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Stability: Conservation laws, Painlevé analysis and exact solutions for S-KP equation in coupled dusty plasma
Results in Physics 7 (2017) 934–946
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Stability: Conservation laws, Painlevé analysis and exact solutions for S-KP equation in coupled dusty plasma O.H. EL-Kalaawy ⇑, S.M. Moawad, Shrouk Wael Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
a r t i c l e
i n f o
Article history: Received 25 September 2016 Received in revised form 7 November 2016 Accepted 3 January 2017 Available online 16 February 2017 Keywords: Coupled dusty plasma Schamel-Kadomtsev-Petviashvili equation Conservation law Painlevé analysis Auto-Bäcklund transformations Exact solutions Mach number and stability
a b s t r a c t The propagation of nonlinear waves in unmagnetized strongly coupled dusty plasma with Boltzmann distributed electrons, iso-nonthermal distributed ions and negatively charged dust grains is considered. The basic set of fluid equations is reduced to the Schamel Kadomtsev-Petviashvili (S-KP) equation by using the reductive perturbation method. The variational principle and conservation laws of S-KP equation are obtained. It is shown that the S-KP equation is non-integrable using Painlevé analysis. A set of new exact solutions are obtained by auto-Bäcklund transformations. The stability analysis is discussed for the existence of dust acoustic solitary waves (DASWs) and it is found that the physical parameters have strong effects on the stability criterion. In additional to, the electric field and the true Mach number of this solution are investigated. Finally, we will study the physical meanings of solutions. Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction Recently, the study of nonlinear waves in plasma is one of the hottest topics in plasma physics, specifically the dusty plasma. A dusty plasma is ordinary plasma with embedded solid microparticles. In dusty plasma, there are essential two types of acoustic waves: high frequency dust ion acoustic waves (DIAWs) involving mobile ions and static dust grains, and a low frequency dust acoustic waves (DAWs) involving mobile dust grains. Both modes have been studied theoretically and experimentally [1–10]. Dusty plasmas are present in astrophysical and space environments, like the tails of comets, interstellar medium and planetary rings [4,5]. The reason for their presence in these regions is the presence of highdensity dust grains different types of plasma wave modes. Ref. [6] was the first report recorded theoretically the existence of DIA waves. The study showed that an unmagnetized weakly coupled dusty plasma supports the DIAW whose phase velocity is much smaller (larger) than the electron (ion) thermal speed. The frequency of DIAWs is much larger (smaller) than the dust (ion) plasma frequency. At the same time, Rao et al. [7] theoretically predicted the existence of DAWs, which is provided by the inertia dust particles mass and are providing power restoration due to pressures of less inertia of electrons and ion. The DAWs have supported a great deal of interest in understanding the basic characteristics ⇑ Corresponding author. E-mail address: [email protected] (O.H. EL-Kalaawy).
and properties of local electrostatic perturbations in space and laboratory dusty plasmas. Strongly coupled dust plasma is of great interest in science because of its applications in the interior of heavy planets, plasmas produced by laser compression of matter and non-ideal plasmas for industrial applications. There are a number of authors who investigated properties of one-dimensional linear and nonlinear DAWs in coupled unmagnetized dusty plasmas. In this paper, we concern the nonlinear propagation of the dust acoustic waves in unmagnetized strongly coupled dusty plasmas with variable dust charge, Boltzmann electrons distribution and iso-nonthermal distributed ions. During the last few years, a number of authors have studied the KP-type equations. Duan [11] had considered transverse perturbations and studied the propagation of DASWs in an unmagnetized plasma in the framework of KP equation. Gill et al. [12] reported the existence of compressive and rarefactive DA solitons from the solution of KP equation derived in two dimensional dusty plasma in the presence of two temperature ions. Masood et al. [13] have studied the formation of two-dimensional non-planar electrostatic shocks in an unmagnetized asymmetric pair-ion plasma and found that the kinematic viscosity enhances the shock strength. In another work, Dorranian et al. [14], derived a KP equation using reductive perturbation method and studied the effects of nonthermal ions on the solitons in a dusty plasma. They also showed that the formation of compressive and rarefactive solitary waves are strongly dependent on the number density and
http://dx.doi.org/10.1016/j.rinp.2017.01.007 2211-3797/Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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temperature of nonthermal ions. The KP, the modified KP (MKP) and the coupled KP equations for dusty plasma with two ions have been obtained in [15]. Furthermore, KP and cylindrical KP equations have studied by many authors [16–30]. Nonlinear partial differential equations (NLPDEs) can describe certain phenomena in plasma physics, fluid mechanics and other fields. Methods used to construct the analytic and exact solutions of the NLPDEs have been proposed, such as the Bäcklund transformation (BT) [31–34], the Hirota bilinear method [35] and Painlevé analysis [36]. Bäcklund transformation is the connection of several analytic solutions and auto-Bäcklund transformation can connect the different solutions of the same NLPDE. In Refs. [37,38], the homogeneous balance (HB) method was improved to investigate the BT, Lax pairs, symmetries, and exact solutions for some nonlinear PDEs [39–42]. He also showed that there is a close connection between the HB method and WTC method. By using his extended homogeneous balance method, the BTs of many nonlinear PDEs have been successfully obtained [40]. The aim of this paper is to obtain a variational principle, conservation laws, exact solutions and their interpretations for the S-KP equation. In addition we use the Painlevé analysis to show that the S-KP equation is not completely integrable. The paper is organized as follows: The introduction is presented in Section ‘Introduction’. In Section ‘Problem formulations’, the basic equations are considered and the problem formulation is derived by using the reductive perturbation method. In Section ‘C onservation laws for S-KP equation’, the variational principle and conservation law of the S-KP are obtained. In Section ‘Painlevé analysis for S-KP equation’, the Painlevé analysis is used to show that the S-KP equation is non-integrable. In Section ‘Auto-Bäcklund transformations for S-KP equation’, auto-Bäcklund transformations and new exact solutions of the S-KP equation are obtained by using an extended homogeneous balance method. In Section ‘Stability and Mach number for S-KP equation’, we investigated the stability, electric field and Mach number of S-KP equation. The results and discussion are investigated in Section ‘Results and Discussion’. Finally, the conclusion is presented in Section ‘Conclusion’. Problem formulations The propagation of dust acoustic waves in collisionless, unmagnetized strongly coupled dusty plasma, negatively charged dust grains, Boltzmann distributed electrons and iso-nonthermal distributed ions are considered. The nonlinear dynamics of this case is governed by the following set equations [5]: @nd @ @ þ ðnd ud Þ þ ðnd td Þ ¼ 0 @t @x @y ! @ @ud @ud @ud @/ @ 2 ud @ 2 ud 1 þ sm þ ud þ td zd nd ¼ g1 þ @t @x @y @x2 @y2 @t @x ! @ @ td @ td @ td @/ @ 2 td @ 2 td 1 þ sm þ ud þ td zd þ nd ¼ g1 @t @x @y @x2 @y2 @t @y
and inverse of dust plasma frequency x1 pd ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T i =ð4pnod Z od e2 Þ.
sm
is the viscoelastic relaxation time normalized by the dust plasma period x1 pd ; g1 is the normalized viscosity coefficient and form the equilibrium charge neutrality condition, Z od nod þ noe ¼ noi . The normalized densities of the Boltzmann distributed electron and isononthermal distributed ion are given by
l
eri / ; 1l 3 1 4ð1 aÞ 1 pffiffiffiffi ð/Þ2 þ /2 þ . . . ; ni ¼ 1/ 1l 2 3 p ne ¼
ð2Þ
where ri ¼ T i =Te; T e is temperature of electrons, l ¼ n0e =n0i and a is ratio of the free temperature to the trapped ion temperature ða ¼ T h =T ht Þ. If a < 0, it represents a vortex-like excavated trapped ion distribution, and if a ¼ 1ða ¼ 0Þ we have Maxwellian (flattopped) ion distribution. Now, to derive the S-KP equation for the propagation of small but finite amplitude DASW, we use the standard reductive perturbation technique in which the independent variables n ¼ 1=4 ðx ktÞ; g ¼ 1=2 y; s ¼ 3=4 t; g1 ¼ 1=4 g0 ; sm ¼ 1=4 smo where is a small dimensionless expansion parameter which characterizes the strength of nonlinearity in the system and k is the phase velocity of the wave along the x direction and normalized by dust acoustic velocity. Now we expand dependent variables as follows:
nd ¼ 1 þ n1d þ 3=2 n2d þ . . . ; ud ¼ u1d þ 3=2 u2d þ . . . ;
td ¼ 5=4 t1d þ 7=4 t2d þ . . . ; / ¼ /1 þ 3=2 /2 þ . . . ;
Z d ¼ 1 þ c1 /1 þ 3=2 c1 /2 þ . . . : ð3Þ
Finally, after the usual elimination of unwanted terms we get the yields S-KP equation
@ @/1 @ @ 3 /1 A ð/1 Þ3=2 þ B @n @ s @n @n3
!
