to
m the standard Burgers and NLS equations respectively and are integrable
[ at any order of relativistic corrections.

1
v 1 General Burgers-Schr¨odinger Hierarchy
9
9
TherelativisticlinearSchr¨odingerequationhasbeendiscussedattheearlyyears
3
1 of quantum mechanics but was dismissed promptly by the Klein-Gordon and
. the Dirac equations. Recently, relativistic versions of the Schr¨odinger equation
1
have been considered in the study of relativistic quark-antiquark bound states
0
9 [1], and gravitational collapse of a boson star [2]. A nonlinear version of the
0 model has appeared in the form of semi-relativistic Hartree-Fock equation [3].
: But none of those models is known to be integrable. In the present paper we
v
i construct an integrable relativistic nonlinear Schr¨odinger equation, preserving
X
integrability at any order of 1/c approximation.
r We start from the Schr¨odinger equation in 1 + 1 dimensions
a
∂Ψ
i~ = (P )Ψ (1)
1
∂t H
for a free particle with classical dispersion of the general analytic form E =
E(p). Here P = i~∂ and P = i~ ∂ are operators of the time and space
0 ∂t 1 − ∂x
translationsrespectively,commuting with the Schr¨odingeroperatorS =i~∂
∂t−
(P ): [P ,S] = 0, µ = 0,1. The general boost operator, defined as K =
1 µ
H
x t ′(P ), is also commuting with S, [K,S] = 0. Commuting it with space
1
− H
and time translations we have the algebra of symmetry operators
[P ,P ]=0, [P ,K]= ~ ′(P ), [P ,K]= i~. (2)
0 1 0 1 1
− H −
1
Then, if Ψ is a solution of (1) and W is an operator from this algebra, so that
[W,S]=0, then SΨ is also solution of (1).
ForagivenclassicaldispersionE =E(p),E E(0)wedefineE-polynomials
0
≡
H(E)(x,t) by the generating function
n
∞ i n pn
e~i(px−(E(p)−E0)t) = H(E)(x,t). (3)
~ n! n
n=0(cid:18) (cid:19)
X
It is equivalent to
H(E)(x,t)=e−~i(H(−i~∂/∂x)−E0)txn (4)
n
so that H(E)(x,t) is a solution of
n
∂
i~ H(E)(x,t)=( E )H(E)(x,t) (5)
∂t n H− 0 n
with the initial value H(E)(x,0) = xn. From commutativity [S,K] = 0, time
n
evolution of the operator K satisfies
∂K
i~ =[ ,K] (6)
∂t H
and has the form K(t) = e−~iHtK(0)e~iHt = e−~iHtxe~iHt. Then as follows,
operator K generates the infinite hierarchy of polynomials according to
KHn(x,t)=Ke−~i(H−E0)txn =e−~i(H−E0)txn+1 =Hn+1(x,t) (7)
1.1 Non-relativistic Schr¨odinger equation
Thenon-relativisticdispersionE(p)=p2/2mimplies the Hamiltonianoperator
~2 ∂2
= (8)
H −2m∂x2
and the Gallilean boost operator
~ ∂
K =x+it . (9)
m∂x
From the generating function
e~i(px−2pm2t) = ∞ ~i n pnn!Hn(S)(x,t) (10)
n=0(cid:18) (cid:19)
X
we have the Schr¨odinger polynomials
Hn(S)(x,t)=e~it2~m2 ∂∂x22xn. (11)
IfH(KF)(x,t)=exp(td2 )xnistheKampedeFerietpolynomial,thenH(S)(x,t)=
n dt2 n
HKF(x, i~ t) or in terms of the Hermit polynomial
2m
i~ n/2 x
H(S)(x,t)= t H . (12)
(cid:18)−2m (cid:19) n −2i~t/m!
p
2
1.2 Semi-relativistic Schr¨odinger equation
The relativistic dispersion E(p)= m2c4+c2p2 implies the Hamiltonian
p ~2 ∂2
=mc2 1 (13)
H − m2c2∂x2
r
and the semi-relativistic boost operator
i~ ∂
K =x+ t ∂x . (14)
m 1 ~2 ∂2
− m2c2∂x2
q
In the non-relativistic limit, when c , it reduces to the Galilean boost (9).
