closed-form solutions of a reduced form of the gross-pitaevskii equation
DESCRIPTION
The Riccati equation method with variable expansion coefficients is used to find explicit solutions to a nonlinear equation appearing in the reduction of the Gross-Pitaevskii equation. This equation appears in two articles by Dalfovo, Pitaevskii and Stringari [Phys. Rev A 54 (1996) 4213] and by Lundh, Pethick and Smith [Phys Rev A55 (1997) 2126] and has the form U''(x)=x*U(x)+U(x)^3 . The solutions are expressed in terms of the Airy functions.TRANSCRIPT
Closed-form solutions of a reduced form
of the Gross-Pitaevskii equation
Solomon M. Antoniou
SKEMSYS Scientific Knowledge Engineering
and Management Systems
Corinthos 20100, Greece [email protected]
New Version: 04/08/2014
Abstract
The Riccati equation method with variable expansion coefficients is used to find
explicit solutions to a nonlinear equation appearing in the reduction of the Gross-
Pitaevskii equation. This equation appears in two articles by Dalfovo, Pitaevskii
and Stringari [Phys. Rev A 54 (1996) 4213] and by Lundh, Pethick and Smith
[Phys Rev A55 (1997) 2126] and has the form )x(U)x(Ux)x(U 3+⋅=′′ . The
solutions are expressed in terms of the Airy functions.
Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, extended
Riccati equation method, nonlinear differential equations, closed-form solutions,
Painlevé P-II equation, Airy functions.
2
1. Introduction.
Bose-Einstein condensate is a new state of matter, predicted theoretically by Bose
and Einstein in 1925. The experimental verification of the theory was made
possible only in 1995 (Anderson et al [2]) and this experimental achievement won
the Nobel Prize for Physics in 2001. Bose-Einstein condensate is a dilute gas of
bosons cooled to temperature very close to the absolute zero. This is a new form of
matter with unusual properties. The quantitative description of the Bose-Einstein
state is implemented through the Gross-Pitaevskii equation (Gross [9], Pitaevskii
[11], Pitaevskii and Stringari [12]). This is a nonlinear, Schrödinger-like equation,
which determines the ground state 0Ψ of the Bose-Einstein gas. In this paper we
solve one form of the Gross-Pitaevskii equation. This equation appears in Dalfovo,
Pitaevskii and Stringari [6] and in Lundh, Pethick and Smith [10].
The paper is organized as follows: In Section 2 we describe the extended Riccati
equation method with variable expansion coefficients. This method was
introduced in Antoniou ([3] and [4]) and applied successfully in two cases of
nonlinear PDEs, the Burgers and KdV equations. In Section 3 we solve the
equation )x(U)x(Ux)x(U 3+⋅=′′ . We find two families of solutions, expressed
in terms of the Airy functions. In Section 4 we consider the −′ )G/G( expansion
method with variable expansion coefficients. Using this last method we establish
two third order ordinary differential equations which when solved, provide more
families of solutions.
2. The Method.
We suppose that a nonlinear ordinary differential equation
0),u,u,u,x(F xxx =L (2.1)
with unknown function )x(u admits solution expressed in the form
∑∑==
+=n
1kkk
n
0k
kk
Y
bYa)x(u (2.2)
3
where all the expansion coefficients depend on the variable x,
)x(aa kk ≡ , )x(bb kk ≡ for every n,,2,1,0k L=
The function )x(YY ≡ satisfies Riccati’s equation
2YBA)x(Y ⋅+=′ (2.3)
where the coefficients A and B depend on the variable x too.
In solving the nonlinear ODE (2.1), we consider the expansion (2.2) and then we
balance the nonlinear term with the highest derivative term of the function )x(u
which determines n (the number of the expansion terms). Equating the
coefficients of the different powers of the function )x(Y to zero, we can
determine the various expansion coefficients )x(ak , )x(bk and the functions
)x(A , )x(B . We finally solve Riccati's equation and then find the solutions of the
equation considered.