þC
@ 2 /1 ¼0 @y2
ð4Þ
where, the nonlinear coefficient A, the dispersion coefficient B and the transverse coefficient C
2ð1 aÞk3 k3 k 1 lri þ 1 ; B ¼ ; C ¼ ; 2 ¼ c1 þ : A ¼ pffiffiffiffi 2 k 1l 2 3 pð1 lÞ
ð5Þ
Eq. (4) shows that the well known S-KP equation which describe the nonlinear propagation of the DIA solitary waves in a coupling dusty plasma with iso-nonthermal ions. Conservation laws for S-KP equation
@2/ @2 / þ ¼ nd Z d þ ne ni ; @x2 @y2 ð1Þ
here nd ; ne and ni are the number density of dust particle, electrons and ions that normalized with nod . Z d is the number of charges residing onto the dust grain surface normalized by its equilibrium value Z 0d . ud and td are velocity components of the dust particle in x and y-direction, respectively. Velocities normalized by the dust pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi acoustic speed C d ¼ Z d T i =md ; T i is temperature of ions and md is dust particle mass. / is the electrostatic wave potential normalized by T i =e (e is the magnitude of the electron charge). Space and time qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi variables are normalized by Debye length kD ¼ md =ð4pnod Z 2d e2 Þ
Mathematical expressions of physical laws are the conservation laws, such as conservation of mass, momentum, and energy. So, the conservation laws play a very important role in the solution and the reduction of NLPDEs. The conservation laws can be further used to study the properties of the existence, uniqueness and stability of solutions and also in the design of numerical integrators for NLPDEs [43–46]. Consider the S-KP Eq. (4) and we can be written as
uxt þ
3A 1=2 2 3A 1=2 ux þ u u uxx þ Bu4x þ Cuyy ¼ 0: 4 2
ð6Þ
The Lagrangian formal for the S-KP equation
E uxt þ
3A 1=2 2 3A 1=2 u u uxx þ Bu4x þ Cuyy ¼ 0 ux þ 4 2
is defined by
ð7Þ
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L ¼ # uxt þ
3A 1=2 2 3A 1=2 ux þ u u uxx þ Bu24x þ Cuyy ¼ 0; 4 2
ð8Þ
here # is a new dependent variable. Consequently the adjoint equation to (8) has the form
dL ¼ 0; F du
ð9Þ
the variational derivatives of the Lagrangian defined by
dL @L @L @L @L ¼ Dx þ Dx Dt þ D2x ; du @u @ux @uxt @uxx
ð10Þ
then, we obtain the adjoint equation of (7)
E ¼ #xt þ
3A 1=2 u #xx þ B#4x þ C#yy ¼ 0: 2
ð11Þ
Suppose that E jv ¼ #ðt; x; y; uÞ ¼ qE, where q is a certain function, and E is given by (7), from the coefficient of term uxt it’s obtained that q ¼ #u and
We rewrite the formal Lagrangian L in the symmetric form
L¼#
1 1 3A 1=2 2 3A 1=2 uxt þ uxt þ u u uxx þ Bu4x þ Cuyy ¼ 0 ux þ 2 2 4 2 ð21Þ t
x
y
by using the relation W ¼ g n ut n ux n uy , we obtain
1 1 C t ¼ ft L w#x þ wx #; 2 2 3A 1=2 1 u ux #t B#xxx C x ¼ f xL þ W 2 2 3A 1=2 1 þ Wx u # þ B#xx þ #W t B#x W xx þ B#W xxx ; 2 2 C y ¼ f y L cW#y þ cW y #:
ð22Þ
ð23Þ ð24Þ
Dx ð#Þ ¼ #x þ ux #u ;
ð12Þ
For the generator X 1 ¼ @t@ , we have nt ¼ 1; n x ¼ 0; n y ¼ 0; g ¼ 0 and W ¼ ut , thus we obtain the following conserved vector
D2x ð#Þ ¼ #xx þ 2ux #ux þ uxx #u þ u2x #uu ;
ð13Þ
Ct ¼
D2y ð#Þ ¼ #yy þ 2uy #uy þ uyy #u þ u2y #uu ;
ð14Þ
Dx Dt ð#Þ ¼ #xt þ ux #ut þ uxt #u þ ux ut #uu þ ut #xu ;
ð15Þ
D3x ð#Þ ¼ #xxxx þ 4ux #xxxu þ 6uxx #xxu þ 6u2x #xxuu þ 4uxxx #xu
ð25Þ
þ 12ux uxx #xuu þ 4u3x #xuuu þ 4ux uxxx #uu þ uxxxx #uu þ 6u2x uxx #uuu þ u3x #xuuu þ u4x #uuuu ; #xt þ ux #ut þ uxt #u þ ux ut #uu þ ut #xu 3A þ u1=2 #xx þ 2ux #ux þ uxx #u þ u2x #uu 2
þ B #yy þ 2uy #uy þ uyy #u þ u2y #uu þ C #xxxx þ 4ux #xxxu þ 6uxx #xxu þ 6u2x #xxuu þ 4uxxx #xu þ12ux uxx #xuu þ 4u3x #xuuu þ 4ux uxxx #uu þ uxxxx #uu þ 6u2x uxx #uuu þu3x #xuuu þ u4x #uuuu 3A 3A ¼ q uxt þ u1=2 u2x þ u1=2 uxx þ Bu4x þ Cuyy ; 4 2
ð16Þ
the
generator
X4 ¼
2ct
@ , @y
we
have
nt ¼ 0; n x ¼ y; n y ¼ 2ct; g ¼ 0 and W ¼ yux þ 2ctuy , thus we obtain the following conserved vector
1 1 yu #x ct#x uy y#uxx þ ct#uxy ; 2 x 2 1 3A 1 C x ¼ y#uxt y#u1=2 u2x þ cy#uyy þ yux #t þ Byux #xxx 2 4 2 ctuy #t 2cBtuy #xxx Byuxx #xx þ 2ctBuxy #xx þ c#uy þ ct#uyt þ Byuxxx #x 2cBt#x uxxy þ 2ctBt#uxxxy ;
ð17Þ
Observing that the coefficient of uu2x yield #uu ¼ 0, we obtain consequently from the coefficients of u2x that #u ¼ 0 the function # must satisfy #xx ¼ 0. Then, we obtain the conservation law
ð19Þ
from the classical Lie group theory, we can obtain the symmetries of (9) as follows:
@ @ @ @ @ ; X2 ¼ ; X3 ¼ ; X 4 ¼ y 2ct ; @t @x @y @x @y 3 @ 3 @ @ @ At þ 2Ax þ cAt 4Ay þ 2Au : X5 ¼ 4 @t 4 @x @y @u
For
@ y @x
Ct ¼
u4x #uuuu þ ð4#xuuu þ #uuu Þu3x 3A þ u1=2 ð#uu þ 6#xxuu þ 6uxx #uuu Þu2x 2
1 þ ut #uu þ 3Au2 #ux þ 4C#xxxu þ 12#xuu þ 4uxxx #uu ux 3A 1=2 u #u þ 6C#xxu uxx þ #xt þ 4Cuxxx #xu þ 3u2xx #uu þ 2 3A 1=2 3A þ u #xx þ B#yy þ C#xxxx þ #u uxt þ u1=2 uxx þ Buyy þ Cuu4x 2 2 þ ut #xu þ 2Buy #yu þ Bu2y #uu 3A 3A ¼ q uxt þ u1=2 u2x þ u1=2 uxx þ Bu4x þ Cuyy : ð18Þ 4 2
Dt ðC t Þ þ Dx ðC x Þ þ Dx ðC y Þ ¼ 0;
1 3A 1=2 2 3A 1=2 1 #uxt þ #u #u uxx þ B#uxxxx þ c#uyy þ #x ut ; ux þ 2 4 2 2 3A 1=2 1 x ux ut þ #t ut þ But #xxx C ¼ u 2 2 3A 1=2 1 u #uxt B#xx uxt #utt þ B#x uxxt B#euxxxt ; 2 2 C y ¼ cut #y cuyt #:
X1 ¼
ð20Þ
ð26Þ
3A ct#u1=2 u2x 3Act#u1=2 uxx 2cBt#uxxxx 2 þ cyux #y 2c2 t#y uy c#ux cy#uxy :
C y ¼ 2ct#uxt
Painlevé analysis for S-KP equation Considering the S-KP (4) and by using /ðx; y; tÞ ¼ u2 ðx; y; tÞ, Eq. (4) becomes: 2 ! @2u @u @u @2u @u @ 4 u2 @ 2 u2 2u ð27Þ þ B 4 þ C 2 ¼ 0: þ2 þ A 3u2 2 þ 6u @x @y @x@t @x @t @x @x The Painlevé analysis for PDEs was suggested in [36], which required that the solutions should be single value around movable singularity manifolds. To be precise, if the singularity manifold is determined by uðz1 ; z2 ; z3 ; . . .Þ ¼ 0, and u ¼ uðz1 ; z2 ; z3 ; . . . ; zn Þ is a solution of the PDEs, then we assume that
uðx; y; tÞ ¼ /a ðx; y; tÞ
1 X uj ðx; y; tÞ/ j ðx; y; tÞ;
ð28Þ
j¼0
where uðz1 ; z2 ; z3 ; . . . ; zn Þ; uj ¼ uj ðz1 ; z2 ; z3 ; . . . ; zn Þ, and u0 – 0, are analytic functions of zj in a neighborhood of the manifold [40] and a is an integer. Substitution of (28) into (27), we see that a ¼ 2 and (28) takes the form as:
uðx; y; tÞ ¼
1 X uj ðx; y; tÞuj2 ðx; y; tÞ j¼0
ð29Þ
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then we have the following recursion relation: 2 ðum1;x þ ðm 2Þum /x Þ ðj m 4Þujm2 /t þ ujm3;t
i.e
ðj þ 1Þðj 6Þðj 7Þðj 10Þ ¼ 0:
þum ujm4;xt þ ðj m 5Þum ujm3;x /t þ ðj m 5Þum ujm3;t /x þðj m 4Þðj m 5Þum ujm2 /t /x þ ðj m 5Þum ujm3 /xt þA 6 ðj m 2Þum ujmn /x þ un ujmn1;x ððm 2Þum /x þ um1;x Þ þ3 un um ujmn2;xx þ 2ðj m n 3Þun um ujmn1;x /x
Then, the resonant points read from (34)
j ¼ 1; 6; 7; 10
"
þ6ðj 7Þðum ujm32 Þxx /xx þ 12ðj 6Þðj 7Þðum ujm2 Þx /x /xx þ4ðj 7Þðum ujm3 Þx /3x þ 4ðj 6Þðj 7Þðum ujm2 Þ/x /3x
u3 ¼
þ6ðj 5Þðj 6Þðj 7Þðum ujm2 Þ/2x /xx þ 3ðj 6Þðj 7Þðum ujm2 Þ/2xx þðj 4Þðj 5Þðj 6Þðj 7Þðum ujm Þ/4x
ð30Þ
where m ¼ 0; 1; 2; . . . ; j and n ¼ 0; 1; 2; . . . ; ðj mÞ. For j ¼ 0, in equation (30), we obtain
ð31Þ
For j ¼ 1, in equation (30), we obtain
20B / : A xx
ð32Þ
Substituting Eq. (31) into Eq. (30), and collecting coefficients of uj we obtain
40B2 ðj þ 1Þðj 6Þðj 7Þðj 10Þuj /6x A ¼ F j uj1 ; . . . ; u0 ; /t ; /x ; /xx ; . . . ; j ¼ 1; 2; 3; . . . :
# /t 2 þ þ 16wx þ 4w ¼ H: /x
ð36Þ
" # 2 ! /2y /y /xy /y K C / þ 10 þ 12 w þ t ð1 wÞ þ 4Bðwxx wwx Þ ¼ G=/x ; 2 /x /x /x /x /x /x 7 ð37Þ
þC ðum ujm4 Þyy þ 2ðj 7Þðum ujm3 Þy /y
u1 ¼
2
For j ¼ 3, in Eq. (30), we obtain
þðj 7Þðum ujm3 Þ/4x þ 4ðj 5Þðj 6Þðj 7Þðum ujm2 Þx /3x
20B 2 / : A x
/y u2 ¼ K 2C /x
þ4ðj 7Þðum ujm3 Þ3x /x þ 6ðj 6Þðj 7Þðum ujm2 Þxx /2x
u0 ¼
ð35Þ
j ¼ 1 is often called the universal. Putting j ¼ 1; 2; 3; . . . ; 10 in (33) or (30) and using (31), we get. For j ¼ 2, in Eq. (30), we obtain
þðj m n 2Þðj m n 3Þun um ujmn /2x þðj m n 3Þun um ujmn1 /xx þ B ðum ujm4 Þ4x
þðj 6Þðj 7Þðum ujm2 Þ/2y þ ðj 7Þðum ujm3 Þ/yy ;
ð34Þ
ð33Þ
where K ¼ 5=ð12AÞ; w ¼ /xx =/x . We can see that (37) is satisfied equivalently if (36) is satisfied, because G ¼ ðH=/x Þx . The compatibility condition at resonance j ¼ 6 is unsatisfied. From this analysis we see that S-KP equation is non-Painlevé and because of Painlevé conjecture it is non-integrable. Auto-Bäcklund transformations for S-KP equation. Consider the S-KP Eq. (27), according the idea of improved HB [39], we seek for ABT of Eq. (27), when balancing u2 uxx with uuxxxx then gives N ¼ 2. therefore, we may choose
uðx; y; tÞ ¼
@ 2 f ðwÞ 00 0 ¼ f ðwÞw2x þ f ðwÞwxx þ a; @x2
ð38Þ
where a is a function in x; y; t (or constant), f ; w are functions to be determined later. We assume the solution as the form f ðwÞ ¼ clnðwÞ, substituting from this solution and (38) into (27), then we obtain
1 1 14 7 2 ð7Þ 2 Bcw8x þ Bcw6x wxx þ Ac w6x wxx Ac w8x þ f 6 120 3 30
1 1 1 71 27 26 1 1 ð6Þ 1 þf cwt w5x þ 2aBw6x aAcw6x cCw6x Bcw4x w2xx þ Ac2 w4x w2xx Bcw5x wxxx þ Ac2 w5x wxxx c 6Acw4x w2xx Acw5x wxxx cCw4x w2y 6 2 10 2 40 3 10 5 15 5 4 5 15 1 9 2 2 3 ð5Þ 4 4 3 4 2 3 3 þf cwx wxt cwt wx wxx þ 30aBwx wx wxx aAcwx wxx Ccwx wxx 73Bcwx wxx þ Ac wx wxx 6 3 2 2 2 272 7 2 3 53 1 2 4 4 1 1 3 2 3 4 Bcwx wxx wxxx þ Ac wx wxx wxxx Bcwx w4x þ Ac wx w4x Ccwx wxy wy Ccwx wxx w2y Ccw4x wyy 3 2 6 8 3 2 6 3 2 ð4Þ 4 2 2 4 2 2 3 2 2 2 þf 2awt wx þ 3a Awx þ 2aCwx 5cwx wxt wxx 3cwt wx wxx þ 90aBwx wxx 24aAcwx wxx 19Bcw4xx þ Ac w4xx 2 5 3 3 2 3 2 2 2 2 2 cwx wxxt cwt wx wxxx þ 40aBwx wxxx 8aAcwx wxxx 144Bcwx wxx wxxx þ 8Ac wx wx wxxx 46Bcwx wxxx 3
14 2 2 2 þ Ac w2x w2xxx 59Bcw2x wxx w4x þ Ac w2x wxx w4x Bcw3x w5x 2Dcw2x w2xy Ccw3x wxyy 2Ccw2x wxxy wy 4Ccwx wxx wxy wy Ccw2xx w2y Ccw2x wxx wyy 3 3 f
ð8Þ
þf
000
3
6aw2x wxt þ 6awt wx wxx þ 18a2 w2x wxx 3cwxt w2xx þ 30aBw3xx 9aAcwxx 6xwx wxx wxxt 2cwx wxt wxxx
2cwt wxx wxxx þ 120aBwx wxx wxxx 30aAcwx wxx wxxx 50Bcwxx w2xxx þ 3Ac2 wxx w2xxx cw2x wxxxt þ 30aBw2x w4x 3 3aAcw2x w4x 33Bcw2xx w4x þ Ac2 w2xx w4x 44Bcwx wxxx w4x 18Bcwx wxx w5x Bcw2x w6x Ccw2x wxxyy 2 4Ccwx wxxy wxy 2Ccwxx w2xy 2Ccwx wxx wxyy 4Ccwxx wxxy wy þ 8aCwx wxy wy þ 2aCwxx w2y þ2aCw2x wyy Ccw2xx wyy 00 2 2 þ f 6awxt wxx þ 9a2 Awxx þ 6awx wxxt þ 2awt wxxx þ 12a2 Awx wxxx 2cwxxt wxxx þ 20aBw2xxx 6aAcwxxx
2cwxx wxxxt þ 30aBwxx w4x 6aAcwxx w4x 6Bcw24x þ 12aBwx w5x 8Bcwxxx w5x 2Bcwxx w6x
0 2Ccw2xxy 2Ccwxx wxxyy þ 4aCw2xy þ 4aCwx wxyy þ 4aCwxxy wy þ 2aCwxx wyy þ f 2awxxxt þ 3a2 Aw4x þ 2aBw6x þ 2aCwxxyy ¼ 0: ð39Þ
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To ð8Þ
ð7Þ
obtain ð6Þ
f ;f ;f ;f the form:
the
ð5Þ;...
