Solitons and other solutions to quantum ZakharovâKuznetsov equation in quantum magneto-plasmas

ORIGINAL PAPER

Solitons and other solutions to quantum Zakharov–Kuznetsov equationin quantum magneto-plasmas

A H Bhrawy1,2, M A Abdelkawy2, S Kumar3, S Johnson4,5 and A Biswas4*1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni Suef, Egypt

3Department of Applied Sciences, Bahra Faculty of Engineering, Patiala 147 001, Punjab, India

4Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA

5Lake Forest High School, 5407 Killens Pond Road, Felton, DE 19943, USA

Received: 26 October 2012 / Accepted: 01 January 2013 / Published online: 17 January 2013

Abstract: In this paper, the soliton solutions to the quantum Zakharov–Kuznetsov equation in quantum dusty plasmas

have been investigated. A few integration tools have been applied to extract the soliton solutions to the governing equation.

The topological, non-topological, singular soliton and complexiton solutions are obtained. Additionally, the periodic and

singular periodic solutions are listed. Finally, the ansatz method has been employed to retrieve a singular 1-soliton solution

to the equation for power law nonlinearity. In this case, a constraint condition naturally falls out and it serves as a pre-

conditioning criteria for singular soliton to exist.

Keywords: Extended F-expansion (EFE) method; Nonlinear partial differential equations; Nonlinear physical

phenomena; Quantum Zakharov–Kuznetsov equation; Quantum plasmas

PACS Nos.: 02.30.Jr; 05.45.Yv; 02.30.Ik

1. Introduction

The study of solitons in applied mathematics and theoret-

ical physics has been continued for the past few decades

and still a major area of research in these fields [1–34].

There are always many ways to explore in this direction.

There are several avenues where research in soliton theory

is still very demanding. Solitons are studied in nonlinear

optics in the context of optical fibers and metamaterials

[32, 33]. Then, solitons are special solutions in the area of

geophysical phenomena that can assist in the study of

earthquake phenomena, although, unfortunately, the time

interval for earthquake prediction cannot be done. Solitons

are also studied in the context of shallow water waves

along ocean shores and in lakes and beaches, where in

addition to solitary waves, cnoidal waves are observed.

Furthermore, chiral solitons are studied in nuclear physics

along with Bohm potential. In biophysics, solitons are

observed in a-helix proteins.

This paper sheds some light on the solitons in quantum

magneto-plasmas that is governed by the quantum Zakha-

rov–Kuznetsov (QZK) equation. While this equation has

been studied earlier on several occasions by several

authors, this paper is on a different flavor than those pub-

lished earlier. The formal derivation of the QZK equation

is given, initially, in a succinct manner. Subsequently, the

integration of the QZK equation is carried out by the

extended F-expansion (EFE) method. Later, the soliton and

complexiton solutions of the QZK equation in (3 ? 1)-

dimensions have been obtained as a limiting case of the

elliptic functions. In the other limiting case, the periodic

functions as well as singular periodic functions are also

listed. Finally, the QZK equation with power law nonlin-

earity has been studied. Here, the ansatz method is applied

to retrieve a singular 1-soliton solution along with a rele-

vant constraint condition in order for the singular soliton to

exist.*Corresponding author, E-mail: [email protected]

Indian J Phys (May 2013) 87:455–463

DOI 10.1007/s12648-013-0248-x

� 2013 IACS

2. Governing equations

The dynamics of low frequency ion-acoustic waves in a

dense magneto-plasma is governed by

oni

otþ5 � niuð Þ ¼ 0; ð1Þ

oui

otþ ui � 5ui ¼ �5 /þ ui � z; ð2Þ

o/oz� ne

one

ozþ H2

e

2

o

oz

52 ffiffiffiffiffi

nepffiffiffiffiffi

nep

� �

¼ 0; ð3Þ

ne ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2/þ H2e

52ffiffiffiffiffi

nepffiffiffiffiffi

nep

s

;

He ¼ eH0h=2cffiffiffiffiffiffiffiffiffiffi

MiMp

KBTFe;

ð4Þ

and Poisson’s equation

52/ ¼ ne � ni; ð5Þ

where ni, ui, /, ne, H0, h, c, Mi, M, TFe and KB are the

ion number density, the ion fluid velocity, the electro-

static potential, electron number density, the strength of

the magnetic field, the Planck constant divided by 2p,

the speed of light in vacuum, the ion mass, the electron/

positron mass, the electron/positron Fermi temperature

and the Boltzmann constant respectively.

