EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1750-1759
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Analytical solution of the Ginzburg-Landau equation

Yanick Alain Servais Wellot1,∗, Gires Dimitri Nkaya2

1 Mathematics, Ecole Normale Supérieure, Université Marien Ngouabi, Brazzaville, Congo
2 Mathematics, Faculté des Sciences et Technique, Université Marien Ngouabi, Brazzaville,
Congo

Abstract. In this paper, we will construct the solution of the Landau-Ginzburg equation by the
Adomian decomposition method. This method avoids linearization of space and discretization of
time, it often gives a good approximation of the exact solution.

2020 Mathematics Subject Classifications: 47H14, 34G20, 47J25, 65J15

Key Words and Phrases: Adomian method, Ginzburg-Landau equation.

1. Introduction

The Landau equation, also called the Fokker-Planck-Landau equation, is a nonlinear
partial differential equation that describes the motion of particles in a plasma. This
equation was obtained by Lev Davidovitch Landau in 1936 from the Bolzmann equation
[3, 4].

In 1950, Ginzburg and Landau proposed an extension of the Landau energy to describe
the superconductor in the presence of a magnetic field. In this article, we are interested
in the difference in these two models try to describe the field applied to systems of partial
differential equations with initial conditions. Then we apply the Adomian Decomposition
Method [1, 2], an analytical method that allows us to obtain an exact solution when it
exists without space linearization and time discretization. However, the existence and
uniqueness of solution of the Ginzburg-Landau equation has been proved in [5–7].
In the mathematical and physical context, explanations are being made available to the
research community.
During the last two decades, the mathematics of superconductivity has been the subject
of intense activity [8, 10, 13, 14].

The Ginzburg-Landau function is a commonly used model to describe the behavior of
a superconductor involving, a wave function (called order parameter) and a vector field
(called magnetic potential), both defined on an open set [9].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4551

Email addresses: yanick.wellot@umng.cg (Y.A.S. Wellot), mail:giresnkaya@gmail.com (G. D. Nkaya)

https://www.ejpam.com 1750 © 2022 EJPAM All rights reserved.



Y. A. S. Wellot, G. D. Nkaya / Eur. J. Pure Appl. Math, 15 (4) (2022), 1750-1759 1751

2. Proposition

On the basis of a suitable choice of physical data the model can be written in a
mathematical way in the form [11]:

∂u(x, t)

∂t
= (1 + iα)

∂2u(x, t)

∂x2
+ γu(x, t)− (1 + ib)|u(x, t)|2nu(x, t)− (1− 4i)|u(x, t)|4nu(x, t)

n ⩾ 1, i2 = −1, α, γ, b are real

u(x, 0) = f(x).

(1)

3. About Adomian Decomposition Method

The Adomian Decomposition Method allows to solve functional problems of different
types: algebraic equations, differential, integral, integro-differential and partial differential
equations. It was introduced in the 1980s by Professor , GEORGES ADOMIAN ( 1922
- 1996).
It is a decomposition method which consists in seeking the solution in the form of a con-
vergent series .
It has the advantage of not linearizing and discretizing the equations. This method allows
to determine the exact solution of the problem if it exists or to give an approximation
of the solution of the problem by preserving the physical properties of the studied phe-
nomenon.

Let us solve the functional equation :

Au = h, (2)

where A is an operator of a real Hilbert space H, u ∈ H an unknown function and h ∈ H
a known function.

The principle of the Adomian Decomposition Method consists in decomposing the
operator A into two parts : one linear and the other non-linear:

A = L+R︸ ︷︷ ︸
linear

+ N︸︷︷︸
nonlinear

, (3)

where L is assumed to be invertible in the Adomian sense. A in (2) leads to

Lu+Ru+Nu = h ; (4)

By applying L−1 to (4), we obtain the canonical form of Adomian:

u = θ + L−1h − L−1Ru− L−1Nu, (5)



Y. A. S. Wellot, G. D. Nkaya / Eur. J. Pure Appl. Math, 15 (4) (2022), 1750-1759 1752

where θ verifies the relationship Lθ = 0 ( θ is the integration constant if L is a differential
operator).