→∞
The generating function in the form of relativistic plane wave
e~i(px−(√m2c4+c2p2−mc2)t) = ∞ i n pnH(SRS)(x,t) (15)
~ n! n
n=0(cid:18) (cid:19)
X
then gives semi-relativistic polynomials
HnSRS(x,t)=e−~imc2t(q1−m~22c2∂∂x22−1)xn. (16)
In the non-relativistic limit HSRS HS. The first three polynomials coincide
n → n
exactly with the Schr¨odinger polynomials HSRS(x,t)=x, H(SRS)(x,t)=x2+
1 2
i~t, H(SRS) =x3+i~3xt, while starting from the fourth one
m 3 m
~ ~2 ~3
H(SRS)(x,t)=x4+i 6x2t 3t2+i 3t (17)
4 m − m2 m3c2
we have relativistic corrections of order 1/c2. For complex valued space coor-
dinate x, as it appears in 2+1 dimensional Chern-Simons theory [4], zeros of
these polynomials describe a motion of point vortices in the plane. Equations
of motion for N vortices are (k = 1,...,N):
i ~ ∂ N 1
x˙ = Res + 1 (18)
k ~ |x=xkH i ∂x x xl!!·
l=1 −
X
1.3 Relativistic Burgers-Schr¨odinger Equation
Using Schr¨odinger’s log Ψ transform [5], Ψ=elnΨ, and identity
∂n ∂ ∂lnΨ n
e−lnΨ elnΨ = + 1 (19)
∂xn ∂x ∂x ·
(cid:18) (cid:19)
the Schr¨odinger equation (1) can be rewritten in the form
∂ ∂ ∂lnΨ
i~ lnΨ= i~ + 1 (20)
∂t H − ∂x ∂x ·
(cid:18) (cid:18) (cid:19)(cid:19)
3
For complex function Ψ=e~iF =eR+~iS we introduce a new complex function
V = i~ ∂ lnΨ = 1 ∂ F, with dimension of velocity. Then F = S i~R, is
− m∂x m∂x −
the complex potential, real and imaginary part of which are the classical and
quantum velocities V = V +iV = 1S i~R . Hence (20) becomes of the
c q m x− m x
complex Hamilton-Jacobi form (quantum Hamilton-Jacobi equation)
∂F ∂
+ i~ +F 1=0 (21)
x
∂t H − ∂x ·
(cid:18) (cid:19)
In the classical (dispersionless) limit ~ 0, the quantum velocity V vanishes
q
→
and complex potential F reduces to the real velocity potential S, playing the
role of Hamilton’s principal function. In this case (21) becomes the classical
Hamilton-Jacobi equation ∂S + ∂S =0. By differentiation of (20) we have
∂t H ∂x
equation for the complex velocity
(cid:0) (cid:1)
∂V ~ ∂ ∂
i~ = i i~ +mV 1 (22)
∂t − m∂x H − ∂x ·
(cid:20) (cid:18) (cid:19) (cid:21)
This equation is the Madelung fluid type representation of the Schr¨odinger
equation (1). In the classical limit it gives the Newton equation m∂Vc =
∂t
∂ [ (mV )] or in the hydrodynamic type form
−∂x H c
∂V ∂V
c + ′(mV ) c =0 (23)
c
∂t H ∂x
whichisjustdifferentiationoftheclassicalHamilton-Jacobiequation. Equation
(23) has implicit general solution as V (x,t)= f(x ′(mV )t), where f is an
c c
−H
arbitraryfunction,anditdevelopsshockatacriticaltimewhenderivative(V )
c x
is blowing up.