solution,
we
set
the
coefficients
of
Case 1.
0
and f equal zero and we assumed wðx; y; tÞ as
uðx; y; tÞ ¼
wðx; y; tÞ ¼ 1 þ e ; h
where h ¼ kðtÞ þ k1 x þ k2 y:
ð40Þ
20B ; k1 ¼ k2 A 2 kðtÞ ¼ k1 ð4Bk1 CÞt;
Case1: c ¼
20B ; k1 ¼ k2 ; A 2 kðtÞ ¼ k1 ð4Bk1 þ CÞt; Case2: c ¼
a¼
3A
5Bk1 2 k1 2 sech x þ y 4Bk1 þ C t ; A 2 2
uðx; y; tÞ ¼ ; ð41aÞ
a ¼ 0;
then the solution of Eq. (27) take the form:
ð42Þ
Case 2.
Substituting from (40) into (39) we get 2 10Bk1
2
5Bk1 2 k1 2 2 þ 3sech x þ y þ 4Bk1 C t ; 3A 2
by retrain the original variable we get the solution of the S-KP equation. Case 1. 2
ð41bÞ
ð43Þ
/1 ðx; y; tÞ ¼
5Bk1 3A
!2 2 3sech
2
2 k1 2 ; x þ y þ 4Bk1 C t 2 ð44Þ
Fig. 1a. The variation of rarefactive two-soliton solutions profile /1 (44) with the iso-nonthermal ion for the parameters y ¼ 0:1; k ¼ 0:09; a ¼ 0:3; l ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 1b. The variation of rarefactive solitary wave profile /2 (45) with the iso-nonthermal ion for the parameters y ¼ 0:1; k ¼ 0:09; a ¼ 0:3; l ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5.
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Case 2.
Stability and Mach number for S-KP equation
!2
ð45Þ
To study stability and the traveling wave solutions of the S-KP Eq. (4), we define a new variable as follows: v ¼ qðlx þ my XtÞ, where l and m are direction cosines of the angles made by the wave
We have represented these solutions (44), (45) for a set of parameter values in Figs (1a,1b). Where the amplitudes
X is the speed of the traveling wave. Substituting /ðvÞ ¼ /1 ðx; y; tÞ
2
/2 ðx; y; tÞ ¼
2
5Bk1 A
2
sech
4
k1 2 x þ y 4Bk1 þ C t : 2
2
2
/1m ¼ ð5Bk1 =3AÞ ; /2m ¼ ð5Bk1 =AÞ . The solution (44) is a new exact solution for S-KP equation which describe the solitary waves solution of S-KP equation. It clear from /1 and /2 that the solitary waves will be associated with positive potential ð/1m Þ > 0; ð/2m Þ > 0 only, then the solution of S-KP equation gives rarefactive solitary waves. Furthermore, the nonlinear coefficient A is assumed to be positive to ensure the reality of the solutions.
2
propagation with x axis and y axis ðl þ m2 ¼ 1Þ, respectively. Here
into Eq. (4), integrating twice and using the vanishing boundary 2
conditions for / ! 0; d/=dv ! 0 and d /=dv2 ! 0 for jvj ! 1, one can obtain the following ordinary differential equation [19, 47, 48]: 2
Bq2 l
4
d / 2 þ Al ð/Þ3=2 ðlX Cm2 Þ/ ¼ 0: dv2
ð46Þ
Fig. 1c. The periodic travelling wave solution /3 of (48) have plotted for different values of the parameters l ¼ 0:62; m ¼ 0:38; X ¼ 0:06; y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 2a. Variation of amplitude /1m (44) of DA solitary waves with ion iso-nonthermal parameter and the other parameters are k ¼ 0:09; ri ¼ 0:2, and c1 ¼ 0:5.
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From (46), we have
2
d/ 1 4 2 5=2 2 2 : Al ¼ l X Cm ð/Þ / 4 dv 5 Bq2 l
ð47Þ
Eq. (47) has solitonic solutions and one-soliton solution for this equation is given by
/3 ðx; y; tÞ ¼ /3m sech
4
v
: D
ð48Þ
We have represented this solution (48) for a set of parameter values in Fig (1c) which corresponds to a periodic traveling wave
2 2 and solution of Eq. (4). where /3m ¼ 5ðlX Cm2 Þ=ð4Al Þ pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 D ¼ 4 Bl =ð lX Cm Þ are the amplitude and width of the solitary waves, respectively. /3m is always positive, the solution of S-KP equation gives rarefactive solitary waves only. In order to investigate the stability solution (48) of the S-KP, we use the following energy-like equation
Fig. 2b. Variation of amplitude /3m (48) of DA solitary waves with ion iso-nonthermal parameter and the other parameters are y ¼ 0:1; l ¼ 0:62; m ¼ 0:38; X ¼ 1:1; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 2c. Variation of width D (48) of DA solitary waves with ion iso-nonthermal parameter and the other parameters are y ¼ 0:1; X ¼ 1:1; l ¼ 0:62; m ¼ 0:38; ri ¼ 0:2, and c1 ¼ 0:5.
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Fig. 2d. Plot of the solitary wave amplitude /3m of (48) is increasing with increasing X and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; l ¼ 0:62; m ¼ 0:38; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 2e. Plot of the solitary wave amplitude /3m of (48) is increasing with increasing X and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; l ¼ 0:62; m ¼ 0:38; ri ¼ 0:2, and c1 ¼ 0:5.
2 1 d/ þ wð/Þ ¼ 0; 2 dv
ð49Þ
where wð/Þ is the pseudopotential or Sagdeev potential (is called the potential energy) and is given by
wð/Þ ¼
1 Bq2 l
4
4 2 lX Cm2 /2 Al ð/Þ5=2 ; 5
ð50Þ
while the first term in Eq. (49) can be considered as the kinetic energy. It is clear that wð/Þ ¼ 0 and dwð/Þ=d/ ¼ 0 at ð/Þ ¼ 0. A stable solution must satisfy the following conditions
(iii) The existence condition of solitary wave solution (48) requires that 2
d wð/Þ d/2
j/¼0 ¼
< 0, which implies that
lX Cm2 2Bl
4
¼
S 2Bl
4
< 0;
ð51Þ
where the parameters l and B are positive. Therefore the above expression shows that the solitary wave solution (48) exists whenever the condition
S ¼ lX Cm > 0 or X > 2
(i) The kinetic energy is exist it requires that wð/Þ < 0 for the interval /min < / < 0 for the rarefactive solitary waves. Where /min is the minimum value of / for which wð/Þ < 0. (ii) There must exist a non-vanishing point / ¼ /min such that wð/ ¼ /min Þ ¼ 0.
d2 wð/Þ j/¼0 d/2
1l l
2
! C;
ð52Þ
is satisfied and the width D of a stable solitary wave is real. The magnitude of the solionic electric field can be found by ~ E ¼ r/,
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Fig. 2f. The variation of width D of (48) is decreasing with increasing X for and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; l ¼ 0:62; m ¼ 0:38; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 3a. The variation of amplitude /3m of (48) is decreasing with increasing l and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; X ¼ 1:1; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 3b. The variation of width D of (48) is increasing with increasing l for and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; X ¼ 1:1; ri ¼ 0:2, and c1 ¼ 0:5.
O.H. EL-Kalaawy et al. / Results in Physics 7 (2017) 934–946 5=2 !
50ðl þ mÞðlX Cm2 Þ 4 ~ sech kv ðtanhkvÞ; where E¼ pffiffiffi 6 2 2A Bl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lX m2 C ¼ ðlx þ my XtÞ; k ¼ pffiffiffi 2 : 4 Bl
Results and discussion
v
ð53Þ The true Mach number is defined as
MT ¼
Xþk
ð54Þ
k
1=2 i þ1 where k ¼ c1 þ lr is the phase speed and X is the incremen1l tal soliton speed. Consequently, we find that
X ¼ c1 þ
lri þ 1 1l
1=2 ðM T 1Þ
943
ð55Þ
In order to investigate the nonlinear properties of small amplitude DAWs including negatively charged dust grains variations and Boltzmann distributed electrons, we have numerically analyzed /1m or /2m ; /3m and D for the same parameters corresponding to space strongly coupled dusty plasma. By taking these parameters, the DA solitary wave solutions (44), (45) and (48) obtained in the small amplitude approximation clearly indicate the existence of the rarefactive solitary waves, corresponding to a hump in the ion density. It turns out that criteria for these are independent of the sign of the nonlinear coefficient A. Obviously, this type of solitary solution which is due to the combined effects of nonlinear pffiffiffiffi term / (containing A) arises due to the trapped ion distribution, and the dispersion term (containing B). We analyzed numerically
Fig. 4a. Plot of the solitary wave electric field E3 magnitude for different values of, based on (53): (4a) The black line is a ¼ 0:5, the red line is a ¼ 0:35, the blue line is a ¼ 0:2, the green line is a ¼ 0 and bold black line is a ¼ 0:5 and the other parameters are l ¼ 0:62; m ¼ 0:38; X ¼ 0:65; y ¼ 0:1; t ¼ 1; l ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4b. Plot of the solitary wave electric field E3 magnitude for different values of, based on (53): (4b) The black line is l ¼ 0:9, the red line is l ¼ 0:7, the blue line is l ¼ 0:5, the green line is l ¼ 0:3 and bold black line is l ¼ 0:1 and the other parameters are l ¼ 0:62; m ¼ 0:38; X ¼ 0:65; y ¼ 0:1; t ¼ 1; a ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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the wave amplitude /1m or /2m ; /3m , the width of the wave D, and the profile for the potential / of the solitary wave due to the following group of parameters: l, ri ; c, and considering a, the ratio of the free temperature to the trapped ion temperature.