As for the Adomian algorithm, it is obtained by assuming that the solution of (2)
has the form of a series

u =
∞∑
n=0

un. (6)

The nonlinear operator also has the form of a series

Nu =

∞∑
n=0

An. (7)

An are Adomian polynomials. That are defined by the formula [5]:

An =
1

n!

[
dn

dλn
N

(
+∞∑
i=0

λiui

)]
λ=0

, n = 0, 1, 2, ... (8)

λ is a real parameter introduced by convenience then we obtain

∞∑
n=0

un = θ + L−1h − L−1

[
R

( ∞∑
n=0

un

)]
− L−1

[ ∞∑
n=0

An

]
. (9)

Assuming that the
∞∑
n=0

un series and
∞∑
n=0

An{
u0 = θ + L−1(h )
un+1 = −L−1 [R (un)]− L−1 (An) ,∀n ≥ 0

(10)

Equation (10) allows to compute all un recursively, then the analytical solution of (2) is
defined by the sum of un:

u =

∞∑
n=0

un (11)

If

φn =

n−1∑
n=0

un

is the truncated series, and if

∞∑
n=0

un is convergent, we have :

u = lim
n→+∞

φn (12)



Y. A. S. Wellot, G. D. Nkaya / Eur. J. Pure Appl. Math, 15 (4) (2022), 1750-1759 1753

4. Application

4.1. Problem 1

We consider the following initial value problem [11]:


∂u(x, t)

∂t
− (1 + i)

∂2u(x, t)

∂x2
+ (1 + 2i)|u(x, t)|2u(x, t)− 3u(x, t) + (1− 4i)|u(x, t)|4u(x, t) = 0

u(x, 0) = eix,

(13)

with u(x, t) a complex function.
Notation:
|u|2 = uu; ⇒ |u|2u = u2u

|u|4 = u2u2 ⇒ |u|4u = u3u2

The Adomian decomposition method consists to look the solution of u(x, t) in the form:

u(x, t) = u(x, 0) + (1 + i)

∫ t

0

∂2u(x, s)

∂x2
ds− (1 + 2i)

∫ t

0
|u(x, s)|2u(x, s)ds+

3

∫ t

0
u(x, s)ds− (1− 4i)

∫ t

0
|u(x, s)|4u(x, s)ds (14)

Let’s look for the solution in the following form:

u(x, t) =
+∞∑
n=0

un(x, t) (15)

with N = N1 +N2, and let us note

N1u = |u(x, t)|2u(x, t)
N2u = |u(x, t)|4u(x, t)

}
(16)

Suppose that

N1u =

∞∑
n=0

An(x, t)

and

N2u =
∞∑
n=0

Bn(x, t)


(17)

with

An =
1

n!

 dn

dλn
N1

+∞∑
j=0

λjuj


λ=0

, n = 0, 1, 2, ... (18)



Y. A. S. Wellot, G. D. Nkaya / Eur. J. Pure Appl. Math, 15 (4) (2022), 1750-1759 1754

and

Bn =
1

n!

 dn

dλn
N2

+∞∑
j=0

λjuj


λ=0

, n = 0, 1, 2, ... (19)

λ is a real parameter introduced by convenience, we obtain the following Adomian poly-
nomials:

An =
1

n!

dn

dλn


(

+∞∑
i=0

λiui

)2
(

+∞∑
j=0

λjuj

) 
λ=0

(20)

and

Bn =
1

n!

dn

dλn


(

+∞∑
i=0

λiui

)3
(

+∞∑
j=0

λjuj

)2

λ=0

(21)

This gives
A0 = u20u0. (22)

A1 = u20u1 + 2u0u1u0. (23)

A2 = u20u2 + 2u0u0u2 + 2u1u1u0 + u21u0. (24)

A3 = u20u3 + 2u0u0u3 + 2u0u1u2 + 2u0u1u2 + 2u0u1u2 + u21u1. (25)

and
B0 = u30u0

2. (26)

B1 = 3u1u
2
0u

2
0 + 2u30u1u0. (27)

B2 = 3u2u
2
0u

2
0 + 3u21u0u

2
0 + 6u1u

2
0u1u0 + 2u30u2u0 + u30u

2
1. (28)