3. The Gross-Pitaevskii equation )x(U)x(Ux)x(U 3+⋅=′′ and its
solutions.
We start considering the Gross-Pitaevskii equation
)r()r(|)r(|m
a4)r(V
m22
2
ext2
2rrrhrh ψµ=ψ
ψπ++∇−
where a is the −s wave scattering length and µ is the chemical potential.
In case of a spherical trap, the Gross-Pitaevskii equation takes the form
0m
a4])r(V[
drd
rmdr
dm2
32
ext
2
2
22=ψπ+ψµ−+ψ−ψ− hhh
(3.1)
We suppose that the boundary of the system is determined by the equation
)R(Vext=µ . In this case we can consider the approximation
)Rr(OF)Rr()r(Vext −+−=µ−
4
where F is the modulus of the external attractive force. Therefore equation (3.1)
becomes
0m
a4F)rR(
dr
dm2
32
2
22=ψπ+ψ−+ψ− hh
(3.2)
Introducing the change of variables
δ−= Rr
x , 3/1
2F
m2−
=δh
and )x(U)a8(
1)r(
2/1πδ=ψ
equation (3.2) becomes an ordinary nonlinear differential equation
)x(U)x(Ux)x(U 3+⋅=′′ (3.3)
This equation appears in Dalfovo, Pitaevskii and Stringari ([6], equation (10)) and
in Lundh, Pethick and Smith ([10], equation (3)). It is a remarkable fact that
equation (3.3) is the Painleve PII equation. This equation was recently solved
(Antoniou [5]) in its general form wz)z(w2)z(w 3 ⋅+=′′ .
We consider the extended Riccati equation method in solving equation (3.3). In
this case we consider the expansion ∑∑==
+=n
1kkk
n
0k
kk
Y
bYa)x(U and balance the
second order derivative term with the second order nonlinear term of (3.3). We
then find 1n = and thus
Y
bYaa)x(U 1
10 ++= (3.4)
where all the coefficients 0a , 1a and 1b depend on x, and Y satisfies Riccati’s
equation 2BYAY +=′ . The prime will always denote derivative with respect to
the variable x. From equation (3.4) we obtain, taking into account 2BYAY +=′
2
2112
110Y
)BYA(b
Y
b)BYA(aYaa)x(U
+−′
+++′+′=′ (3.5)
)BYA(YB2YBA(a)BYA(a2Yaa)x(U 221
2110 ++′+′++′+′′+′′=′′
5
2
21
21
211
Y
)BYA(BYb2)YBA(b)BYA(b2
Y
b ++′+′++′−
′′+
3
221
Y
)BYA(b2 ++ (3.6)
Therefore equation (3.3), under the substitution (3.4) and (3.6), becomes
)BYA(YB2YBA(a)BYA(a2Yaa 221
2110 ++′+′++′+′′+′′
2
21
21
211
Y
)BYA(BYb2)YBA(b)BYA(b2
Y
b ++′+′++′−
′′+
31
101
103
221
Y
bYaa
Y
bYaax
Y
)BYA(b2
+++
++⋅=++ (3.7)
Upon expanding and equating the coefficients of Y to zero in the above equation,
we obtain a system of seven differential equations from which we can determine
the various expansion coefficients. We obtain
coefficient of 3Y :
0aBa2 31
21 =− (3.8)
coefficient of 2Y :
0Ba2Baaa3 11210 =′+′+− (3.9)
coefficient of Y:
0aaa3ba3axABa2 11201
2111 =′′+−−− (3.10)
coefficient of 0Y :
0Aa2BbAaBb2baa6aaxa 11111103000 =′+′−′+′−−−−′′ (3.11)
coefficient of 1Y − :
0ABb2ba3ba3bxb 12111
2011 =+−−−′′ (3.12)
coefficient of 2Y − :
0Abba3Ab2 12101 =′−−′− (3.13)
6
coefficient of 3Y − :
0bAb2 31
21 =− (3.14)
We are to solve the system of equations (3.8)-(3.14), supplemented by Riccati's
equation 2BYAY +=′ .