The three-dimensional QZK equation can be obtained

from Eqs. (1)–(5), using reductive technique [29, 30], as

o/otþ d/

o/ozþ k

o3/oz3þ � o

oz

o2/ox2þ o2/

oy2

� �

¼ 0; ð6Þ

where d ¼ 2; k ¼ 12ð1� H2

e

4Þ; � ¼ 1� H2

e

8:

3. Extended F-expansion method

In this section, we have introduced a simple description of

the EFE method, for a given partial differential equation

(PDE)

G u; ux; uy; uz; ut; uxy; . . .� �

¼ 0: ð7Þ

We would like to know whether travelling waves (or

stationary waves) are solutions of Eq. (7). The first step is

to unite the independent variables x, y, z and t into one

particular variable through the new variable

f ¼ axþ byþ czþ mt; uðx; y; z; tÞ ¼ UðfÞ;

where m is the wave speed and reduce Eq. (7) to an ordinary

differential equation (ODE)

GðU; U0; U

00; U

000; . . .Þ ¼ 0: ð8Þ

Our main goal is to derive exact or at least approximate

solutions, if possible, for this ODE. For this purpose, let us

simply U as the expansion in the form,

uðx; y; z; tÞ ¼ UðfÞ ¼X

N

i¼0

aiFi þX

N

i¼1

a�iF�i; ð9Þ

where

F0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Aþ BF2 þ CF4p

; ð10Þ

the highest degree of dpUdfp is taken as

OdpU

dfp

� �

¼ N þ p; p ¼ 1; 2; 3; . . .; ð11Þ

O UqdpU

dfp

� �

¼ ðqþ 1ÞN þ p; q ¼ 0; 1; 2; . . .;

p ¼ 1; 2; 3; . . .ð12Þ

where A, B and C are constants, and N in Eq. (9) is a

positive integer that can be determined by balancing the

nonlinear term(s) and the highest order derivatives. Nor-

mally N is a positive integer, so that an analytic solution in

closed form may be obtained. Substituting Eqs. (7)–(10)

into Eq. (8) and comparing the coefficients of each power

of FðfÞ in both sides, we get an over-determined system of

nonlinear algebraic equations with respect to m; a0; a1; . . .

Solving the over-determined system of nonlinear algebraic

equations by use of Mathematica. The relations between

values of A, B, C and corresponding Jacobian elliptic

Table 1 The relation between values of (A, B, C) and the corre-

sponding FðfÞ

A B C FðfÞ

1 -1 - m2 m2snðfÞ or cdðfÞ ¼ cnðfÞ

dnðfÞ

1 - m2 2m2 - 1 -m2 cnðfÞm2 - 1 2 - m2 -1 dnðfÞm2 -1 - m2 1 nsðfÞ 1

snðfÞ or

dcðfÞ ¼ dnðfÞcnðfÞ

-m2 2m2 - 1 1 - m2ncðfÞ ¼ 1

cnðfÞ

-1 2 - m2 m2 - 1 ndðfÞ ¼ 1dnðfÞ

1 2 - m2 1 - m2scðfÞ ¼ snðfÞ

cnðfÞ

1 2m2 - 1 -m2(-1 - m2) sdðfÞ ¼ snðfÞdnðfÞ

1 - m2 2 - m2 1 csðfÞ ¼ cnðfÞsnðfÞ

-m2(1 - m2) 2m2 - 1 1 dsðfÞ ¼ dnðfÞsnðfÞ

14

1�2m2

2

14

nsðfÞ þ csðfÞ1�m2

41þm2

21�m2

2ncðfÞ þ scðfÞ

14

m2�22

m2

4nsðfÞ þ dsðfÞ

m2

4m2�2

2m2

4snðfÞ þ icsðfÞ

456 A H Bhrawy et al.

function (JEF) solution FðfÞ of Eq. (9) are given in the

Appendix Table 1. Substituting the values of A, B, C and

the corresponding JEF solution FðfÞ chosen from Table 1

into the general form of solution, an ideal periodic wave

solution expressed by JEF is obtained.