(ii) The solitary waves for Fig. 2a implies that the amplitude /1m of (44) is decreased with l increase while the amplitude /1m is increased with a increase. (iii) It is shown from Figs. 2b, 2c and 2d, 2e 2f that both the amplitude /3m and width D of these traveling structures increases and decreases with the increase of l and a at X ¼ 0:3. Also, the amplitude /3m increases with the increase of l at jaj ¼ 0:3 and X ¼ 0:3. But both the amplitude /3m and width D of these traveling structures increase and decrease with the increase in X at jaj ¼ 0:3 and l ¼ 0:5. We note that the two figures (2d, 2e) are very similar, except
(i) Figs. 1a,1b,1c shows the variation of the solutions of S-KP equation and the wave potential /, as a function of the variables n; s. It can be related to the behavior of a solitary wave from the figures with only negative potential. The solitary wave structures which are due to the negatively charged dust grains caused by the ions following the vortex-like distribution.
Fig. 5a. Variation of amplitude /3m (44) of DA solitary l ¼ 0:62; m ¼ 0:38; X ¼ 0:06; k ¼ 0:09; ri ¼ 0:2; a ¼ 0:3 and c1 ¼ 0:5.
waves
with
ion
iso-nonthermal
parameter
and
the
other
parameters
are
Fig. 5b. Variation of amplitude /3m (44) of DA solitary l ¼ 0:62; m ¼ 0:38; X ¼ 0:06; k ¼ 0:09; ri ¼ 0:2; a ¼ 0:3 and c1 ¼ 0:5.
waves
with
ion
iso-nonthermal
parameter
and
the
other
parameters
are
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Fig. 5c. Variation of width D (44) of DA solitary l ¼ 0:62; m ¼ 0:38; X ¼ 0:06; k ¼ 0:09; ri ¼ 0:2; a ¼ 0:3 and c1 ¼ 0:5.
waves
that the soliton amplitudes are reduced by a factor of about 3. (iv) From Figs. 2d,2e,2f we can see that the amplitude /3m (width D of the soliton increases (decreases) when X is increased. On the other hand, from the definition of the soliton amplitude and its width, one can find that the amplitude (width) decreases (increases) with an increasing value for the parameter l. The values of l that satisfy the existence condition are strongly controlled by arbitrary parameter X see Figs. 3a and 3b. (v) Figs. 4a and 4b show how the electric field varies with isononthermal and the proportion of trapped ions present. With increased iso-nonthermal we see the field becomes less localized with lower amplitude. As the proportion of trapped ions is decreased (lower value of), the magnitude of the electric field decreases. (vi) It is shown from Figs. 5a, 5b and 5c that both the amplitude /3m and width D of these solitary structures increase and decrease with the increase of M T at jaj ¼ 0:3 and as l is increased, their amplitude increases, but their width decreases, whenever the amplitude increases and width decreases at M T increase. Conclusion In this present work, we have investigated the nonlinear propagation of DA solitary wave in an unmagnetized strongly coupled dusty plasma. A S-KP equation is derived which governs the dynamics of small amplitude DA solitary wave in strongly coupled dusty plasma using the standard reductive perturbation technique. The variational principle and conservation laws are obtained. In this paper, we used the Painlevé analysis to show that the equations S-KP is not complectly integrable. The exact solutions of the S-KP equation was solved by well-known auto-Bäcklund transformation. These solutions are numerically analyzed and the effect of various dusty plasma constituents DA solitary wave propagation is taken into account. The stability condition and electric field of S-KP equation are also presented. The solutions of S-KP Eqs. (44), (45) and (48) give rarefactive solitary waves only.
with
ion
iso-nonthermal
parameter
and
the
other
parameters
are
References [1] Hase Y, Watanabe S, Tanaca H. Cylindrical ion acoustic soliton in plasma with negative ion. J Phys Soc Jpn 1985;54:4115–25. [2] Duan WS, Hong XR, Shi YR, Sun JA. Envelop solitons in dusty plasmas for warm dust. Chaos Soliton Fractals 2003;16:767–77. [3] Mamun AA, Shukla PK. Cylindrical and spherical dust ion-acoustic solitary waves. Phys Plasmas 2002;9:1468–70. [4] Waleed MM. Dust-ion-acoustic solitons and shocks in dusty plasmas. Chaos Soliton Fractals 2006;28:994–9. [5] Shukla PK, Mamun AA. Introduction to Dusty Plasma Physics (Bristol U K: Inst. Phys. Publ.); 2002. [6] Shukla PK, Silin VP. Dust ion-acoustic wave. Phys Scr 1992;45:508. [7] Rao NN, Shukla PK, Yu M. Dust-acoustic waves in dusty plasmas. Planet Space Sci 1990;38:543–6. [8] Shah A, Saeed R. Nonlinear Korteweg-de Vries-Burger equation for ionacoustic shock waves in the presence of kappa distributed electrons and positrons. Plasma Phys Control Fusion 2011;53:095006–16. [9] Ghosh S, Bharuthram R. Ion acoustic solitons and double layers in electronpositron-ion plasmas with dust particulates. Astrophys Space Sci 2008;314:121–7. [10] Duha SS, Mamun AA. Dust-ion-acoustic shock waves due to dust charge fluctuation. Phys Lett A 2009;373:1287–9. [11] Duan WS. Weakly two-dimensional dust acoustic waves. Phys Plasmas 2001;8:3583–6. [12] Gill TS, Saini NS, Kaur H. The Kadomstev-Petviashvili equation in dusty plasma with variable dust charge and two temperature ions. Chaos Solitons Fractals 2006;28:1106–11. [13] Masood W, Rizvi H. Two dimensional nonplanar evolution of electrostatic shock waves in pair-ion plasmas. Phys Plasmas 2012;19:012119–26. [14] Dorranian D, Sabetkar A. Dust acoustic solitary waves in a dusty plasma with two kinds of nonthermal ions at different temperatures. Phys Plasmas 2012;19:013702–6. [15] Duan WS. The Kadomtsev-Petviashvili equation of dust acoustic waves for hot dust plasmas. Chaos Solitons Fractals 2002;14(3):503–6. [16] Saha A, Chatterjee P. Dust ion acoustic travelling waves in the framework of a modified Kadomtsev-Petviashvili equation in a magnetized dusty plasma with superthermal electrons. Astrophys Space Sci 2014;349:813–20. [17] Adhikary NC, Deka MK, Dev AN, Sarma J. Modified Korteweg-de Vries equation in a negative ion rich hot adiabatic dusty plasma with non-thermal ion and trapped electron. Phys Plasmas 2014;21:083703–0837708. [18] El-Shewy EK, Abo el Maaty MI, Abdelwahed HG, Elmessary MA. Solitary solution and energy for the Kadomstev-Petviashvili equation in two temperatures charged dusty grains. Astrophys Space Sci 2011;332:179–86. [19] Pakzad HR. Dust acoustic solitary and shock waves in coupled dusty plasmas with variable dust charge and vortex-like ion distribution. Astrophys Space Sci 2010;330:301–10. [20] Liu Hai-Feng, Wang Shi-Qing, Yang Fa-Zhan. Propagation of two dimensional cylindrical fast magnetoacoustic solitary waves in a warm dust plasma. Astrophys Space Sci 2013;347:139143.
946
O.H. EL-Kalaawy et al. / Results in Physics 7 (2017) 934–946
[21] Ata-ur-Rahman, Ali S, Moslem WM, Mushtaq A. Two-dimensional cylindrical ion-acoustic solitary and rogue waves in ultrarelativistic plasmas. Phys Plasmas 2013;20:072103–12. [22] Darvishi MT, Najafi M, Arbabi S, Kavitha L. Exact propagating multi-anti-kink soliton solutions of a (3+1)-dimensional B-type KadomtsevPetviashvili equation. Nonlinear Dyn 2016;83:1453–62. [23] Shukla PK, Stenflo L. Transverse shear waves generating vortex-like dust fluid motions in strongly coupled dusty plasmas. Phys Lett A 2003;315:244–7. [24] El-Wakil SA, Abulwafa EM, El-Shewy EK, Gomaa H, Abd-El-Hamid HM. Nonlinear ion acoustic waveforms for Kadomstev-Petviashvili equation. Astrophys Space Sci 2013;346(1):141–7. [25] Biswas A. 1-Soliton of the generalized Camass-Holm Kadomtsev-Petviashvili equation. Commun Nonlinear Sci Numer Simul 2009;14(6):2524–7. [26] Biswas A, Ranasinghe A. 1-Soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity. Appl Math Comput 2009;214(2):645–7. [27] Biswas A, Ranasinghe A. Topological 1-soliton solution of KadomtsevPetviashvili equation with powe law nonlineaity. Appl Math Comput 2010;217(4):1771–3. [28] Triki H, Sturdevant BJM, Hayat T, Aldossary OM, Biswas A. Shock wave solutions of the variants of Kadomtsev-Petviashvili equation. Can J Phys 2011;89(9):979–84. [29] Ebadi G, Yousefzadeh NF, Triki H, Biswas A. Exact solutions of the (2+1)dimensional Camassa-Holm Kadomtsev-Petviashvili equation. Nonl Anal Modelling Control 2012;17(3):280–96. [30] Jawad AJM, Mirzazadeh M, Biswas A. Dynamics of shallow water waves with Gardner Kadomtsev-Petviashvili equation. Discrete Continuous Dyn Syst Series S 2015;8(6):1155–64. [31] Wadati M, Sunukt H, Konno K. Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws. Progress Theor Phys 1975;53:419–36. [32] Rogers C, Schief WK. Bäcklund and Darboux transformations geometry and modern applications in soliton theory. Cambridge: Cambridge University Press; 2002. [33] Ablowitz MJ, Clarkson PA. Solitons; nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press; 1991. [34] Khater AH, Helal MA, El-Kalaawy OH. Two new classes of exact solutions for the KdV equation via Bäcklund transformations. Chaos Solitons Fractals 1997;8:1901–7.
[35] Hirota R. Direct methods in soliton theory. Berlin: Springer; 2004. [36] Weiss J. The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative. J Math Phys 1983;24:1405–13. [37] Wang ML. Exact solutions for a compound KdV-Burgers equation. Phys Lett A 1996;213(5–6):279–87. [38] Fan E, Zhang HQ. New exact solutions to a system of coupled KdV equations. Phys Lett A 1998;245:389–92. [39] Chun-Ping Liu, Zhou L. A new auto-Bäacklund transformation and two-soliton solution for (3+1)-dimensional Jimbo-Miwa equation. Commun Theor Phys 2011;55:213–6. [40] Yi Qin, Yi-Tian Gao, Xin Yu, Gao-Qing Meng. Bell polynomial approach and Nsoliton solutions for a coupled KdV-mKdV system. Commun Theor Phys 2012;58(1):73–7. [41] Gao YT, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulons and symbolic computation for cosmic dust ion-acoustic waves. Phys Lett A 2006;349:314–9. [42] Gao YT, Tian B. Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion acoustic waves with symbolic computation. Phys Lett A 2007;361:523–8. [43] Ibragimov NH. A new conservation theorem. J Math Anal Apple 2007;333:311–28. [44] Abdullahi RA, Xing L. Travelling wave solutions of a two-dimensional generalized Sawada-Kotera equation. Nonlinear Dyn 2016;84:915922. [45] Abdullahi RA, Muatjetjeja Ben. Conservation laws and exact solutions for a 2D Zakharov–Kuznetsov equation. Appl Math Lett 2015;48:109–17. [46] El-Kalaawy OH. Variational principle, conservation laws and exact solutions for dust ion acoustic shock waves modeling modified Burger equation. Comput Math Appl 2016;72:1031–41. [47] Masood W, Rizvi H, Hussain S. Two dimensional electromagnetic shock structures in dense electron-positron-ion magnetoplasmas. Astrophys Space Sci 2011;332:287–99. [48] Shahmansouri M, Astaraki E. Transverse perturbation on three-dimensional ion acoustic waves in electronpositronion plasma with high-energy tail electron and positron distribution. J Theor Appl Phys 2014;8:189–201.