The equation of problem (1), becomes:

+∞∑
n=0

un(x, t) = u(x, 0) + (1 + i)
+∞∑
n=0

∫ t

0

∂2un(x, s)

∂x2
ds− (1 + 2i)

+∞∑
n=0

∫ t

0
An(x, s)ds+

3
+∞∑
n=0

∫ t

0
un(x, s)ds− (1− 4i)

+∞∑
n=0

∫ t

0
Bn(x, s)ds. (29)

From the above equation (4.1) we get the Adomian canonical form. This leads us to obtain
the following Adomian algorithm (30) below:

u0(x, t) = u(x, 0)

un+1(x, t) = (1 + i)

∫ t

0

∂2un(x, s)

∂x2
ds− (1 + 2i)

∫ t

0
An (x, s) ds+

3

∫ t

0
un(x, s)ds− (1− 4i)

∫ t

0
Bn (x, s) ds

(30)



Y. A. S. Wellot, G. D. Nkaya / Eur. J. Pure Appl. Math, 15 (4) (2022), 1750-1759 1755

Either: 

u0(x, t) = eix

un+1(x, t) = (1 + i)

∫ t

0

∂2un(x, s)

∂x2
ds− (1 + 2i)

∫ t

0
An (x, s) ds+

3

∫ t

0
un(x, s)ds− (1− 4i)

∫ t

0
Bn (x, s) ds

(31)

u1(x, t) = (1 + i)

∫ t

0

∂2u0(x, s)

∂x2
ds− (1 + 2i)

∫ t

0
A0ds+ 3

∫ t

0
u0(x, s)ds− (1− 4i)

∫ t

0
B0ds

(32)
With A0 = B0 = eix, we can easily calculate u1(x, t)

u1(x, t) = −(1 + i)teix + 3teix − (1 + 2i)teix − (1− 4i)teix = iteix (33)

The same calculation procedure leads us to obtain A1 as follows

A1 = u20u1 + 2u0u0u0 = iteix. (34)

Then

B1 = 3u1u
2
0u

2
0 + 2u30u1u0 = iteix. (35)

Thus, the simple formula to calculate the expression u2(x, t) follows from this. Let it be:

u2(x, t) = (1 + i)

∫ t

0

∂2u1(x, s)

∂x2
ds− (1 + 2i)

∫ t

0
A1ds+ 3

∫ t

0
u1(x, s)ds− (1− 4i)

∫ t

0
B1ds

(36)

u2(x, t) = [−i(1 + i) + 3i− (1 + 2i)(i)− (1− 4i)(i)]
t2

2
eix (37)

Thus:

u2(x, t) = − t2

2
eix (38)

We proceed in the same way for the expression of u3(x, t).

u3(x, t) = (1 + i)

∫ t

0

∂2u2(x, s)

∂x2
ds− (1 + 2i)

∫ t

0
A2ds+ 3

∫ t

0
u2(x, s)ds− (1− 4i)

∫ t

0
B2ds

(39)
With

A2 = u20u2 + 2u0u0u2 + 2u1u1u0 + u21u0 = − t2

2
eix, (40)

and

B2 = 3u2u
2
0u

2
0 + 3u21u0u

2
0 + 6u1u

2
0u1u0 + 2u30u2u0 + u30u

2
1 = − t2

2
eix. (41)

Therefore:

u3(x, t) = [1 + i+ 1 + 2i− 3 + 1− 4i]
t3

3!
eix = −i

t3

3!
eix. (42)



Y. A. S. Wellot, G. D. Nkaya / Eur. J. Pure Appl. Math, 15 (4) (2022), 1750-1759 1756

Thus we have:



u0(x, t) = eix

u1(x, t) = iteix

u2(x, t) =
(it)2

2!
eix

u3(x, t) =
(it)3

3!
eix

...

un(x, t) =
(it)n

n!
eix

(43)

Hence, the exact solution of the Ginzburg-Landau equation is:

u(x, t) =

+∞∑
n=0

un (x, t) =

+∞∑
n=0

(it)n

n!
eix = ei(x+t). (44)