From equations (3.8) and (3.14), ignoring the trivial solutions, we obtain
B2a1 ±= and A2b1 ±= (3.15)
respectively.
3.I. Case I. First Solution.
We first consider the case
B2a1 = and A2b1 = (3.16)
We then obtain from (3.9) and (3.13)
B
B
2
2a0
′= and
A
A
2
2a0
′−= (3.17)
respectively. Equating the two different expressions of 0a and integration, we find
that
pAB = (3.18)
where p is a constant.
From equation (3.10), we get
0)xp4(B
B
2
3
B
B 2
=+−
′−
′′ (3.19)
From equation (3.12) we get
0)xp4(AA
23
AA 2
=+−
′−
′′ (3.20)
From equation (3.11) we get
0BB
)xp8(BB
21
BB 3
=
′+−
′−
″
′ (3.21)
We first solve equation (3.19). Under the substitution
7
BB
F′
= (3.22)
equation (3.19) transforms into
)xp4(F21
F 2 ++=′ (3.23)
which is a Riccati differential equation. Under the standard substitution
uu
2F′
−= (3.24)
Riccati's equation becomes
0)x(u2
xp4)x(u =⋅++′′ (3.25)
The previous equation can be transformed into the Airy equation. In fact, under
the substitution µ+λ= zx (λ and µ are parameters to be determined) equation
(3.25) takes on the form
0)z(u2
)p4(z
2dz
)z(ud 23
2
2=
+µλ+λ+ (3.26)
The choice 23 −=λ and p4−=µ transforms (3.26) into
0)z(uzdz
)z(ud2
2=− (3.27)
which is the Airy differential equation with general equation
)z(BiC)z(AiC)z(u 21 += (3.28)
where )z(Ai and )z(Bi are Airy's functions of the first and second kind
respectively (see for example Abramowitz and Stegun [1]). Going back to the
original variable, i.e. taking into account z)2(p4x 3/1−=+ , we obtain the general
solution of equation (3.25):
−++
−+=
3/123/11)2(
p4xBiC
)2(
p4xAiC)x(u (3.29)
8
We now have to determine the function Y which satisfies Riccati's equation
2BYAY +=′ . Under the substitution
vv
B1
Y′
⋅−= (3.30)
Riccati's equation becomes
0v)AB(vBB
v =+′
′−′′ (3.31)
Since uu
2BB ′
−=′
and pBA = , equation (3.31) becomes
0uvp2vu2vu =+′′+′′
which can be written as
0v)uup2()vu( =′′−+′′⋅ (3.32)
The substitution
vuZ ⋅= (3.33)
transforms (3.32) into
0Zuu
p2Z =
′′−+′′ (3.34)
Since x21
p2uu −−=
′′ (notice that this relation comes from (3.25)), equation
(3.34) takes on the form
0Zx21
p4Z =
++′′ (3.35)
The above equation can be transformed into the Airy's equation – the same way
(3.25) was transformed into (3.27) – and admits general solution
−++
−+=
3/123/11)2(
p8xBiC
~
)2(
p8xAiC
~Z (3.36)
Integrating uu
2BB ′
−=′
we obtain
9
)x(u
KB
2= (3.37)
where K is a constant. Since uu
ZZ
vv ′
−′
=′
(this relation comes by differentiation
of u/Zv = ), we obtain from (3.30), using also (3.37), that
′−
′−=
u
u
Z
Z
K
uY
2 (3.38)
where Z and u are given by (3.36) and (3.29) respectively.
So far we have not taken into account equations (3.20) and (3.21). It is obvious
that not every solution of (3.19) satisfies both (3.20) and (3.21). We thus have to
find the range of values the parameters and the constants should attain in order to
have compatible equations. The coefficients A and B of Riccati's equations are
connected through the relation 0BB
AA =
′+
′ and satisfy equations (3.19) and (3.20)
respectively. We thus have to examine the compatibility condition between (3.19)
and (3.20) first, taking into account 0BB
AA =
′+
′. We state and prove the following
Lemma. If 0BB
AA =
′+
′ then
2
BB
2BB
AA
′=
′′+
′′.