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Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics
Stability: Conservation laws, Painlevé analysis and exact solutions for S-KP equation in coupled dusty plasma O.H. EL-Kalaawy ⇑, S.M. Moawad, Shrouk Wael Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
a r t i c l e
i n f o
Article history: Received 25 September 2016 Received in revised form 7 November 2016 Accepted 3 January 2017 Available online 16 February 2017 Keywords: Coupled dusty plasma Schamel-Kadomtsev-Petviashvili equation Conservation law Painlevé analysis Auto-Bäcklund transformations Exact solutions Mach number and stability
a b s t r a c t The propagation of nonlinear waves in unmagnetized strongly coupled dusty plasma with Boltzmann distributed electrons, iso-nonthermal distributed ions and negatively charged dust grains is considered. The basic set of fluid equations is reduced to the Schamel Kadomtsev-Petviashvili (S-KP) equation by using the reductive perturbation method. The variational principle and conservation laws of S-KP equation are obtained. It is shown that the S-KP equation is non-integrable using Painlevé analysis. A set of new exact solutions are obtained by auto-Bäcklund transformations. The stability analysis is discussed for the existence of dust acoustic solitary waves (DASWs) and it is found that the physical parameters have strong effects on the stability criterion. In additional to, the electric field and the true Mach number of this solution are investigated. Finally, we will study the physical meanings of solutions. Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction Recently, the study of nonlinear waves in plasma is one of the hottest topics in plasma physics, specifically the dusty plasma. A dusty plasma is ordinary plasma with embedded solid microparticles. In dusty plasma, there are essential two types of acoustic waves: high frequency dust ion acoustic waves (DIAWs) involving mobile ions and static dust grains, and a low frequency dust acoustic waves (DAWs) involving mobile dust grains. Both modes have been studied theoretically and experimentally [1–10]. Dusty plasmas are present in astrophysical and space environments, like the tails of comets, interstellar medium and planetary rings [4,5]. The reason for their presence in these regions is the presence of highdensity dust grains different types of plasma wave modes. Ref. [6] was the first report recorded theoretically the existence of DIA waves. The study showed that an unmagnetized weakly coupled dusty plasma supports the DIAW whose phase velocity is much smaller (larger) than the electron (ion) thermal speed. The frequency of DIAWs is much larger (smaller) than the dust (ion) plasma frequency. At the same time, Rao et al. [7] theoretically predicted the existence of DAWs, which is provided by the inertia dust particles mass and are providing power restoration due to pressures of less inertia of electrons and ion. The DAWs have supported a great deal of interest in understanding the basic characteristics ⇑ Corresponding author. E-mail address: [email protected] (O.H. EL-Kalaawy).
and properties of local electrostatic perturbations in space and laboratory dusty plasmas. Strongly coupled dust plasma is of great interest in science because of its applications in the interior of heavy planets, plasmas produced by laser compression of matter and non-ideal plasmas for industrial applications. There are a number of authors who investigated properties of one-dimensional linear and nonlinear DAWs in coupled unmagnetized dusty plasmas. In this paper, we concern the nonlinear propagation of the dust acoustic waves in unmagnetized strongly coupled dusty plasmas with variable dust charge, Boltzmann electrons distribution and iso-nonthermal distributed ions. During the last few years, a number of authors have studied the KP-type equations. Duan [11] had considered transverse perturbations and studied the propagation of DASWs in an unmagnetized plasma in the framework of KP equation. Gill et al. [12] reported the existence of compressive and rarefactive DA solitons from the solution of KP equation derived in two dimensional dusty plasma in the presence of two temperature ions. Masood et al. [13] have studied the formation of two-dimensional non-planar electrostatic shocks in an unmagnetized asymmetric pair-ion plasma and found that the kinematic viscosity enhances the shock strength. In another work, Dorranian et al. [14], derived a KP equation using reductive perturbation method and studied the effects of nonthermal ions on the solitons in a dusty plasma. They also showed that the formation of compressive and rarefactive solitary waves are strongly dependent on the number density and
http://dx.doi.org/10.1016/j.rinp.2017.01.007 2211-3797/Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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temperature of nonthermal ions. The KP, the modified KP (MKP) and the coupled KP equations for dusty plasma with two ions have been obtained in [15]. Furthermore, KP and cylindrical KP equations have studied by many authors [16–30]. Nonlinear partial differential equations (NLPDEs) can describe certain phenomena in plasma physics, fluid mechanics and other fields. Methods used to construct the analytic and exact solutions of the NLPDEs have been proposed, such as the Bäcklund transformation (BT) [31–34], the Hirota bilinear method [35] and Painlevé analysis [36]. Bäcklund transformation is the connection of several analytic solutions and auto-Bäcklund transformation can connect the different solutions of the same NLPDE. In Refs. [37,38], the homogeneous balance (HB) method was improved to investigate the BT, Lax pairs, symmetries, and exact solutions for some nonlinear PDEs [39–42]. He also showed that there is a close connection between the HB method and WTC method. By using his extended homogeneous balance method, the BTs of many nonlinear PDEs have been successfully obtained [40]. The aim of this paper is to obtain a variational principle, conservation laws, exact solutions and their interpretations for the S-KP equation. In addition we use the Painlevé analysis to show that the S-KP equation is not completely integrable. The paper is organized as follows: The introduction is presented in Section ‘Introduction’. In Section ‘Problem formulations’, the basic equations are considered and the problem formulation is derived by using the reductive perturbation method. In Section ‘C onservation laws for S-KP equation’, the variational principle and conservation law of the S-KP are obtained. In Section ‘Painlevé analysis for S-KP equation’, the Painlevé analysis is used to show that the S-KP equation is non-integrable. In Section ‘Auto-Bäcklund transformations for S-KP equation’, auto-Bäcklund transformations and new exact solutions of the S-KP equation are obtained by using an extended homogeneous balance method. In Section ‘Stability and Mach number for S-KP equation’, we investigated the stability, electric field and Mach number of S-KP equation. The results and discussion are investigated in Section ‘Results and Discussion’. Finally, the conclusion is presented in Section ‘Conclusion’. Problem formulations The propagation of dust acoustic waves in collisionless, unmagnetized strongly coupled dusty plasma, negatively charged dust grains, Boltzmann distributed electrons and iso-nonthermal distributed ions are considered. The nonlinear dynamics of this case is governed by the following set equations [5]: @nd @ @ þ ðnd ud Þ þ ðnd td Þ ¼ 0 @t @x @y ! @ @ud @ud @ud @/ @ 2 ud @ 2 ud 1 þ sm þ ud þ td zd nd ¼ g1 þ @t @x @y @x2 @y2 @t @x ! @ @ td @ td @ td @/ @ 2 td @ 2 td 1 þ sm þ ud þ td zd þ nd ¼ g1 @t @x @y @x2 @y2 @t @y
and inverse of dust plasma frequency x1 pd ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T i =ð4pnod Z od e2 Þ.
sm
is the viscoelastic relaxation time normalized by the dust plasma period x1 pd ; g1 is the normalized viscosity coefficient and form the equilibrium charge neutrality condition, Z od nod þ noe ¼ noi . The normalized densities of the Boltzmann distributed electron and isononthermal distributed ion are given by
l
eri / ; 1l 3 1 4ð1 aÞ 1 pffiffiffiffi ð/Þ2 þ /2 þ . . . ; ni ¼ 1/ 1l 2 3 p ne ¼
ð2Þ
where ri ¼ T i =Te; T e is temperature of electrons, l ¼ n0e =n0i and a is ratio of the free temperature to the trapped ion temperature ða ¼ T h =T ht Þ. If a < 0, it represents a vortex-like excavated trapped ion distribution, and if a ¼ 1ða ¼ 0Þ we have Maxwellian (flattopped) ion distribution. Now, to derive the S-KP equation for the propagation of small but finite amplitude DASW, we use the standard reductive perturbation technique in which the independent variables n ¼ 1=4 ðx ktÞ; g ¼ 1=2 y; s ¼ 3=4 t; g1 ¼ 1=4 g0 ; sm ¼ 1=4 smo where is a small dimensionless expansion parameter which characterizes the strength of nonlinearity in the system and k is the phase velocity of the wave along the x direction and normalized by dust acoustic velocity. Now we expand dependent variables as follows:
nd ¼ 1 þ n1d þ 3=2 n2d þ . . . ; ud ¼ u1d þ 3=2 u2d þ . . . ;
td ¼ 5=4 t1d þ 7=4 t2d þ . . . ; / ¼ /1 þ 3=2 /2 þ . . . ;
Z d ¼ 1 þ c1 /1 þ 3=2 c1 /2 þ . . . : ð3Þ
Finally, after the usual elimination of unwanted terms we get the yields S-KP equation
@ @/1 @ @ 3 /1 A ð/1 Þ3=2 þ B @n @ s @n @n3
!