4. Three-dimensional QZK equation

In this section, we have applied the extended method to

study the three-dimensional QZK equation (6)

o/otþ d/

o/ozþ k

o3/oz3þ � o

oz

o2/ox2þ o2/

oy2

� �

¼ 0: ð13Þ

If we use f ¼ axþ byþ czþ mt; /ðx; y; z; tÞ ¼ UðfÞcarries PDE Eq. (13) into the ODE

mU0 þ dUU

0 þ kc2 þ � b2 þ a2� ��

U000 ¼ 0; ð14Þ

where by integrating once we obtain, upon setting the

constant of integration to zero,

2mU þ dU2 þ 2 kc2 þ � b2 þ a2� ��

U00 ¼ 0: ð15Þ

Balancing the term U00 with the term U2 we obtain N = 2

then

UðfÞ ¼ a0 þ a1F þ a�1F�1 þ a2F2 þ a�2F�2;

F0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Aþ BF2 þ CF4p

:ð16Þ

Substituting Eq. (16) in Eq. (15) and comparing the

coefficients of each power of F in both sides, we get an

over-determined system of nonlinear algebraic equations

with respect to m, ai, i = 1, -1, -2, 2. Solving the over-

determined system of nonlinear algebraic equations by use

of Mathematica, we obtain three groups of constants:

(i)

a�1 ¼ a1 ¼ 0; a0 ¼ �4 Br�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðB2 þ 12ACÞðrÞ2q

� �

d;

a2 ¼ �12Cr

d; a�2 ¼ �

12Ard

;

m ¼ �4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðB2 þ 12ACÞr2p

; and r ¼ a2 þ b2� �

�þ c2k;

ð17Þ

(ii)

a�1 ¼ a1 ¼ a2 ¼ 0; a0 ¼ �4 Br�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðB2 � 3ACÞr2p

d;

m ¼ �4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðB2 � 3ACÞr2p

; a�2 ¼ �12Ar

dand r ¼ a2 þ b2

� �

�þ c2k;

ð18Þ

(iii)

a�1 ¼ a�2 ¼ a1 ¼ 0; a0 ¼ �4 Br�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðB2 � 3ACÞr2p

d;

m ¼ �4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðB2 � 3ACÞr2p

; a2 ¼ �12Cr

dand r ¼ a2 þ b2

� �

�þ c2k:

ð19Þ

If we use group of constants given by Eqs. (17)–(19) we

obtained the electrostatic potentials of Eq. (13) as:

/1 ¼4 1þm2ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12m2 þ 1þm2ð Þ2

r2

r� �

d

� 12rd

m2sn2 axþ byþ cz� 4


12m2 þ 1þm2ð Þ2

r2

r

t

� ��

þns2 axþ byþ cz� 4


12m2 þ 1þm2ð Þ2

r2

r

t

� ��

;

ð20Þ

/2 ¼4 1þm2ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12m2þ 1þm2ð Þ2

r2

r� �

d� 12r

dm2cd2 axþ byþ cz� 4


12m2þ 1þm2ð Þ2

r2

r

t

� ��

þdc2 axþ byþ cz� 4


12m2þ 1þm2ð Þ2

r2

r

t

� ��

;

ð21Þ

/3 ¼�4 2m2 � 1ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12m2 m2 � 1ð Þ þ 1� 2m2ð Þ2

r2

r� �

d

þ 12rd

m2cn2 axþ byþ cz� 4


12m2 m2 � 1ð Þ þ 1� 2m2ð Þ2

r2

r

t

� ��

� 1� m2� �

nc2 axþ byþ cz� 4


12m2 m2 � 1ð Þ þ 1� 2m2ð Þ2

r2

r

t

� ��

;

ð22Þ

Solitons and other solutions 457

/4 ¼�4 2� m2ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2� m2ð Þ2�12 �1þ m2ð Þ

r2

r� �

d

þ 12rd

dn2 axþ byþ cz� 4


2� m2ð Þ2�12 �1þ m2ð Þ

r2

r

t

� ��

þ 1� m2� �

nd2 axþ byþ cz� 4


2� m2ð Þ2�12 �1þ m2ð Þ

r2

r

t

� ��

;