Proof. We let AA
H′
= . Since 0FH =+ , we also have FH ′−=′ and then
22
BB
FAA
HAA
′+′−=
′+′=
′′ and
2
BB
FBB
′+′=
′′. Adding the last two
equations, we obtain 2
BB
2BB
AA
′=
′′+
′′ and the Lemma is proved. ■
10
Adding now equations (3.19) and (3.20) and taking into account the previous
Lemma, we obtain the equation 0)xp4(2BB 2
=+−
′− from which we obtain in
view of uu
2BB ′
−=′
,
02x
p2uu 2
=
++
′ (3.39)
The above equation is the compatibility condition between (3.19) and (3.20) and
should be satisfied for every x. We finally consider equation (3.21). This equation
takes the form
F)p8x(F21
F 3 +=−′′ (3.40)
where F is defined in (3.22). This equation should be combined with (3.23).
From (3.23) multiplying by F we obtain F)xp4(FFF21 3 ++′−=− and because
of that, equation (3.40) takes on the form Fp4FFF =′−′′ . Differentiating (3.23)
we get 1FFF =′−′′ . We thus obtain the equation 1Fp4 = . Since uu
2F′
−= , we
derive the compatibility condition p8
1uu −=
′, i.e.
0uup8 =+′ (3.41)
This last equation should hold for every x.
Equations (3.39) and (3.41) should also be compatible each other. Equations
(3.39) and (3.41) are considered in Appendix A. In that Appendix, expanding the
function )x(u given by (3.29), we find the conditions between the parameters and
the various constants in order equations (3.39) and (3.41) should be true for every
value of x. According to the results of Appendix A, the compatibility equations
(3.39) and (3.41) lead to the same conditions
0);C,C(X 21 =ω and 0);C,C(Z 21 =ω (3.42)
11
where );C,C(X 21 ω and );C,C(Z 21 ω are defined by
)(BiC)(AiC);C,C(X 2121 ω+ω=ω (3.43)
)(iBC)(iAC);C,C(Z 2121 ω′+ω′=ω (3.44)
p)3i1(2 3/2 −=ω (3.45)
and the prime denotes the usual derivative
ω==ω′x
)x(Aidxd
)(iA and ω==ω′x
)x(Bidxd
)(iB
The two equations (3.42) hold simultaneously, in view of (3.43) and (3.44), if
)(Ai)(Bi
C
C
2
1
ωω−= and
)(iA)(iB
C
C
2
1
ω′ω′
−= (3.46)
Equating the two different expressions of the ratio 21 C/C , we arrive at the
condition
0)(Bi)(iA)(iB)(Ai =ω⋅ω′−ω′⋅ω (3.47)
The above condition determines the constant p. On the other hand, because of
(3.46), the solution (3.29) becomes
−+ω−
−+ω=
3/13/1 )2(
p4xBi)(Ai
)2(
p4xAi)(BiC)x(u (3.48)
We thus obtain the following
Conclusion I. The solution of the nonlinear equation )x(U)x(Ux)x(U 3+⋅=′′ is
given by Y
bYaa)x(U 1
10 ++= where BB
22
a0′
= , B2a1 = , A2b1 = with
pBA = , )x(u
K)x(B
2= and
′−
′−=
u
u
Z
Z
K
uY
2, where Z and u are given by
(3.36) and (3.48) respectively. Therefore )x(U is given by
1
ZZ
uu
p2ZZ
2)x(U−
′−
′+
′−= (3.49)
12
where p is determined by (3.47), and Z, u are given by (3.36) and (3.48)
respectively.
3.II. Case II. Second Solution.
We next consider the case
B2a1 −= and A2b1 −= (3.50)
We then obtain from (3.9) and (3.13)
BB
22
a0′
−= and AA
22
a0′
= (3.51)
respectively. Equating the two different expressions of 0a , we find that
pAB = (3.52)
where p is a constant.