1.3.1 Non-relativistic Burgers-Schro¨dinger equation
In this case the Schr¨odinger equation (1) with non-relativistic Hamiltonian (8)
is equivalent to the nonlinear equation for complex velocity V
∂V ~2 ∂2V ∂V
i~ = i~V (24)
∂t −2m ∂x2 − ∂x
whichwe callthe Burgers-Schr¨odingerequation. In terms ofthe realandimag-
inary parts it gives the Madelung fluid with density ρ=eR and velocity V . In
c
the classical limit it reduces to the one real equation for classical velocity V ,
c
namely the ordinary dispersionless Burgers equation.
1.3.2 Semi-relativistic Burgers-Schro¨dinger equation
For Hamiltonian (13), the ”Burgerization”procedure described above gives the
semi-relativistic Burgers-Schr¨odingerequation
2
1∂V ∂ 1 ∂
+c 1+ i~ +mV 1 =0 (25)
c ∂t ∂xs m2c2 − ∂x · 
(cid:18) (cid:19)
 
4
Inthe non-relativisticlimit itreducesto (24)withrelativisticcorrectionsinthe
lowest order as
∂V ~2 ∂2V ∂V 1
i~ = i~V + ~4V (26)
∂t −2m ∂x2 − ∂x 8m3c2 − xxxx
+ 1 im~3(10V V +4VV )+m2~2(12VV2+6V2V )+4(cid:2)im3~V3V (cid:3) (27)
8m3c2 − x xx xxx x xx x
(cid:2) (cid:3)
Intheclassical(dispersionless)limit(25)becomesequationofthehydrodynamic
type
V
c
(V ) + (V ) =0 (28)
c t c x
1+V2/c2
c
In the non-relativistic limit it repduces to the dispersionless Burgers equation
with the lowest relativistic correction
1
(V ) +V (V ) V3(V ) =0 (29)
c t c c x− 2c2 c c x
The general implicit solution of (28) is
V t
c
V (x,t)=f x (30)
c
− 1+Vc2/c2!
p
and it develops shock at a finite time.
1.4 B¨acklund Transformation for Burgers-Schr¨odinger equa-
tion
By the boost transformation of Section 1, from a given solution Ψ of the
1
Schr¨odinger equation (1) we can generate another solution as
∂
Ψ =KΨ = x t ′ i~ Ψ . (31)
2 1 1
− H − ∂x
(cid:20) (cid:18) (cid:19)(cid:21)
Using identity
∂ ∂
Ψ−1G i~ Ψ=G i~ +mV 1 (32)
− ∂x − ∂x ·
(cid:18) (cid:19) (cid:18) (cid:19)
~
for complex velocities V = i lnΨ , (a = 1,2), we obtain the B¨acklund
a − m a
transformation
~ ∂ ∂
V =V i ln x tH′ i~ +mV 1 (33)
2 1
− m∂x − − ∂x ·
(cid:20) (cid:18) (cid:19) (cid:21)
For the non-relativistic quantum mechanics (8) it gives complex B¨acklund
transformation
~ 1 (V ) t
1 x
V =V i − (34)
2 1
− m x V t
1
−
5
for the Burgers-Schr¨odingerequation (24).
For the semi-relativistic quantum mechanics (13) we have the B¨acklund
transformation of the form
~ ∂ 1
V =V i ln x V t (35)
2 1 1
− m∂x  − 1+ 1 ( i~ ∂ +mV )2 
m2c2 − ∂x 1
Itisworthtonotethatintheclassicqallimit~ 0,V V theaboveB¨acklund
c
→ →
transformations reduce to the trivial identity V =V .
c1 c2
2 Integrable General NLS Hierarchy
In previous sections we studied the so called C-integrable relativistic Burgers-
Schr¨odingerequation. Now usingthe AKNS hierarchyfor theNLSequationwe
are going to construct relativistic NLS.