4.2. Problem 2

We consider the following initial value problem [12, 15]:


∂u(x, y, t)

∂t
− (1 + 2i)

[
∂2u(x, y, t)

∂x2
+

∂2u(x, y, t)

∂y2

]
+ (1 + 2i)|u(x, y, t)|2u(x, y, t)− γu(x, y, t) = 0

u(x, y, 0) = e
i
π

3
(x+ y)

,

(45)

With u(x, y, t) a complex function, γ ∈ R.
According to the Adomian decomposition method, the equation (45)

∂u(x, y, t)

∂t
− (1 + 2i)∆u(x, y, t) + (1 + 2i)|u(x, y, t)|2u(x, y, t)− γu(x, y, t) = 0 (46)

u(x, y, t) = u(x, y, 0)+

∫ t

0
(1+2i)∆u(x, y, s)ds−(1+2i)

∫ t

0
|u(x, y, s)|2u(x, y, s)ds+γ

∫ t

0
u(x, y, s)ds) = 0

(47)
Let’s put Nu = |u(x, y, t)|2u(x, y, t), we have

u(x, y, t) = u(x, y, 0)+

∫ t

0
(1+2i)∆u(x, y, s)ds−(1+2i)

∫ t

0
Nu(x, y, s)ds+γ

∫ t

0
u(x, y, s)ds) = 0

(48)



Y. A. S. Wellot, G. D. Nkaya / Eur. J. Pure Appl. Math, 15 (4) (2022), 1750-1759 1757

Let us look for the solution of (45) in the form of a series and the nonlinear part

u(x, y, t) =
∞∑
n=0

un(x, y, t) and Nu(x, y, t) =
∞∑
n=0

An(x, y, t) (49)

with
A0 = u20u0. (50)

A1 = u20u1 + 2u0u1u0. (51)

A2 = u20u2 + 2u0u0u2 + 2u1u1u0 + u21u0. (52)

We obtain the following canonical form:

∞∑
n=0

un(x, y, t) = u(x, y, 0)+(1+2i)

∫ t

0
∆un(x, y, s)ds−(1+2i)

∫ t

0
Ands+γ

∫ t

0
un(x, y, s)ds

(53)

From (53), for γ = 1 + 2π2

9 (because γ is a constant), we obtain the following Adomian
algorithm:

u0(x, y, t) = e
i
π

3
(x+ y)

un+1(x, y, t) =

∫ t

0
(1 + 2i)∆un(x, y, s)ds− (1 + 2i)

∫ t

0
Ands+ (1 +

2π2

9
)

∫ t

0
un(x, y, s)ds

(54)

For n = 0,

u1(x, y, t) =

∫ t

0
(1+2i)∆u0(x, y, s)ds− (1+2i)

∫ t

0
A0ds+(1+

2π2

9
)

∫ t

0
u0(x, y, s)ds (55)

With ∆u0(x, y, t) = −2π2

9 e
i
π

3
(x+ y)

and A0 = e
i
π

3
(x+ y)

(55), gives us

u1(x, y, t) = −2i

(
1 +

2π2

9

)
te

i
π

3
(x+ y)

For n = 2,

u2(x, y, t) =

∫ t

0
(1+2i)∆u1(x, y, s)ds− (1+2i)

∫ t

0
A1ds+(1+

2π2

9
)

∫ t

0
u1(x, y, s)ds (56)

With ∆u1(x, y, t) = 4iπ
2

9 (1+ 2π2

9 )te
i
π

3
(x+ y)

and A1 = −2i(1+ 2π2

9 )te
i
π

3
(x+ y)

So

u2(x, y, t) = −4

(
1 +

2π2

9

)2
t2

2
e
i
π

3
(x+ y)

(57)



REFERENCES 1758

For n = 3,

u3(x, y, t) =

∫ t

0
(1+2i)∆u2(x, y, s)ds− (1+2i)

∫ t

0
A2ds+(1+

2π2

9
)

∫ t

0
u2(x, y, s)ds (58)