From equation (3.10), we get
0)xp4(BB
23
BB 2
=+−
′−
′′ (3.53)
From equation (3.12) we get
0)xp4(A
A
2
3
A
A 2
=+−
′−
′′ (3.54)
From equation (3.11) we get
0BB
)xp8(BB
21
BB 3
=
′+−
′−
″
′ (3.55)
Using the same reasoning as in Case I, we obtain the following
Conclusion II. The solution of the nonlinear equation )x(U)x(Ux)x(U 3+⋅=′′ is
given by Y
bYaa)x(U 1
10 ++= where B
B
2
2a0
′−= , B2a1 −= , A2b1 −=
with pBA = , )x(u
K)x(B
2= and
′−
′−=
u
u
Z
Z
K
uY
2, where Z and u are given
by (3.36) and (3.48) respectively. Therefore )x(U is given by
13
1
ZZ
uu
p2ZZ
2)x(U−
′−
′+
′= (3.56)
where p is determined by (3.47), and Z, u are given by (3.36) and (3.48)
respectively.
4. The −
′GG expansion method with variable expansion coefficients.
We consider that equation (3.3) admits a solution of the form
′+=
)x(G)x(G
)x(a)x(a)x(U 10 (4.1)
Upon substituting (4.1) into (3.3) we obtain an equation, which when arranged
properly, can take the form
]Gaa3Ga2GaGaxGa[)x(G
1)aaxa( 2
0111113000 ′−′′′+′′′+′−′′′+−−′′
])G(aa3GGa3)G(a2[)x(G
1 20
211
212
′−′′′−′′−+
0])G(a2)G(a[)x(G
1 31
3313
=′+′−+ (4.2)
Equating to zero all the coefficients of )x(G in the above equation, we obtain the
following system of ordinary differential equations
0aaxa 3000 =−−′′ (4.3)
0Gaa3Ga2GaGaxGa 2011111 =′−′′′+′′′+′−′′′ (4.4)
0)G(aa3GGa3)G(a2 20
211
21 =′−′′′−′′− (4.5)
0)G(a2)G(a 31
331 =′+′− (4.6)
Equation (4.3) is essentially equation (3.3) and thus admits all the solutions found
in Section 4. From equation (4.6), ignoring the trivial solution, we obtain that
2a1 ±= (4.7)
14
From equation (4.4), taking into account the above values of 1a , we derive the
equation
xa3GG 2
0 +=′′′′
(4.8)
From equation (4.5), we obtain
0a2GG −=
′′′
for 2a1 = and 0a2G
G =′′′
for 2a1 −= (4.9)
We thus derive the following solutions, by dividing (4.8) by (4.9):
Solution I. For 2a1 = , the function )x(G is determined as solution of the
equation 0
20
a
xa3
22
GG +⋅−=
′′′′′
where 0a is any solution of equation (4.3).
Solution II. For 2a1 −= , the function )x(G is determined as solution of the
equation 0
20
a
xa3
22
GG +⋅=
′′′′′
where 0a is any solution of equation (4.3).
In both the above solutions, 0a is given by (3.49) and (3.56) (simply substitute
)x(U by 0a ). This method pressuposes the Riccati method with variable
expansion coefficients and the final solution might require supercomputing
facilities with symbolic capabilities. A remarkable feature of the
−′ )G/G( expansion with variable expansion coefficients is that a repeated use of
the method in determining 0a , leads to a proliferation of solutions, a rather unique
feature of the Riccati – −′ )G/G( expansion.
Appendix A. In this Appendix we find the conditions the various parameters
and the constants should satisfy so as equations (3.39) and (3.41) should be true
for every value of the parameter x.
We first consider equation (3.39) which can be written as a couple of equations
15
0u2x
p2iu =++′ (A.1)
and
0u2x
p2iu =+−′ (A.2)
where )x(u is given by (3.29).