ð23Þ

/5 ¼�4ðð2� m2Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð12ð1� m2Þ þ ð2� m2Þ2Þr2

q

Þd

� 12rdð1� m2Þsc2ðaxþ byþ cz� 4



q

tÞ�

þ cs2ðaxþ byþ cz� 4



q

tÞ�

;

ð24Þ

/6 ¼�4 2m2 � 1ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12m2 1þ m2ð Þ þ �1þ 2m2ð Þ2

r2

r� �

d

� 12rd

m2 1þ m2� �

sd2 axþ byþ cz� 4


12m2 1þ m2ð Þ þ �1þ 2m2ð Þ2

r2

r

t

� ��

þ ds2 axþ byþ cz� 4


12m2 1þ m2ð Þ þ �1þ 2m2ð Þ2

r2

r

t

� ��

;

ð25Þ

/7 ¼�4 0:5� m2ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0:75þ 0:5� m2ð Þ2

r2

r� �

d� 3r

dns axþ byþ cz� 4


0:75þ 0:5� m2ð Þ2

r2

r

t

� ��

þ cs axþ byþ cz� 4


0:75þ 0:5� m2ð Þ2

r2

r

t

� ��2

þ ns axþ byþ cz� 4


0:75þ 0:5� m2ð Þ2

r2

r

t

� ��



0:75þ 0:5� m2ð Þ2

r2

r

t

� ��2#

;

ð26Þ

/8 ¼�4 0:5þ 0:5m2ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12 0:5� 0:5m2ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r� �

d

� 12rd

0:5� 0:5m2� �

nc axþ byþ cz� 4


12 0:5� 0:5m2ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r

t

� ��

þ sc axþ byþ cz� 4


12 0:5� 0:5m2ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r

t

� ��2

þ 0:25� 0:25m2� �

nc axþ byþ cz� 4


12 0:5� 0:5m2ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r

t

� ��



12 0:5� 0:5m2ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r

t

� ��2#

;

ð27Þ


/9 ¼�4 0:5m2� 1ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0:75m2þ �1þ 0:5m2ð Þ2

r2

r� �

d

� 3rd

m2 ns axþbyþ cz� 4


0:75m2þ �1þ 0:5m2ð Þ2

r2

r

t

� ��

þ ds axþbyþ cz� 4


0:75m2þ �1þ 0:5m2ð Þ2

r2

r

t

� ��2

þ ns axþbyþ cz� 4


0:75m2þ �1þ 0:5m2ð Þ2

r2

r

t

� ��

þ ds axþbyþ cz� 4


0:75m2þ �1þ 0:5m2ð Þ2

r2

r

t

� ��2#

;

ð28Þ

/10¼�4 0:5m2� 1ð Þr�


0:75m4þ �1þ0:5m2ð Þ2

r2

r� �

d

�3rd

m2 sn axþbyþ cz�4


0:75m4þ �1þ0:5m2ð Þ2

r2

r

t

� ��

þ ics axþbyþ cz�4


0:75m4þ �1þ0:5m2ð Þ2

r2

r

t

� ��2

þ sn axþbyþ cz�4


0:75m4þ �1þ0:5m2ð Þ2

r2

r

t

� ��

þ ics axþbyþ cz�4


0:75m4þ �1þ0:5m2ð Þ2

r2

r

t

� ��2#

;

ð29Þ

/11 ¼4 1þm2ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þm2ð Þ2�3m2

r2

r� �

d

� 12rd

m2sn2 axþbyþ cz� 4


1þm2ð Þ2�3m2

r2

r

t

� ��

;

ð30Þ

/12 ¼4 1þm2ð Þr�


1þm2ð Þ2�3m2

r2

r� �

d

� 12rd

m2cd2 axþbyþ cz� 4


1þm2ð Þ2�3m2

r2

r

t

� ��

;