þC
@ 2 /1 ¼0 @y2
ð4Þ
where, the nonlinear coefficient A, the dispersion coefficient B and the transverse coefficient C
2ð1 aÞk3 k3 k 1 lri þ 1 ; B ¼ ; C ¼ ; 2 ¼ c1 þ : A ¼ pffiffiffiffi 2 k 1l 2 3 pð1 lÞ
ð5Þ
Eq. (4) shows that the well known S-KP equation which describe the nonlinear propagation of the DIA solitary waves in a coupling dusty plasma with iso-nonthermal ions. Conservation laws for S-KP equation
@2/ @2 / þ ¼ nd Z d þ ne ni ; @x2 @y2 ð1Þ
here nd ; ne and ni are the number density of dust particle, electrons and ions that normalized with nod . Z d is the number of charges residing onto the dust grain surface normalized by its equilibrium value Z 0d . ud and td are velocity components of the dust particle in x and y-direction, respectively. Velocities normalized by the dust pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi acoustic speed C d ¼ Z d T i =md ; T i is temperature of ions and md is dust particle mass. / is the electrostatic wave potential normalized by T i =e (e is the magnitude of the electron charge). Space and time qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi variables are normalized by Debye length kD ¼ md =ð4pnod Z 2d e2 Þ
Mathematical expressions of physical laws are the conservation laws, such as conservation of mass, momentum, and energy. So, the conservation laws play a very important role in the solution and the reduction of NLPDEs. The conservation laws can be further used to study the properties of the existence, uniqueness and stability of solutions and also in the design of numerical integrators for NLPDEs [43–46]. Consider the S-KP Eq. (4) and we can be written as
uxt þ
3A 1=2 2 3A 1=2 ux þ u u uxx þ Bu4x þ Cuyy ¼ 0: 4 2
ð6Þ
The Lagrangian formal for the S-KP equation
E uxt þ
3A 1=2 2 3A 1=2 u u uxx þ Bu4x þ Cuyy ¼ 0 ux þ 4 2
is defined by
ð7Þ
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L ¼ # uxt þ
3A 1=2 2 3A 1=2 ux þ u u uxx þ Bu24x þ Cuyy ¼ 0; 4 2
ð8Þ
here # is a new dependent variable. Consequently the adjoint equation to (8) has the form
dL ¼ 0; F du
ð9Þ
the variational derivatives of the Lagrangian defined by
dL @L @L @L @L ¼ Dx þ Dx Dt þ D2x ; du @u @ux @uxt @uxx
ð10Þ
then, we obtain the adjoint equation of (7)
E ¼ #xt þ
3A 1=2 u #xx þ B#4x þ C#yy ¼ 0: 2
ð11Þ
Suppose that E jv ¼ #ðt; x; y; uÞ ¼ qE, where q is a certain function, and E is given by (7), from the coefficient of term uxt it’s obtained that q ¼ #u and
We rewrite the formal Lagrangian L in the symmetric form
L¼#
1 1 3A 1=2 2 3A 1=2 uxt þ uxt þ u u uxx þ Bu4x þ Cuyy ¼ 0 ux þ 2 2 4 2 ð21Þ t
x
y
by using the relation W ¼ g n ut n ux n uy , we obtain
1 1 C t ¼ ft L w#x þ wx #; 2 2 3A 1=2 1 u ux #t B#xxx C x ¼ f xL þ W 2 2 3A 1=2 1 þ Wx u # þ B#xx þ #W t B#x W xx þ B#W xxx ; 2 2 C y ¼ f y L cW#y þ cW y #:
ð22Þ
ð23Þ ð24Þ
Dx ð#Þ ¼ #x þ ux #u ;
ð12Þ
For the generator X 1 ¼ @t@ , we have nt ¼ 1; n x ¼ 0; n y ¼ 0; g ¼ 0 and W ¼ ut , thus we obtain the following conserved vector
D2x ð#Þ ¼ #xx þ 2ux #ux þ uxx #u þ u2x #uu ;
ð13Þ
Ct ¼
D2y ð#Þ ¼ #yy þ 2uy #uy þ uyy #u þ u2y #uu ;
ð14Þ
Dx Dt ð#Þ ¼ #xt þ ux #ut þ uxt #u þ ux ut #uu þ ut #xu ;
ð15Þ
D3x ð#Þ ¼ #xxxx þ 4ux #xxxu þ 6uxx #xxu þ 6u2x #xxuu þ 4uxxx #xu
ð25Þ
þ 12ux uxx #xuu þ 4u3x #xuuu þ 4ux uxxx #uu þ uxxxx #uu þ 6u2x uxx #uuu þ u3x #xuuu þ u4x #uuuu ; #xt þ ux #ut þ uxt #u þ ux ut #uu þ ut #xu 3A þ u1=2 #xx þ 2ux #ux þ uxx #u þ u2x #uu 2
þ B #yy þ 2uy #uy þ uyy #u þ u2y #uu þ C #xxxx þ 4ux #xxxu þ 6uxx #xxu þ 6u2x #xxuu þ 4uxxx #xu þ12ux uxx #xuu þ 4u3x #xuuu þ 4ux uxxx #uu þ uxxxx #uu þ 6u2x uxx #uuu þu3x #xuuu þ u4x #uuuu 3A 3A ¼ q uxt þ u1=2 u2x þ u1=2 uxx þ Bu4x þ Cuyy ; 4 2
ð16Þ
the
generator
X4 ¼
2ct
@ , @y
we
have
nt ¼ 0; n x ¼ y; n y ¼ 2ct; g ¼ 0 and W ¼ yux þ 2ctuy , thus we obtain the following conserved vector
1 1 yu #x ct#x uy y#uxx þ ct#uxy ; 2 x 2 1 3A 1 C x ¼ y#uxt y#u1=2 u2x þ cy#uyy þ yux #t þ Byux #xxx 2 4 2 ctuy #t 2cBtuy #xxx Byuxx #xx þ 2ctBuxy #xx þ c#uy þ ct#uyt þ Byuxxx #x 2cBt#x uxxy þ 2ctBt#uxxxy ;
ð17Þ
Observing that the coefficient of uu2x yield #uu ¼ 0, we obtain consequently from the coefficients of u2x that #u ¼ 0 the function # must satisfy #xx ¼ 0. Then, we obtain the conservation law
ð19Þ
from the classical Lie group theory, we can obtain the symmetries of (9) as follows:
@ @ @ @ @ ; X2 ¼ ; X3 ¼ ; X 4 ¼ y 2ct ; @t @x @y @x @y 3 @ 3 @ @ @ At þ 2Ax þ cAt 4Ay þ 2Au : X5 ¼ 4 @t 4 @x @y @u
For
@ y @x
Ct ¼
u4x #uuuu þ ð4#xuuu þ #uuu Þu3x 3A þ u1=2 ð#uu þ 6#xxuu þ 6uxx #uuu Þu2x 2
1 þ ut #uu þ 3Au2 #ux þ 4C#xxxu þ 12#xuu þ 4uxxx #uu ux 3A 1=2 u #u þ 6C#xxu uxx þ #xt þ 4Cuxxx #xu þ 3u2xx #uu þ 2 3A 1=2 3A þ u #xx þ B#yy þ C#xxxx þ #u uxt þ u1=2 uxx þ Buyy þ Cuu4x 2 2 þ ut #xu þ 2Buy #yu þ Bu2y #uu 3A 3A ¼ q uxt þ u1=2 u2x þ u1=2 uxx þ Bu4x þ Cuyy : ð18Þ 4 2
Dt ðC t Þ þ Dx ðC x Þ þ Dx ðC y Þ ¼ 0;
1 3A 1=2 2 3A 1=2 1 #uxt þ #u #u uxx þ B#uxxxx þ c#uyy þ #x ut ; ux þ 2 4 2 2 3A 1=2 1 x ux ut þ #t ut þ But #xxx C ¼ u 2 2 3A 1=2 1 u #uxt B#xx uxt #utt þ B#x uxxt B#euxxxt ; 2 2 C y ¼ cut #y cuyt #:
X1 ¼
ð20Þ
ð26Þ
3A ct#u1=2 u2x 3Act#u1=2 uxx 2cBt#uxxxx 2 þ cyux #y 2c2 t#y uy c#ux cy#uxy :
C y ¼ 2ct#uxt
Painlevé analysis for S-KP equation Considering the S-KP (4) and by using /ðx; y; tÞ ¼ u2 ðx; y; tÞ, Eq. (4) becomes: 2 ! @2u @u @u @2u @u @ 4 u2 @ 2 u2 2u ð27Þ þ B 4 þ C 2 ¼ 0: þ2 þ A 3u2 2 þ 6u @x @y @x@t @x @t @x @x The Painlevé analysis for PDEs was suggested in [36], which required that the solutions should be single value around movable singularity manifolds. To be precise, if the singularity manifold is determined by uðz1 ; z2 ; z3 ; . . .Þ ¼ 0, and u ¼ uðz1 ; z2 ; z3 ; . . . ; zn Þ is a solution of the PDEs, then we assume that
uðx; y; tÞ ¼ /a ðx; y; tÞ
1 X uj ðx; y; tÞ/ j ðx; y; tÞ;
ð28Þ
j¼0
where uðz1 ; z2 ; z3 ; . . . ; zn Þ; uj ¼ uj ðz1 ; z2 ; z3 ; . . . ; zn Þ, and u0 – 0, are analytic functions of zj in a neighborhood of the manifold [40] and a is an integer. Substitution of (28) into (27), we see that a ¼ 2 and (28) takes the form as:
uðx; y; tÞ ¼
1 X uj ðx; y; tÞuj2 ðx; y; tÞ j¼0
ð29Þ
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O.H. EL-Kalaawy et al. / Results in Physics 7 (2017) 934–946
then we have the following recursion relation: 2 ðum1;x þ ðm 2Þum /x Þ ðj m 4Þujm2 /t þ ujm3;t
i.e
ðj þ 1Þðj 6Þðj 7Þðj 10Þ ¼ 0:
þum ujm4;xt þ ðj m 5Þum ujm3;x /t þ ðj m 5Þum ujm3;t /x þðj m 4Þðj m 5Þum ujm2 /t /x þ ðj m 5Þum ujm3 /xt þA 6 ðj m 2Þum ujmn /x þ un ujmn1;x ððm 2Þum /x þ um1;x Þ þ3 un um ujmn2;xx þ 2ðj m n 3Þun um ujmn1;x /x
Then, the resonant points read from (34)
j ¼ 1; 6; 7; 10
"
þ6ðj 7Þðum ujm32 Þxx /xx þ 12ðj 6Þðj 7Þðum ujm2 Þx /x /xx þ4ðj 7Þðum ujm3 Þx /3x þ 4ðj 6Þðj 7Þðum ujm2 Þ/x /3x
u3 ¼
þ6ðj 5Þðj 6Þðj 7Þðum ujm2 Þ/2x /xx þ 3ðj 6Þðj 7Þðum ujm2 Þ/2xx þðj 4Þðj 5Þðj 6Þðj 7Þðum ujm Þ/4x
ð30Þ
where m ¼ 0; 1; 2; . . . ; j and n ¼ 0; 1; 2; . . . ; ðj mÞ. For j ¼ 0, in equation (30), we obtain
ð31Þ
For j ¼ 1, in equation (30), we obtain
20B / : A xx
ð32Þ
Substituting Eq. (31) into Eq. (30), and collecting coefficients of uj we obtain
40B2 ðj þ 1Þðj 6Þðj 7Þðj 10Þuj /6x A ¼ F j uj1 ; . . . ; u0 ; /t ; /x ; /xx ; . . . ; j ¼ 1; 2; 3; . . . :
# /t 2 þ þ 16wx þ 4w ¼ H: /x
ð36Þ
" # 2 ! /2y /y /xy /y K C / þ 10 þ 12 w þ t ð1 wÞ þ 4Bðwxx wwx Þ ¼ G=/x ; 2 /x /x /x /x /x /x 7 ð37Þ
þC ðum ujm4 Þyy þ 2ðj 7Þðum ujm3 Þy /y
u1 ¼
2
For j ¼ 3, in Eq. (30), we obtain
þðj 7Þðum ujm3 Þ/4x þ 4ðj 5Þðj 6Þðj 7Þðum ujm2 Þx /3x
20B 2 / : A x
/y u2 ¼ K 2C /x
þ4ðj 7Þðum ujm3 Þ3x /x þ 6ðj 6Þðj 7Þðum ujm2 Þxx /2x
u0 ¼
ð35Þ
j ¼ 1 is often called the universal. Putting j ¼ 1; 2; 3; . . . ; 10 in (33) or (30) and using (31), we get. For j ¼ 2, in Eq. (30), we obtain
þðj m n 2Þðj m n 3Þun um ujmn /2x þðj m n 3Þun um ujmn1 /xx þ B ðum ujm4 Þ4x
þðj 6Þðj 7Þðum ujm2 Þ/2y þ ðj 7Þðum ujm3 Þ/yy ;
ð34Þ
ð33Þ
where K ¼ 5=ð12AÞ; w ¼ /xx =/x . We can see that (37) is satisfied equivalently if (36) is satisfied, because G ¼ ðH=/x Þx . The compatibility condition at resonance j ¼ 6 is unsatisfied. From this analysis we see that S-KP equation is non-Painlevé and because of Painlevé conjecture it is non-integrable. Auto-Bäcklund transformations for S-KP equation. Consider the S-KP Eq. (27), according the idea of improved HB [39], we seek for ABT of Eq. (27), when balancing u2 uxx with uuxxxx then gives N ¼ 2. therefore, we may choose
uðx; y; tÞ ¼
@ 2 f ðwÞ 00 0 ¼ f ðwÞw2x þ f ðwÞwxx þ a; @x2
ð38Þ
where a is a function in x; y; t (or constant), f ; w are functions to be determined later. We assume the solution as the form f ðwÞ ¼ clnðwÞ, substituting from this solution and (38) into (27), then we obtain
1 1 14 7 2 ð7Þ 2 Bcw8x þ Bcw6x wxx þ Ac w6x wxx Ac w8x þ f 6 120 3 30
1 1 1 71 27 26 1 1 ð6Þ 1 þf cwt w5x þ 2aBw6x aAcw6x cCw6x Bcw4x w2xx þ Ac2 w4x w2xx Bcw5x wxxx þ Ac2 w5x wxxx c 6Acw4x w2xx Acw5x wxxx cCw4x w2y 6 2 10 2 40 3 10 5 15 5 4 5 15 1 9 2 2 3 ð5Þ 4 4 3 4 2 3 3 þf cwx wxt cwt wx wxx þ 30aBwx wx wxx aAcwx wxx Ccwx wxx 73Bcwx wxx þ Ac wx wxx 6 3 2 2 2 272 7 2 3 53 1 2 4 4 1 1 3 2 3 4 Bcwx wxx wxxx þ Ac wx wxx wxxx Bcwx w4x þ Ac wx w4x Ccwx wxy wy Ccwx wxx w2y Ccw4x wyy 3 2 6 8 3 2 6 3 2 ð4Þ 4 2 2 4 2 2 3 2 2 2 þf 2awt wx þ 3a Awx þ 2aCwx 5cwx wxt wxx 3cwt wx wxx þ 90aBwx wxx 24aAcwx wxx 19Bcw4xx þ Ac w4xx 2 5 3 3 2 3 2 2 2 2 2 cwx wxxt cwt wx wxxx þ 40aBwx wxxx 8aAcwx wxxx 144Bcwx wxx wxxx þ 8Ac wx wx wxxx 46Bcwx wxxx 3
14 2 2 2 þ Ac w2x w2xxx 59Bcw2x wxx w4x þ Ac w2x wxx w4x Bcw3x w5x 2Dcw2x w2xy Ccw3x wxyy 2Ccw2x wxxy wy 4Ccwx wxx wxy wy Ccw2xx w2y Ccw2x wxx wyy 3 3 f
ð8Þ
þf
000
3
6aw2x wxt þ 6awt wx wxx þ 18a2 w2x wxx 3cwxt w2xx þ 30aBw3xx 9aAcwxx 6xwx wxx wxxt 2cwx wxt wxxx
2cwt wxx wxxx þ 120aBwx wxx wxxx 30aAcwx wxx wxxx 50Bcwxx w2xxx þ 3Ac2 wxx w2xxx cw2x wxxxt þ 30aBw2x w4x 3 3aAcw2x w4x 33Bcw2xx w4x þ Ac2 w2xx w4x 44Bcwx wxxx w4x 18Bcwx wxx w5x Bcw2x w6x Ccw2x wxxyy 2 4Ccwx wxxy wxy 2Ccwxx w2xy 2Ccwx wxx wxyy 4Ccwxx wxxy wy þ 8aCwx wxy wy þ 2aCwxx w2y þ2aCw2x wyy Ccw2xx wyy 00 2 2 þ f 6awxt wxx þ 9a2 Awxx þ 6awx wxxt þ 2awt wxxx þ 12a2 Awx wxxx 2cwxxt wxxx þ 20aBw2xxx 6aAcwxxx
2cwxx wxxxt þ 30aBwxx w4x 6aAcwxx w4x 6Bcw24x þ 12aBwx w5x 8Bcwxxx w5x 2Bcwxx w6x
0 2Ccw2xxy 2Ccwxx wxxyy þ 4aCw2xy þ 4aCwx wxyy þ 4aCwxxy wy þ 2aCwxx wyy þ f 2awxxxt þ 3a2 Aw4x þ 2aBw6x þ 2aCwxxyy ¼ 0: ð39Þ
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To ð8Þ
ð7Þ
obtain ð6Þ
f ;f ;f ;f the form:
the
ð5Þ;...