Upon expanding )x(u in power series, we find that (A.1) can be written as
);C,C(Z)3i1(4
2);C,C(Xp2i 21
3/2
21 ω−+ω
x)};C,C(Z)i1(p24);C,C(X)p162i({p8
121
6/121
2/3 ω+⋅+ω−+
)3i()p2[(8);C,C(X)p2128ip322i({p128
1 6/121
32/32/3
++ω++−+
221
3/22/5 x)};C,C(Z]2p)13i(4 ω−+
+ω−++ );C,C(X)pi2640i23p2048({p3072
121
32/92/5
)i3(p2512)13i(2p128[ 46/13/22/5 +⋅−−+
321
6/1 x)};C,C(Z]p)3i(212 ω+⋅−
)x(O 4+ (A.3)
where we have introduced the notation
)(BiC)(AiC);C,C(X 2121 ω+ω=ω (A.4)
)(iBC)(iAC);C,C(Z 2121 ω′+ω′=ω (A.5)
p)3i1(2 3/2 −=ω (A.6)
and the prime denotes the usual derivative
ω==ω′x
)x(Aidxd
)(iA and ω==ω′x
)x(Bidxd
)(iB
Similarly expanding )x(u in power series, we find that (A.2) can be written as
16
);C,C(Z)3i1(4
2);C,C(Xp2i 21
3/2
21 ω−+ω−
x)};C,C(Z)i1(p24);C,C(X)p162i({p8
121
6/121
2/3 ω+⋅+ω+−
)3i()p2([8);C,C(X)p2128ip322i({p128
1 6/121
32/32/3
+−+ω+−+
221
3/22/5 x)};C,C(Z]2p)13i(4 ω−+
+ω+−+ );C,C(X)pi2640i23p2048({p3072
121
32/92/5
)i3(p2512)13i(2p128[ 46/13/22/5 +⋅+−+
321
6/1 x)};C,C(Z]p)3i(212 ω+⋅+
)x(O 4+ (A.7)
Expanding (3.41) we obtain similarly
+ω−⋅+ω });C,C(Z)3i1(p22);C,C(X{ 213/2
21
+
ω⋅−+ω⋅−+ x);C,C(Z)3i1(4
2);C,C(Xp16 21
3/2
212
+ω⋅−⋅+ω⋅−+ 221
23/221 x});C,C(Z)13i(p22);C,C(Xp3{
+
ω⋅−⋅−ω⋅
−+ 3
21
3/2
21
3x);C,C(Z)3i1(p
1225
);C,C(X121
3p16
0)x(O 4 =+ (A.8)
In all the higher order expansion terms, there appears the same linear combination
of the quantities );C,C(X 21 ω and );C,C(Z 21 ω . Therefore the compatibility
conditions (3.39) and (3.41) are true for every x , if and only if
0);C,C(X 21 =ω and 0);C,C(Z 21 =ω (A.9)
The two equations (A.9) hold simultaneously if
17
)(Ai)(Bi
C
C
2
1
ωω−= and
)(iA)(iB
C
C
2
1
ω′ω′
−= (A.10)
Equating the two different expressions of the ratio 21 C/C , we arrive at the
condition
0)(Bi)(iA)(iB)(Ai =ω⋅ω′−ω′⋅ω (A.11)
The above condition determines the constant p.
Appendix B.
In this Appendix we consider the two cases ( B2a1 = , A2b1 −= ) and
( B2a1 −= , A2b1 = ). We show that in these two cases we get incompatible
equations.
Case III. We first consider the case
B2a1 = and A2b1 −= (B.1)
We then obtain from (3.9) and (3.13)
BB
22
a0′
= and AA
22
a0′
= (B.2)
respectively. Equating the two different expressions of 0a and integration, we find
that
BsA 2−= (B.3)
where s is a real constant. We do not consider the case BsA 2= , since in that
case Riccati's equation gives complex-valued solutions.