ð31Þ

/14 ¼�4 2� m2ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2� m2ð Þ2�3 1� m2ð Þ

r2

r� �

d

þ 12rd

dn2 axþ byþ czð�

� 4


2� m2ð Þ2�3 1� m2ð Þ

r2

r

t

��

; ð33Þ

/15 ¼�4 2� m2ð Þr�


3 m2 � 1ð Þ þ 2� m2ð Þ2

r2

r� �

d

� 12rd

1� m2� �

sc2 axþ byþ czð�

� 4


3 m2 � 1ð Þ þ 2� m2ð Þ2

r2

r

tÞ�; ð34Þ

/16 ¼ �4 2m2 � 1ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 2m2ð Þ2�3m2 1þ m2ð Þ

r2

r� �

d

� 12rd

m2 1þ m2� �

sd2�

�ðaxþ byþ cz� 4


1� 2m2ð Þ2�3m2 1þ m2ð Þ

r2

r

t

��

;

ð35Þ

/17 ¼�4ðð0:5� m2Þr�


ðð0:5� m2Þ2 � 316Þr2

q

Þd

� 3rd

nsðaxþ byþ cz� 4


ð0:5� m2Þ2 � 3

16

� �

r2

s

tÞ "

þ csðaxþ byþ cz� 4


ð0:5� m2Þ2 � 3

16

� �

r2

s

tÞ!23

5;

ð36Þ

/13¼�4 2m2�1ð Þr�


3m2 1�m2ð Þþ 1�2m2ð Þ2

r2

r� �

d

þ12rd

m2cn2 axþbyþ cz�4


3m2 1�m2ð Þþ 1�2m2ð Þ2

r2

r

t

� ��

;

ð32Þ


/19 ¼�4 0:5m2 � 1ð Þr�


1� 0:5m2ð Þ2�3m2

16

r2

r� �

d

� 3rd

m2 ns axþ byþ cz� 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 0:5m2ð Þ2�3m2

16

� �

r2

s

t

! "

þ ds axþ byþ cz� 4


1� 0:5m2ð Þ2�3m2

16

� �

r2

s

t

!!23

5;

ð38Þ

/20 ¼�4 0:5m2 � 1ð Þr�


1� 0:5m2ð Þ2�3m4

16

r2

r� �

d

� 3rd

m2 sn axþ byþ cz� 4


1� 0:5m2ð Þ2�3m4

16

� �

r2

s

t

! "

þ ics axþ byþ cz� 4


1� 0:5m2ð Þ2�3m4

16

� �

r2

s

t

!!23

5;

ð39Þ

/21 ¼4 1þ m2ð Þr�


1þ m2ð Þ2�3m2

r2

r� �

d

� 12rd

ns2 axþ byþ cz� 4


1þ m2ð Þ2�3m2

r2

r

t

� ��

;

ð40Þ

/22 ¼4 1þ m2ð Þr�


1þ m2ð Þ2�3m2

r2

r� �

d

� 12rd

dc2 axþ byþ cz� 4


1þ m2ð Þ2�3m2

r2

r

t

� ��

;

ð41Þ

/23¼�4 2m2�1ð Þr�


3m2 1�m2ð Þþ 1�2m2ð Þ2

r2

r� �

d� 1�m2� �

nc2 axþbyþczð

�4


3m2 1�m2ð Þþ 1�2m2ð Þ2

r2

r

t

��

;

ð42Þ

/24 ¼�4 2� m2ð Þr�


2� m2ð Þ2�3 1� m2ð Þ

r2

r� �

d

þ 12rd

"

1� m2� �

nd2 axþ byþ czð

�4


2� m2ð Þ2�3 1� m2ð Þ

r2

r

t

�

#

;

ð43Þ

/25 ¼�4 2� m2ð Þr�


3 m2 � 1ð Þ þ 2� m2ð Þ2

r2

r� �

d

� 12rd

cs2 axþ byþ czð�

�4


3 m2 � 1ð Þ þ 2� m2ð Þ2

r2

r

t

��

;

ð44Þ

/26¼�4 2m2�1ð Þr�


1�2m2ð Þ2�3m2 1þm2ð Þ

r2

r� �

d

�12rd

�

ds2 axþbyþ czð

�4


1�2m2ð Þ2�3m2 1þm2ð Þ

r2

r

t

��

;