2.1 NLS hierarchy
We consider the Zakharov-Shabatlinear problem
∂ v ip κ2ψ¯ v v
∂x(cid:18) v12 (cid:19)=(cid:18) −ψ2 −2ip (cid:19)(cid:18) v12 (cid:19)=J1(cid:18) v12 (cid:19), (36)
for the space evolution, and the generalized AKNS problem [6]
∂ v iA κ2C¯ v v
∂t v12 = −C − iA v12 =J0 v12 , (37)
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
for the time evolution, where for the real A(x,t,p) and complex C(x,t,p)
functions, determined by the zero-curvature condition, we substitute A =
N
N A(n) p n, C = N C(n) p n. It gives the evolution equation
n=0 −2 N n=0 −2
∂ ψ = ∂ C(0) + 2iA(0)ψ and C(N) = 0, A(N) = a = const.. We fix
PtN x(cid:0) (cid:1) P (cid:0) (cid:1) N
this constant so that a = ( 2)N−1. Then we have the recurrence rela-
N
tions C(n) = 1∂ C(n+1) + A(−n+1)ψ, ∂ A(n) = iκ2(C¯(n)ψ C(n)ψ¯), where
2i x x −
n=0,1,2,...,N 1. Integrating the last equation one has
−
x
A(n) = iκ2 (ψ¯C(n) ψC¯(n)) (38)
− −
Z
Substituting (38) into recursion formula we find
C(n) 1 C(n+1)
= (39)
C¯(n) −2R C¯(n+1)
(cid:18) (cid:19) (cid:18) (cid:19)
where is the matrix integro-differential operator - the recursion operator of
R
the NLS hierarchy [6] -
∂ +2κ2ψ xψ¯ 2κ2ψ xψ
x
−
=iσ (40)
3
R  2κ2ψ¯ xRψ¯ ∂ +2κ2ψR¯ xψ 
x
−
 
R R
6
and σ - the Pauli matrix. Then we get
3
ψ ψ
iσ = N (41)
3 ψ¯ R ψ¯
(cid:18) (cid:19)tN (cid:18) (cid:19)
where t , N = 1,2,3,... is an infinite time hierarchy. In the linear approx-
N
imation, when κ = 0, the recursion operator is just the momentum operator
=iσ ∂ andthe NLS hierarchy(41) becomes the linear Schrodingerhierar-
R0 3∂x
chy
iψ =in∂nψ (42)
tn x
fromSection1. TheMadelungrepresentationforthishierarchy,producedbythe
complex Cole-Hopf transformation, is given by the complex Burgers hierarchy
[4].
Every equation of hierarchy (41) is integrable. The linear problem for the
N-thequationisgivenby the Zakharov-Shabatproblem(36)forthe spacepart
and
∂ v iA κ2C¯ v v
∂tN (cid:18) v12 (cid:19)=(cid:18) −CNN −−iANN (cid:19)(cid:18) v12 (cid:19)=J0N (cid:18) v12 (cid:19), (43)
for the time part. Coefficient functions C can be found conveniently as
N
N
C ψ ψ
N = pN−k k−1 =(pN−1+pN−2 +...+ N−1)
C¯N R ψ¯ R R ψ¯
(cid:18) (cid:19) k=1 (cid:18) (cid:19) (cid:18) (cid:19)
X
(44)
To rewrite this expression in a compact form we introduce notation of the q-
number operator
1+q+q2+...+qN−1 [N] (45)
q
≡
where q is a linear operator. Hence, with operator q /p we have following
≡ R
finite Laurent form in the spectral parameter p
2 N−1
1+ R + R +...+ R [N] (46)
p p p ≡ R/p
(cid:18) (cid:19) (cid:18) (cid:19)
Then we have shortly
C ψ
N =pN−1[N] (47)
C¯ R/p ψ¯
N
(cid:18) (cid:19) (cid:18) (cid:19)
In a similar way
pN x x ψ
A = iκ2pN−1 ψ¯, ψ [N] (48)
N − 2 − − R/p ψ¯
(cid:18)Z Z (cid:19) (cid:18) (cid:19)
Equations (43),(47) and (48) give the time part of the linear problem (the Lax
representation) for the N-th flow of NLS hierarchy (41) in the q-calculus form.