solution,
we
set
the
coefficients
of
Case 1.
0
and f equal zero and we assumed wðx; y; tÞ as
uðx; y; tÞ ¼
wðx; y; tÞ ¼ 1 þ e ; h
where h ¼ kðtÞ þ k1 x þ k2 y:
ð40Þ
20B ; k1 ¼ k2 A 2 kðtÞ ¼ k1 ð4Bk1 CÞt;
Case1: c ¼
20B ; k1 ¼ k2 ; A 2 kðtÞ ¼ k1 ð4Bk1 þ CÞt; Case2: c ¼
a¼
3A
5Bk1 2 k1 2 sech x þ y 4Bk1 þ C t ; A 2 2
uðx; y; tÞ ¼ ; ð41aÞ
a ¼ 0;
then the solution of Eq. (27) take the form:
ð42Þ
Case 2.
Substituting from (40) into (39) we get 2 10Bk1
2
5Bk1 2 k1 2 2 þ 3sech x þ y þ 4Bk1 C t ; 3A 2
by retrain the original variable we get the solution of the S-KP equation. Case 1. 2
ð41bÞ
ð43Þ
/1 ðx; y; tÞ ¼
5Bk1 3A
!2 2 3sech
2
2 k1 2 ; x þ y þ 4Bk1 C t 2 ð44Þ
Fig. 1a. The variation of rarefactive two-soliton solutions profile /1 (44) with the iso-nonthermal ion for the parameters y ¼ 0:1; k ¼ 0:09; a ¼ 0:3; l ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 1b. The variation of rarefactive solitary wave profile /2 (45) with the iso-nonthermal ion for the parameters y ¼ 0:1; k ¼ 0:09; a ¼ 0:3; l ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5.
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O.H. EL-Kalaawy et al. / Results in Physics 7 (2017) 934–946
Case 2.
Stability and Mach number for S-KP equation
!2
ð45Þ
To study stability and the traveling wave solutions of the S-KP Eq. (4), we define a new variable as follows: v ¼ qðlx þ my XtÞ, where l and m are direction cosines of the angles made by the wave
We have represented these solutions (44), (45) for a set of parameter values in Figs (1a,1b). Where the amplitudes
X is the speed of the traveling wave. Substituting /ðvÞ ¼ /1 ðx; y; tÞ
2
/2 ðx; y; tÞ ¼
2
5Bk1 A
2
sech
4
k1 2 x þ y 4Bk1 þ C t : 2
2
2
/1m ¼ ð5Bk1 =3AÞ ; /2m ¼ ð5Bk1 =AÞ . The solution (44) is a new exact solution for S-KP equation which describe the solitary waves solution of S-KP equation. It clear from /1 and /2 that the solitary waves will be associated with positive potential ð/1m Þ > 0; ð/2m Þ > 0 only, then the solution of S-KP equation gives rarefactive solitary waves. Furthermore, the nonlinear coefficient A is assumed to be positive to ensure the reality of the solutions.
2
propagation with x axis and y axis ðl þ m2 ¼ 1Þ, respectively. Here
into Eq. (4), integrating twice and using the vanishing boundary 2
conditions for / ! 0; d/=dv ! 0 and d /=dv2 ! 0 for jvj ! 1, one can obtain the following ordinary differential equation [19, 47, 48]: 2
Bq2 l
4
d / 2 þ Al ð/Þ3=2 ðlX Cm2 Þ/ ¼ 0: dv2
ð46Þ
Fig. 1c. The periodic travelling wave solution /3 of (48) have plotted for different values of the parameters l ¼ 0:62; m ¼ 0:38; X ¼ 0:06; y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 2a. Variation of amplitude /1m (44) of DA solitary waves with ion iso-nonthermal parameter and the other parameters are k ¼ 0:09; ri ¼ 0:2, and c1 ¼ 0:5.
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O.H. EL-Kalaawy et al. / Results in Physics 7 (2017) 934–946
From (46), we have
2
d/ 1 4 2 5=2 2 2 : Al ¼ l X Cm ð/Þ / 4 dv 5 Bq2 l
ð47Þ
Eq. (47) has solitonic solutions and one-soliton solution for this equation is given by
/3 ðx; y; tÞ ¼ /3m sech
4
v
: D
ð48Þ
We have represented this solution (48) for a set of parameter values in Fig (1c) which corresponds to a periodic traveling wave
2 2 and solution of Eq. (4). where /3m ¼ 5ðlX Cm2 Þ=ð4Al Þ pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 D ¼ 4 Bl =ð lX Cm Þ are the amplitude and width of the solitary waves, respectively. /3m is always positive, the solution of S-KP equation gives rarefactive solitary waves only. In order to investigate the stability solution (48) of the S-KP, we use the following energy-like equation
Fig. 2b. Variation of amplitude /3m (48) of DA solitary waves with ion iso-nonthermal parameter and the other parameters are y ¼ 0:1; l ¼ 0:62; m ¼ 0:38; X ¼ 1:1; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 2c. Variation of width D (48) of DA solitary waves with ion iso-nonthermal parameter and the other parameters are y ¼ 0:1; X ¼ 1:1; l ¼ 0:62; m ¼ 0:38; ri ¼ 0:2, and c1 ¼ 0:5.
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O.H. EL-Kalaawy et al. / Results in Physics 7 (2017) 934–946
Fig. 2d. Plot of the solitary wave amplitude /3m of (48) is increasing with increasing X and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; l ¼ 0:62; m ¼ 0:38; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 2e. Plot of the solitary wave amplitude /3m of (48) is increasing with increasing X and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; l ¼ 0:62; m ¼ 0:38; ri ¼ 0:2, and c1 ¼ 0:5.
2 1 d/ þ wð/Þ ¼ 0; 2 dv
ð49Þ
where wð/Þ is the pseudopotential or Sagdeev potential (is called the potential energy) and is given by
wð/Þ ¼
1 Bq2 l
4
4 2 lX Cm2 /2 Al ð/Þ5=2 ; 5
ð50Þ
while the first term in Eq. (49) can be considered as the kinetic energy. It is clear that wð/Þ ¼ 0 and dwð/Þ=d/ ¼ 0 at ð/Þ ¼ 0. A stable solution must satisfy the following conditions
(iii) The existence condition of solitary wave solution (48) requires that 2
d wð/Þ d/2
j/¼0 ¼
< 0, which implies that
lX Cm2 2Bl
4
¼
S 2Bl
4
< 0;
ð51Þ
where the parameters l and B are positive. Therefore the above expression shows that the solitary wave solution (48) exists whenever the condition
S ¼ lX Cm > 0 or X > 2
(i) The kinetic energy is exist it requires that wð/Þ < 0 for the interval /min < / < 0 for the rarefactive solitary waves. Where /min is the minimum value of / for which wð/Þ < 0. (ii) There must exist a non-vanishing point / ¼ /min such that wð/ ¼ /min Þ ¼ 0.
d2 wð/Þ j/¼0 d/2
1l l
2
! C;
ð52Þ
is satisfied and the width D of a stable solitary wave is real. The magnitude of the solionic electric field can be found by ~ E ¼ r/,
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Fig. 2f. The variation of width D of (48) is decreasing with increasing X for and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; l ¼ 0:62; m ¼ 0:38; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 3a. The variation of amplitude /3m of (48) is decreasing with increasing l and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; X ¼ 1:1; ri ¼ 0:2, and c1 ¼ 0:5.
Fig. 3b. The variation of width D of (48) is increasing with increasing l for and the other parameters are y ¼ 0:1; a ¼ 0:3; l ¼ 0:4; X ¼ 1:1; ri ¼ 0:2, and c1 ¼ 0:5.
O.H. EL-Kalaawy et al. / Results in Physics 7 (2017) 934–946 5=2 !
50ðl þ mÞðlX Cm2 Þ 4 ~ sech kv ðtanhkvÞ; where E¼ pffiffiffi 6 2 2A Bl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lX m2 C ¼ ðlx þ my XtÞ; k ¼ pffiffiffi 2 : 4 Bl
Results and discussion
v
ð53Þ The true Mach number is defined as
MT ¼
Xþk
ð54Þ
k
1=2 i þ1 where k ¼ c1 þ lr is the phase speed and X is the incremen1l tal soliton speed. Consequently, we find that
X ¼ c1 þ
lri þ 1 1l
1=2 ðM T 1Þ
943
ð55Þ
In order to investigate the nonlinear properties of small amplitude DAWs including negatively charged dust grains variations and Boltzmann distributed electrons, we have numerically analyzed /1m or /2m ; /3m and D for the same parameters corresponding to space strongly coupled dusty plasma. By taking these parameters, the DA solitary wave solutions (44), (45) and (48) obtained in the small amplitude approximation clearly indicate the existence of the rarefactive solitary waves, corresponding to a hump in the ion density. It turns out that criteria for these are independent of the sign of the nonlinear coefficient A. Obviously, this type of solitary solution which is due to the combined effects of nonlinear pffiffiffiffi term / (containing A) arises due to the trapped ion distribution, and the dispersion term (containing B). We analyzed numerically
Fig. 4a. Plot of the solitary wave electric field E3 magnitude for different values of, based on (53): (4a) The black line is a ¼ 0:5, the red line is a ¼ 0:35, the blue line is a ¼ 0:2, the green line is a ¼ 0 and bold black line is a ¼ 0:5 and the other parameters are l ¼ 0:62; m ¼ 0:38; X ¼ 0:65; y ¼ 0:1; t ¼ 1; l ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4b. Plot of the solitary wave electric field E3 magnitude for different values of, based on (53): (4b) The black line is l ¼ 0:9, the red line is l ¼ 0:7, the blue line is l ¼ 0:5, the green line is l ¼ 0:3 and bold black line is l ¼ 0:1 and the other parameters are l ¼ 0:62; m ¼ 0:38; X ¼ 0:65; y ¼ 0:1; t ¼ 1; a ¼ 0:4; ri ¼ 0:2, and c1 ¼ 0:5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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the wave amplitude /1m or /2m ; /3m , the width of the wave D, and the profile for the potential / of the solitary wave due to the following group of parameters: l, ri ; c, and considering a, the ratio of the free temperature to the trapped ion temperature.