ð45Þ

/27 ¼�4 0:5� m2ð Þr�


0:5� m2ð Þ2�316

r2

r� �

d

� 3rd

ns axþ byþ cz� 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0:5� m2ð Þ2� 3

16

� �

r2

s

t

! "


ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0:5� m2ð Þ2� 3

16

� �

r2

s

t

!!�23

5;

ð46Þ

/18 ¼�4 0:5þ 0:5m2ð Þr�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3 0:5m2 � 0:5ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r� �

d

� 12rd

0:5� 0:5m2� �

� nc axþ byþ cz� 4


3 0:5m2 � 0:5ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r

t

� ��



3 0:5m2 � 0:5ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r

t

� ��2

;

ð37Þ


/29 ¼�4 0:5m2 � 1ð Þr�


1� 0:5m2ð Þ2�3m2

16

r2

r� �

d

� 3rd

ns axþ byþ cz� 4


1� 0:5m2ð Þ2�3m2

16

� �

r2

s

t

! "

þ ds axþ byþ cz� 4


1� 0:5m2ð Þ2�3m2

16

� �

r2

s

t

!!�23

5;

ð48Þ

/30 ¼�4 0:5m2 � 1ð Þr�


1� 0:5m2ð Þ2�3m4

16

r2

r� �

d

� 3rd

sn axþ byþ cz� 4


1� 0:5m2ð Þ2�3m4

16

� �

r2

s

t

! "

þ ics axþ byþ cz� 4


1� 0:5m2ð Þ2�3m4

16

� �

r2

s

t

!!�23

5:

ð49Þ

4.1. Soliton solutions

Some solitary wave solutions are obtained, when the

modulus m approaches to 1 in Eqs. (20)–(49):

/31 ¼4ð2r� 4rÞ

d� 12r

dtanh2ðaxþ byþ cz� 16rtÞ�

þ coth2ðaxþ byþ cz� 16rtÞ�;ð50Þ

/32 ¼ �4ðr� rÞ

dþ 12r

dsech2ðaxþ byþ cz� 4rtÞ� �

;

ð51Þ

/33 ¼ �4ðr� rÞ

d� 12r

dcsch2ðaxþ byþ cz� 4rtÞ� �

;

ð52Þ

/34 ¼ �4ðr� 5rÞ

d� 12r

d2 sinh2ðaxþ byþ cz� 20rtÞ�

þ csch2ðaxþ byþ cz� 20rtÞ�; ð53Þ

/35 ¼�4ð�0:5r� rÞ

d� 3r

dðcothðaxþ byþ cz� 4rtÞ½

þ cschðaxþ byþ cz� 4rtÞÞ2:

þ ðcothðaxþ byþ cz� 4rtÞ

þ cschðaxþ byþ cz� 4rtÞÞ�2�;ð54Þ

/36 ¼�4ð�0:5r� rÞ

d� 3r

dðtanhðaxþ byþ cz� 4rtÞ½

þ icschðaxþ byþ cz� 4rtÞÞ2:

þ ðtanhðaxþ byþ cz� 4rtÞ

þ icschðaxþ byþ cz� 4rtÞÞ�2�; ð55Þ

/37 ¼4ð2r� rÞ

d� 12r

dtanh2ðaxþ byþ cz� 4rtÞ� �

;

ð56Þ

/38 ¼ �4ðr� rÞ

dþ 12r

dsech2ðaxþ byþ cz� 4rtÞ� �

;

ð57Þ

/39 ¼ �4ðr� ir

ffiffiffi

5pÞ

d

� 24rd

sinh2ðaxþ byþ cz� 4irffiffiffi

5p

tÞh i

; ð58Þ

/40 ¼ �4 �r

2� r

4

� �

d� 3r

dcoth axþ byþ cz� 4

r4

t h

þ csch axþ byþ cz� 4r4

t

Þ2�; ð59Þ

/41 ¼ �4 �r

2� r

4

� �

d� 3r

dtanh axþ byþ cz� 4

r4

t h

þ icsch axþ byþ cz� 4r4

t

Þ2�; ð60Þ

/42 ¼4ð2r� rÞ

d� 12r

dcoth2ðaxþ byþ cz� 4rtÞ� �

;