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2.2 General NLS hierarchy equation
For time t determined by the formal series ∂ = ∞ E ∂ where E are
t N=0 N tN N
arbitrary constants, the general NLS hierarchy equation is
P
ψ ψ
iσ = E +E +...+E N +... (49)
3 ψ¯ 0 1R NR ψ¯
(cid:18) (cid:19)t (cid:18) (cid:19)
(cid:0) (cid:1)
Integrability of this equation is associated with the Zakharov-Shabat prob-
lem (36) and the time evolution
∞ iA κ2C¯
J0 = ENJ0N = −C − iA (50)
N=0 (cid:18) − (cid:19)
X
where
∞ ∞
C C ψ
= E N = E pN−1[N] (51)
C¯ N C¯ N R/p ψ¯
N
(cid:18) (cid:19) N=0 (cid:18) (cid:19) N=1 (cid:18) (cid:19)
X X
In the last equation we have used that for N =0, C =0. Then we have
0
∞ 1 ∞ x x C
A= E A = E pN iκ2 ψ¯, ψ (52)
N N −2 N − − C¯
N=0 N=0 (cid:18)Z Z (cid:19)(cid:18) (cid:19)
X X
Theaboveequation(49)givesintegrablenonlinearextensionoflinearSchrd¨inger
equation with generalanalytic dispersion considered in Section 1. Let one con-
siders the classical particle system with the energy-momentum relation E(p)=
E +E p+ E p2 +.... Then the corresponding time-dependent Schr¨odinger
0 1 2
wave equationis (1) where the Hamiltonian operator results from the standard
substitutionformomentump i~ ∂ inthedispersion. Equation(1)together
→− ∂x
with its complex conjugate can be rewritten as
∂ ψ ∂ ψ
i~σ =H i~σ (53)
3∂t ψ¯ − 3∂x ψ¯
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
The momentum operator here is just the recursion operator in the linear ap-
proximation = iσ ∂ . Hence, (53) is the linear Schr¨odinger equation with
R0 3∂x
arbitrary analytic dispersion. The nonlinear integrable extension of this equa-
tion appears as (49), which corresponds to the replacement , (~ = 1),
0
R → R
so that
ψ ψ
iσ =H( ) (54)
3 ψ¯ R ψ¯
(cid:18) (cid:19)t (cid:18) (cid:19)
Fromthis point ofview, the standardsubstitution forclassicalmomentum p
i~ ∂ orequivalentlyp i~σ ∂ = fortheequationinspinorform,giv→es
− ∂x →− 3∂x R0
quantizationinthe formofthe linearSchr¨odingerequation. While substitution
p gives ”nonlinear quantization” and the nonlinear Schr¨odinger hierarchy
→R
equation.
8
The related Lax representation for equation (54) is given by (51), (52). By
definition of q-derivative D(ζ)f(ζ) = f(qζ)−f(ζ) for operator q = /p, we have
q (q−1)ζ R
relation D(p) ζN =[N] pN−1. Then equation (51) can be rewritten as
R/p R/p
∞ ∞
C ψ ψ
= E pN−1[N] = E D(p) pN (55)
C¯ N R/p ψ¯ N R/p ψ¯
(cid:18) (cid:19) N=1 (cid:18) (cid:19) N=1 (cid:18) (cid:19)
X X
or using linearity of q-derivative and analytic dispersion form
∞
C ψ ψ
=D(p) E pN =D(p) E(p) (56)
C¯ R/p N ψ¯ R/p ψ¯
(cid:18) (cid:19) N=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
Due to above definition it gives simple formula
C E( ) E(p) ψ
C¯ = R −p ψ¯ (57)
(cid:18) (cid:19) R− (cid:18) (cid:19)
Then for A we obtain
1 x x E( ) E(p) ψ
A=−2E(p)−iκ2 ψ¯,− ψ R −p ψ¯ (58)
(cid:18)Z Z (cid:19) R− (cid:18) (cid:19)
Equations (57),(58) give the Lax representation of the general integrable NLS
hierarchy model (54) in a simple and compact form. It is worth to note that
specialformofthedispersionE =E(p)isfixedbyphysicalproblem. Inthenext
Sectionwewilldiscusstherelativisticformofthisdispersionandcorresponding
semi-relativistic NLS equation.