From equation (3.10), we get
0xBs8BB
23
BB 22
2
=−−
′−
′′ (B.4)
From equation (3.12) we get
0xBs8A
A
2
3
A
A 222
=−−
′−
′′ (B.5)
From equation (3.11) we get
18
0)B(s12BB
xBB
21
BB 22
3
=′−
′−
′−
″
′ (B.6)
Equation (B.4) under the substitution
)x(u
1B
2= (B.7)
takes on the form
)x(u
s4)x(u
2x
)x(u3
2−=+′′ (B.8)
which is Ermakov's equation (Ermakov [7]). The equation 0)x(u2x
)x(u =+′′
admits two linearly independent solutions,
− 3/1)2(
xAi and
− 3/1)2(
xBi .
Therefore, using the standard procedure, the general solution of (B.8) is given by
−
++−×
−= ∫
2
2
3/1
122
2
3/12
1
)2(
xAi
dxCCs4
)2(
xAi)x(uC (B.9)
and
−
++−×
−= ∫
2
2
3/1
122
2
3/12
1
)2(
xBi
dxCCs4
)2(
xBi)x(uC (B.10)
Every solution of the equation (B.4) found previously has to be substituted in
(B.6) and thus obtain a compatibility condition. Despite the fact that (B.4) admits a
19
closed-form solution, equations (B.4) and (B.6) are incompatible. The proof goes
as follows: Equations (B.4) and (B.6) can be written in terms of BB
G′
≡ as
xBs8G21
G 222 +=−′ (B.11)
and
0)B(s12GxG21
G 223 =′−−−′′ (B.12)
respectively. Multiplying equation (B.11) by G we derive the equation
GzBGs8GGG21 223 ++′−=− (B.13)
Combining (B.12) with (B.13) we get
0)B(s12BGs8GGG 2222 =′−+′−′′ (B.14)
Differentiating (B.11) with respect to x we obtain
1)B(s8GGG 22 +′=′−′′ (B.15)
Equations (B.14) and (B.15) give
0)B(s12BGs81)B(s8 222222 =′−++′
which is equivalent to the quite remarkable result 01= . We have thus proved that
equations (B.4) and (B.6) are incompatible.
Case IV. We next consider the case
B2a1 −= and A2b1 = (B.16)
Using the same reasoning as before, we obtain again incompatible equations.
20
References
[1] M. Abramowitz and I. A. Stegun: "Handbook of Mathematical
Functions". Dover 1972
[2] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and
E. A. Cornell: "Observation of Bose-Einstein Condensation in a Dilute
Atomic Vapor". Science (New Series) 269 (1995) 198-201.
[3] S. Antoniou: “The Riccati equation method with variable expansion
coefficients. I. Solving the Burgers equation”. submitted for publication
[4] S. Antoniou: “The Riccati equation method with variable expansion
coefficients. II. Solving the KdV equation”. submitted for publication
[5] S. Antoniou: “Some General Solutions of the Painleve P-II Equation".
submitted for publication
[6] F. Dalfovo, L. Pitaevskii and S. Stringari: "Order parameter at the
boundary of a trapped Bose gas". Phys. Rev. A 54 (1996) 4213-4217
[7] V. P. Ermakov: “Second-Order Differential Equations: Conditions of
Complete Integrability”. Appl. Anal. Discr. Math. 2 (2008) 123-145
Translation from the original Russian article:
Universitetskiye Izvestiya Kiev No. 9 (1880) 1-25
[8] A. Griffin, D. W. Snoke and S. Stringari: "Bose Einstein Condensation"
Cambridge University Press 1995
[9] E. P. Gross: "Structure of a quantized vortex in boson systems"
Nuovo Cimento 20 (1961) 454-457
[10] E. Lundh, C. J. Pethick and H. Smith: "Zero-temperature properties of a
trapped Bose-condensed gas: Beyond the Thomas-Fermi approximation".
Phys. Rev. A 55 (1997) 2126-2131
[11] L. P. Pitaevskii: "Vortex Lines in an Imperfect Bose Gas".
Zh. Eksp. Teor. Fiz. 40 (1961) 646 [Sov. Phys. JETP 13 (1961) 451-454]
21
[12] L. Pitaevskii and S. Stringari: "Bose-Einstein Condensation".
Oxford University Press 2003