ð61Þ

/28 ¼�4 0:5þ 0:5m2ð Þr�


3 0:5m2 � 0:5ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r� �

d

� 12rd

0:25� 0:25m2� �

� nc axþ byþ cz� 4


3 0:5m2 � 0:5ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r

t

� ��



3 0:5m2 � 0:5ð Þ 0:25� 0:25m2ð Þ þ 0:5þ 0:5m2ð Þ2

r2

r

t

� ��2

;

ð47Þ


/43 ¼ �4ðr� ir

ffiffiffi

5pÞ

d

� 12rd

csch2ðaxþ byþ cz� 4irffiffiffi

5p

tÞh i

; ð62Þ

/44 ¼ �4 �r

2� r

4

� �

d� 3r

dcoth axþ byþ cz� 4

r4

t h

þ csch axþ byþ cz� 4r4

t

Þ�2i

;

ð63Þ

/45 ¼ �4 �r

2� r

4

� �

d� 3r

dtanh axþ byþ cz� 4

r4

t h

þ icsch axþ byþ cz� 4r4

t

Þ�2i

;

ð64Þ

4.2. Triangular periodic solutions

Some trigonometric function solutions are obtained, when

the modulus m approaches to zero in Eqs. (20)–(49):

/46 ¼4ðr� rÞ

d� 12r

dcsc2ðaxþ byþ cz� 4rtÞ� �

; ð65Þ

/47 ¼4ðr� rÞ

d� 12r

dsec2ðaxþ byþ cz� 4rtÞ� �

; ð66Þ

/48 ¼ �4ð2r� 4rÞ

d� 12r

dtan2ðaxþ byþ cz� 16rtÞ�

þ cot2ðaxþ byþ cz� 16rtÞ�

; ð67Þ

/49 ¼�4ð0:5r� rÞ

d� 3r

dðcscðaxþ byþ cz� 4rtÞ½

þ cotðaxþ byþ cz� 4rtÞÞ2ðcscðaxþ byþ cz

�4rtÞ þ cotðaxþ byþ cz� 4rtÞÞ�2�

; ð68Þ

/50 ¼�2ðr� r

ffiffiffi

7pÞ

d� 12r

d0:5ðsecðaxþ byþ cz½

�2rffiffiffi

7p

tÞ þ tanðaxþ byþ cz� 2rffiffiffi

7p

tÞÞ2

þ 0:25ðsecðaxþ byþ cz� 2rffiffiffi

7p

tÞ

þ tanðaxþ byþ cz� 2rffiffiffi

7p

tÞÞ�2�

; ð69Þ

/51 ¼ �4ðr� rÞ

d� 3r

dsin2ðaxþ byþ cz� 4rtÞ� �

; ð70Þ

/52 ¼ �4ðr� rÞ

d� 3r

dðsinðaxþ byþ cz� 4rtÞ½

þ i cotðaxþ byþ cz� 4rtÞÞ�2�; ð71Þ

/53 ¼ �4ð2r� rÞ

d� 12r

dtan2ðaxþ byþ cz� 4rtÞ� �

;

ð72Þ

/54 ¼ �2ðr� rÞ

d� 3r


þ cotðaxþ byþ cz� 2rtÞÞ2�

;

ð73Þ

/55 ¼ �ð2r� ir

ffiffiffi

2pÞ

d� 6r

dðsecðaxþ byþ cz� ir

ffiffiffi

2p

tÞ

þ tanðaxþ byþ cz� irffiffiffi

2p

tÞÞ2; ð74Þ

/56 ¼ �4ð2r� rÞ

d� 12r

dcot2ðaxþ byþ cz� 4rtÞ� �

;

ð75Þ

/57 ¼ �2ðr� rÞ

d� 3r


þ cotðaxþ byþ cz� 2rtÞÞ�2�;ð76Þ

/58 ¼�ð2r� ir

ffiffiffi

2pÞ

d� 12r

d0:25� 0:25m2� �

� ðsecðaxþ byþ cz� irffiffiffi

2p

tÞþ tanðaxþ byþ cz� ir

ffiffiffi

2p

tÞÞ�2;

ð77Þ

/59 ¼4ðr� rÞ

d� 3r

dðsinðaxþ byþ cz� 4rtÞ½

þi cotðaxþ byþ cz� 4rtÞÞ�2�

:

ð78Þ

5. Singular solitons

This section is devoted to the derivation of a singular

1-soliton solution of the QZK equation that is with power

law nonlinearity. In dimensionless form, the QZK equation

with power law nonlinearity, is given by

qt þ aqnqz þ b qxx þ qyy

� �

zþcqzzz ¼ 0; ð79Þ

where a, b and c are arbitrary non-zero real-valued constants.