(ii) The solitary waves for Fig. 2a implies that the amplitude /1m of (44) is decreased with l increase while the amplitude /1m is increased with a increase. (iii) It is shown from Figs. 2b, 2c and 2d, 2e 2f that both the amplitude /3m and width D of these traveling structures increases and decreases with the increase of l and a at X ¼ 0:3. Also, the amplitude /3m increases with the increase of l at jaj ¼ 0:3 and X ¼ 0:3. But both the amplitude /3m and width D of these traveling structures increase and decrease with the increase in X at jaj ¼ 0:3 and l ¼ 0:5. We note that the two figures (2d, 2e) are very similar, except
(i) Figs. 1a,1b,1c shows the variation of the solutions of S-KP equation and the wave potential /, as a function of the variables n; s. It can be related to the behavior of a solitary wave from the figures with only negative potential. The solitary wave structures which are due to the negatively charged dust grains caused by the ions following the vortex-like distribution.
Fig. 5a. Variation of amplitude /3m (44) of DA solitary l ¼ 0:62; m ¼ 0:38; X ¼ 0:06; k ¼ 0:09; ri ¼ 0:2; a ¼ 0:3 and c1 ¼ 0:5.
waves
with
ion
iso-nonthermal
parameter
and
the
other
parameters
are
Fig. 5b. Variation of amplitude /3m (44) of DA solitary l ¼ 0:62; m ¼ 0:38; X ¼ 0:06; k ¼ 0:09; ri ¼ 0:2; a ¼ 0:3 and c1 ¼ 0:5.
waves
with
ion
iso-nonthermal
parameter
and
the
other
parameters
are
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Fig. 5c. Variation of width D (44) of DA solitary l ¼ 0:62; m ¼ 0:38; X ¼ 0:06; k ¼ 0:09; ri ¼ 0:2; a ¼ 0:3 and c1 ¼ 0:5.
waves
that the soliton amplitudes are reduced by a factor of about 3. (iv) From Figs. 2d,2e,2f we can see that the amplitude /3m (width D of the soliton increases (decreases) when X is increased. On the other hand, from the definition of the soliton amplitude and its width, one can find that the amplitude (width) decreases (increases) with an increasing value for the parameter l. The values of l that satisfy the existence condition are strongly controlled by arbitrary parameter X see Figs. 3a and 3b. (v) Figs. 4a and 4b show how the electric field varies with isononthermal and the proportion of trapped ions present. With increased iso-nonthermal we see the field becomes less localized with lower amplitude. As the proportion of trapped ions is decreased (lower value of), the magnitude of the electric field decreases. (vi) It is shown from Figs. 5a, 5b and 5c that both the amplitude /3m and width D of these solitary structures increase and decrease with the increase of M T at jaj ¼ 0:3 and as l is increased, their amplitude increases, but their width decreases, whenever the amplitude increases and width decreases at M T increase. Conclusion In this present work, we have investigated the nonlinear propagation of DA solitary wave in an unmagnetized strongly coupled dusty plasma. A S-KP equation is derived which governs the dynamics of small amplitude DA solitary wave in strongly coupled dusty plasma using the standard reductive perturbation technique. The variational principle and conservation laws are obtained. In this paper, we used the Painlevé analysis to show that the equations S-KP is not complectly integrable. The exact solutions of the S-KP equation was solved by well-known auto-Bäcklund transformation. These solutions are numerically analyzed and the effect of various dusty plasma constituents DA solitary wave propagation is taken into account. The stability condition and electric field of S-KP equation are also presented. The solutions of S-KP Eqs. (44), (45) and (48) give rarefactive solitary waves only.
with
ion
iso-nonthermal
parameter
and
the
other
parameters
are
References [1] Hase Y, Watanabe S, Tanaca H. Cylindrical ion acoustic soliton in plasma with negative ion. J Phys Soc Jpn 1985;54:4115–25. [2] Duan WS, Hong XR, Shi YR, Sun JA. Envelop solitons in dusty plasmas for warm dust. Chaos Soliton Fractals 2003;16:767–77. [3] Mamun AA, Shukla PK. Cylindrical and spherical dust ion-acoustic solitary waves. Phys Plasmas 2002;9:1468–70. [4] Waleed MM. Dust-ion-acoustic solitons and shocks in dusty plasmas. Chaos Soliton Fractals 2006;28:994–9. [5] Shukla PK, Mamun AA. Introduction to Dusty Plasma Physics (Bristol U K: Inst. Phys. Publ.); 2002. [6] Shukla PK, Silin VP. Dust ion-acoustic wave. Phys Scr 1992;45:508. [7] Rao NN, Shukla PK, Yu M. Dust-acoustic waves in dusty plasmas. Planet Space Sci 1990;38:543–6. [8] Shah A, Saeed R. Nonlinear Korteweg-de Vries-Burger equation for ionacoustic shock waves in the presence of kappa distributed electrons and positrons. Plasma Phys Control Fusion 2011;53:095006–16. [9] Ghosh S, Bharuthram R. Ion acoustic solitons and double layers in electronpositron-ion plasmas with dust particulates. Astrophys Space Sci 2008;314:121–7. [10] Duha SS, Mamun AA. Dust-ion-acoustic shock waves due to dust charge fluctuation. Phys Lett A 2009;373:1287–9. [11] Duan WS. Weakly two-dimensional dust acoustic waves. Phys Plasmas 2001;8:3583–6. [12] Gill TS, Saini NS, Kaur H. The Kadomstev-Petviashvili equation in dusty plasma with variable dust charge and two temperature ions. Chaos Solitons Fractals 2006;28:1106–11. [13] Masood W, Rizvi H. Two dimensional nonplanar evolution of electrostatic shock waves in pair-ion plasmas. Phys Plasmas 2012;19:012119–26. [14] Dorranian D, Sabetkar A. Dust acoustic solitary waves in a dusty plasma with two kinds of nonthermal ions at different temperatures. Phys Plasmas 2012;19:013702–6. [15] Duan WS. The Kadomtsev-Petviashvili equation of dust acoustic waves for hot dust plasmas. Chaos Solitons Fractals 2002;14(3):503–6. [16] Saha A, Chatterjee P. Dust ion acoustic travelling waves in the framework of a modified Kadomtsev-Petviashvili equation in a magnetized dusty plasma with superthermal electrons. Astrophys Space Sci 2014;349:813–20. [17] Adhikary NC, Deka MK, Dev AN, Sarma J. Modified Korteweg-de Vries equation in a negative ion rich hot adiabatic dusty plasma with non-thermal ion and trapped electron. Phys Plasmas 2014;21:083703–0837708. [18] El-Shewy EK, Abo el Maaty MI, Abdelwahed HG, Elmessary MA. Solitary solution and energy for the Kadomstev-Petviashvili equation in two temperatures charged dusty grains. Astrophys Space Sci 2011;332:179–86. [19] Pakzad HR. Dust acoustic solitary and shock waves in coupled dusty plasmas with variable dust charge and vortex-like ion distribution. Astrophys Space Sci 2010;330:301–10. [20] Liu Hai-Feng, Wang Shi-Qing, Yang Fa-Zhan. Propagation of two dimensional cylindrical fast magnetoacoustic solitary waves in a warm dust plasma. Astrophys Space Sci 2013;347:139143.
946
O.H. EL-Kalaawy et al. / Results in Physics 7 (2017) 934–946
[21] Ata-ur-Rahman, Ali S, Moslem WM, Mushtaq A. Two-dimensional cylindrical ion-acoustic solitary and rogue waves in ultrarelativistic plasmas. Phys Plasmas 2013;20:072103–12. [22] Darvishi MT, Najafi M, Arbabi S, Kavitha L. Exact propagating multi-anti-kink soliton solutions of a (3+1)-dimensional B-type KadomtsevPetviashvili equation. Nonlinear Dyn 2016;83:1453–62. [23] Shukla PK, Stenflo L. Transverse shear waves generating vortex-like dust fluid motions in strongly coupled dusty plasmas. Phys Lett A 2003;315:244–7. [24] El-Wakil SA, Abulwafa EM, El-Shewy EK, Gomaa H, Abd-El-Hamid HM. Nonlinear ion acoustic waveforms for Kadomstev-Petviashvili equation. Astrophys Space Sci 2013;346(1):141–7. [25] Biswas A. 1-Soliton of the generalized Camass-Holm Kadomtsev-Petviashvili equation. Commun Nonlinear Sci Numer Simul 2009;14(6):2524–7. [26] Biswas A, Ranasinghe A. 1-Soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity. Appl Math Comput 2009;214(2):645–7. [27] Biswas A, Ranasinghe A. Topological 1-soliton solution of KadomtsevPetviashvili equation with powe law nonlineaity. Appl Math Comput 2010;217(4):1771–3. [28] Triki H, Sturdevant BJM, Hayat T, Aldossary OM, Biswas A. Shock wave solutions of the variants of Kadomtsev-Petviashvili equation. Can J Phys 2011;89(9):979–84. [29] Ebadi G, Yousefzadeh NF, Triki H, Biswas A. Exact solutions of the (2+1)dimensional Camassa-Holm Kadomtsev-Petviashvili equation. Nonl Anal Modelling Control 2012;17(3):280–96. [30] Jawad AJM, Mirzazadeh M, Biswas A. Dynamics of shallow water waves with Gardner Kadomtsev-Petviashvili equation. Discrete Continuous Dyn Syst Series S 2015;8(6):1155–64. [31] Wadati M, Sunukt H, Konno K. Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws. Progress Theor Phys 1975;53:419–36. [32] Rogers C, Schief WK. Bäcklund and Darboux transformations geometry and modern applications in soliton theory. Cambridge: Cambridge University Press; 2002. [33] Ablowitz MJ, Clarkson PA. Solitons; nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press; 1991. [34] Khater AH, Helal MA, El-Kalaawy OH. Two new classes of exact solutions for the KdV equation via Bäcklund transformations. Chaos Solitons Fractals 1997;8:1901–7.
[35] Hirota R. Direct methods in soliton theory. Berlin: Springer; 2004. [36] Weiss J. The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative. J Math Phys 1983;24:1405–13. [37] Wang ML. Exact solutions for a compound KdV-Burgers equation. Phys Lett A 1996;213(5–6):279–87. [38] Fan E, Zhang HQ. New exact solutions to a system of coupled KdV equations. Phys Lett A 1998;245:389–92. [39] Chun-Ping Liu, Zhou L. A new auto-Bäacklund transformation and two-soliton solution for (3+1)-dimensional Jimbo-Miwa equation. Commun Theor Phys 2011;55:213–6. [40] Yi Qin, Yi-Tian Gao, Xin Yu, Gao-Qing Meng. Bell polynomial approach and Nsoliton solutions for a coupled KdV-mKdV system. Commun Theor Phys 2012;58(1):73–7. [41] Gao YT, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulons and symbolic computation for cosmic dust ion-acoustic waves. Phys Lett A 2006;349:314–9. [42] Gao YT, Tian B. Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion acoustic waves with symbolic computation. Phys Lett A 2007;361:523–8. [43] Ibragimov NH. A new conservation theorem. J Math Anal Apple 2007;333:311–28. [44] Abdullahi RA, Xing L. Travelling wave solutions of a two-dimensional generalized Sawada-Kotera equation. Nonlinear Dyn 2016;84:915922. [45] Abdullahi RA, Muatjetjeja Ben. Conservation laws and exact solutions for a 2D Zakharov–Kuznetsov equation. Appl Math Lett 2015;48:109–17. [46] El-Kalaawy OH. Variational principle, conservation laws and exact solutions for dust ion acoustic shock waves modeling modified Burger equation. Comput Math Appl 2016;72:1031–41. [47] Masood W, Rizvi H, Hussain S. Two dimensional electromagnetic shock structures in dense electron-positron-ion magnetoplasmas. Astrophys Space Sci 2011;332:287–99. [48] Shahmansouri M, Astaraki E. Transverse perturbation on three-dimensional ion acoustic waves in electronpositronion plasma with high-energy tail electron and positron distribution. J Theor Appl Phys 2014;8:189–201.