3 Semi-relativistic NLS
In Section 1.2 we have considered the relativistic dispersion relation E(p) =
m2c4+p2c2. Thismaybeusedtoconstructa“semi-relativistic”Schr¨odinger
equationwithHamiltonian(13). Then,combiningtwocomplexconjugateequa-
p
tions together, we have
2
ψ 1 ∂ ψ
iσ =mc2 1+ iσ (59)
3 ψ¯ s m2c2 3∂x ψ¯
(cid:18) (cid:19)t (cid:18) (cid:19) (cid:18) (cid:19)
We liketoemphasizethatifψ describesthe relativisticparticleforwardintime
andwithpositiveenergy,ψ¯correspondstothebackwardtimeortothenegative
energy. From this point of view equation (59) is complete since includes both
states.
Following the generalprocedure described in previous Section one may pro-
ceed further: by replacing the derivative operator = iσ ∂ corresponding
R0 3∂x
to linear momenta p with the full recursion operator (40), one obtains an
R
integrable relativistic nonlinear Schr¨odinger equation
9
ψ 1 ψ
iσ =mc2 1+ 2 (60)
3 ψ¯ m2c2R ψ¯
(cid:18) (cid:19)t r (cid:18) (cid:19)
where the square root operator has meaning of the formal power series so that
ψ 1 1 1 ψ
iσ =mc2 1+ 2 4+ 6 ...
3 ψ¯ 2m2c2R − 8m4c4R 16m6c6R ± ψ¯
(cid:18) (cid:19)t (cid:18) (cid:19)(cid:18) (cid:19)
(61)
For the above relativistic dispersion and equation (60), we have the next linear
problem
∂ v ip κ2ψ¯ v
1 = −2 − 1 , (62)
∂x(cid:18) v2 (cid:19) (cid:18) ψ 2ip (cid:19)(cid:18) v2 (cid:19)
∂ v iA κ2C¯ v
1 = − − 1 , (63)
∂t v2 C iA v2
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)
where
C √m2c4+ 2c2 m2c4+p2c2 ψ
C¯ = R − p ψ¯ (64)
(cid:18) (cid:19) R−p (cid:18) (cid:19)
1 x x √m2c4+ 2c2 m2c4+p2c2 ψ
A= m2c4+p2c2 iκ2 ψ¯, ψ R −
−2 − − p ψ¯
(cid:18)Z Z (cid:19) R−p (cid:18) (cid:19)
p (65)
and the spectral parameter p has meaning of the classical momentum. The
model (60), is an integrable nonlinear Schrodinger equation with relativistic
dispersion:
1 ∂2
iψ =mc2 1 ψ+F(ψ) (66)
t − m2c2∂x2
r
where the nonlinearity expanded in 1/c2 is the infinite sum
1
F(ψ)= [ 2κ2 ψ 2ψ]
2m − | |
1 1
[2κ2(2ψ 2ψ+4ψ 2ψ +ψ¯ ψ2+3ψ¯ψ2)+6κ4 ψ 4ψ]+O( )
−8m3c2 | x| | | xx xx x | | c4
It is interesting to note that if we expand also the dispersion part in 1/c2,
thenateveryorderof1/c2wegetanintegrablesystem. Itmeansthatweobtain
integrable relativistic corrections to the NLS equation at any order. From the
knownrelativisticintegrablemodels likethe Sine-Gordonorthe Liouvilleequa-
tions, neither one has this property. Finally we note that nonlinear relativistic
equationsconsideredinthispaperaredistinctfromthoseobtainedin[1]-[3]and
references therein. They might be useful in analyzing relativistic corrections to
solitons,Bose-Einsteincondensates or other condensed matter systems with an
effective equation of relativistic form.
10