With ∆u2(x, y, t) = 8π2

9

(
1 + 2π2

9

)2 t2
2
e
i
π

3
(x+ y)

and A2 = −4
(
1 + 2π2

9

)2 t2
2
e
i
π

3
(x+ y)

So

u3(x, y, t) = 8i

(
1 +

2π2

9

)3
t3

3!
e
i
π

3
(x+ y)

(59)

Then

u(x, y, t) =
∞∑
n=0

un(x, y, t) = u0(x, y, t) + u1(x, y, t) + u2(x, y, t) + u3(x, y, t) + ... (60)

u(x, y, t) =

[
1− 2i

(
1 +

2π2

9

)
t− 4

(
1 +

2π2

9

)2
t2

2
+ 8i

(
1 +

2π2

9

)3
t3

3
+ ...

]
e
i
π

3
(x+ y)

(61)
This implies

u(x, y, t) =
∞∑
n=0

[
−2i

(
1 + 2π2

9

)]n
n!

e
i
π

3
(x+ y)

. (62)

Then the exact solution of the Landau-Ginzburg equation in dimension two is:

u(x, y, t) = e
i

π
3
(x+ y)− 2

(
1 +

2π2

9

)
t


. (63)

5. Conclusion

In this paper, the Adomian decomposition method has been used to solve the complex
model of the Landau Ginzburg equation.
In order to show the importance and applicability of the proposed method, the Adomian
polynomial manipulated in this article allowed us to find the exact solution of the studied
problem.

References

[1] George Adomian. A review of the decomposition method in applied mathematics.
Journal of mathematical analysis and applications.

[2] George Adomian. Solving frontier problems of physics: the decomposition method.
Springer Science & Business Media.



REFERENCES 1759

[3] Kamel Attar. The Ginzburg-Landau model with a variable magnetic field. PhD thesis,
Université Paris Sud-Paris XI; Université Libanaise. Faculté des Sciences . . . .

[4] Kamel Attar. The ground state energy of the two dimensional ginzburg–landau func-
tional with variable magnetic field. In Annales de l’Institut Henri Poincaré C, Analyse
non linéaire, volume 32, pages 325–345. Elsevier, 2015.

[5] Charles Bu. An initial-boundary value problem for the ginzburg-landau equation.
Applied mathematics letters, 5(5):31–34, 1992.

[6] Qiang Du. Global existence and uniqueness of solutions of the time-dependent
ginzburg-landau model for superconductivity. Applicable Analysis, 53(1-2):1–17, 1994.

[7] Qiang Du. Numerical approximations of the ginzburg–landau models for supercon-
ductivity. Journal of mathematical physics, 46(9):095109, 2005.

[8] Francis Filbet and Lorenzo Pareschi. A numerical method for the accurate solu-
tion of the fokker–planck–landau equation in the nonhomogeneous case. Journal of
Computational Physics, 179(1):1–26, 2002.

[9] Ayman Kachmar and Marwa Nasrallah. On the ginzburg–landau energy with a mag-
netic field vanishing along a curve.

[10] Mohammed Lemou and Luc Mieussens. Fast implicit schemes for the fokker–planck–
landau equation. Comptes Rendus Mathematique, 338(10):809–814, 2004.

[11] Soheila Naghshband and Mohammad Ali Fariborzi Araghi. Solving generalized quin-
tic complex ginzburg–landau equation by homotopy analysis method. Ain Shams
Engineering Journal, 9(4):607–613, 2018.

[12] Eduardo Salete, Antonio M Vargas, Ángel Garćıa, Mihaela Negreanu, Juan J Ben-
ito, and Francisco Ureña. Complex ginzburg–landau equation with generalized finite
differences. Mathematics.

[13] Etienne Sandier and Sylvia Serfaty. The decrease of bulk-superconductivity close to
the second critical field in the ginzburg–landau model. SIAM journal on mathematical
analysis.

[14] Etienne Sandier and Sylvia Serfaty. Vortices in the magnetic Ginzburg-Landau model.
Springer Science & Business Media.

[15] Shao-Wen Yao, Esin Ilhan, P Veeresha, and Haci Mehmet Baskonus. A powerful
iterative approach for quintic complex ginzburg–landau equation within the frame of
fractional operator. Fractals.