The parameter n represents the power law nonlinearity

parameter and represents the strength of nonlinearity with

n [ 0. Also, here in Eq. (79), q(x, y, z, t) represents the

dependent variable. The target in this section is to obtain a

singular 1-soliton solution to Eq. (79) by the aid of ansatz

method. In this method, a trial solution to Eq. (79) is taken

to be

qðx; y; z; tÞ ¼ Acschp B1xþ B2yþ B3z� vtð Þ ¼ Acschps;

ð80Þ

where

s ¼ B1xþ B2yþ B3z� vt: ð81Þ

In Eq. (80), the parameters A, Bj for j = 1, 2, 3 are all free

parameters and v represents the velocity of the soliton.

Substitution of Eq. (80) in Eq. (79) leads to, after

simplification,

v� bp2B3 B21 þ B2

2

� �

� cp2B33

�

cschps� aAnB3cschðnþ1Þps

� ðpþ 1Þðpþ 2Þ bB3 B21 þ B2

2

� �

þ cB33

�

cschpþ2s ¼ 0:

ð82Þ


Then from Eq. (82), by the balancing principle, equating

the exponents (n ? 1)p and p ? 2 leads to

ðnþ 1Þp ¼ pþ 2; ð83Þ

so that

p ¼ 2

n: ð84Þ

From Eq. (82), setting the coefficients of the linearly

independent functions cschp?j where j = 0, 2 to zero leads to

v ¼ 4B3

n2b B2

1 þ B22

� �

þ cB23

�

; ð85Þ

and

A ¼ �2ðnþ 1Þðnþ 2Þ b B2

1 þ B22

� �

þ cB23

�

an2

� �

1n

; ð86Þ

which compels the constraint relation

a b B21 þ B2

2

� �

þ cB23

�

\0; ð87Þ

for even values of n, in order for the singular solitons to exist.

Thus, finally, the singular 1-soliton solution to Eq. (79) is

given by

qðx; y; z; tÞ ¼ Acsch2n B1xþ B2yþ B3z� vtð Þ; ð88Þ

where the free parameters are related as given in Eq. (86)

and the velocity of the soliton is given by Eq. (85). In

addition, the constraint conditions given by Eq. (87) must

remain valid in order for the singular solitons to exist.

6. Conclusions

This paper addresses the integrability aspects of the QZK

equation in a fairly detailed fashion. The EFE method is

implemented to list a plethora of solutions. Later, the

limiting cases of these solutions are obtained that revealed

soliton and complexiton solutions on one hand and the

periodic solutions on the other hand. Finally, the ansatz

method is applied to the QZK equation with power law

nonlinearity where a singular 1-soliton solution was

revealed. This solution comes with a baggage, namely a

constraint condition falls out during the course of deriva-

tion of the solution which must hold in order for the sin-

gular soliton to exist.

Appendix

The relation between values of (A, B, C) and the corre-

sponding JEF solution FðfÞ of Eq. (10) are given in Table 1,

where snðfÞ; cnðfÞ and dnðfÞ are the Jacobian elliptic (JE)

sine function, JE cosine function and the JEF of the third

kind, respectively. And

cn2ðfÞ ¼ 1� sn2ðfÞ; dn2ðfÞ ¼ 1� m2sn2ðfÞ; ð89Þ

with the modulus m (0 \ m \ 1).

When m �! 1; the Jacobi functions degenerate to the

hyperbolic functions, i.e.,

snf �! tanhf; cnf �! sechf; dnf �! sechf;

when m �! 0; the Jacobi functions degenerate to the

triangular functions, i.e.,

snf �! sinf; cnf �! cosf and dnf �! 1:

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Documents

Solitons and other solutions to quantum ZakharovâKuznetsov equation in quantum magneto-plasmas