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New soliton solutions for a Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation
Applied Mathematics and Computation 218 (2012) 5966–5973
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New soliton solutions for a Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation Mohammed K. Elboree Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt
a r t i c l e
i n f o
Keywords: Homogeneous balance method Traveling wave solutions Soliton solutions Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation Riccati equation
a b s t r a c t The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Many phenomena in physics and other fields were described by nonlinear evolution equations. When we want to understand the physical mechanism of phenomena in nature, described by nonlinear evolution equations, we have to explored exact traveling wave solutions. The investigation of the exact traveling wave solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, elastic media, optical fibers, etc. In recent years, the homogeneous balance (HB) method has been widely applied to derive the nonlinear transformation and exact solutions (especially the solitary wave solutions) [1–3] and auto Bäcklund transformations [3,4,15] as well as the similarity reductions [3,15] of nonlinear partial differential equations (PDEs) in mathematical physics. Wang et al. [5–7] and Mohammed Khalfallah and co workers [8–14] applied the (HB) method to obtain the new exact traveling wave solutions of a given nonlinear partial differential equation. Fan [15] showed that there is a close connection among the HB method, Wiess, Tabor, Carnevale (WTC) method and Clarkson, Kruskal (CK) method. The tanh method, developed for years, is one of most direct and effective algebraic method for finding exact solutions of nonlinear equations. Recently, much work has been concentrated on the various extensions and applications of the method [16–20]. As a mathematical model of complex physical phenomena, nonlinear evolution equations are involved in many fields from physics to biology, chemistry, engineering, plasma physics, optical fibers and solid state physics etc. Many methods were developed for finding the exact traveling wave solutions of nonlinear evolution equations, such as Hirota’s method, Bäcklund and Darboux transformation, Painlevé expansions, Homogeneous balance method, Jacobi elliptic function, Extended tanh-function method, F-expansion method and extended F-expansion method. In this paper, we use the HB method to solve the Riccati equation /0 = a/2 + b/ + c and the reduced nonlinear ordinary differential equation for the KP like equation coupled to a Schrödinger equation, respectively. It makes the HB method use more extensively. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.074
M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973
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We aim to in this paper to obtain for many new exact traveling wave solutions which contain rational and periodic-like solutions. 2. Summary of the method This method consists of the following steps. Step1: For a given nonlinear evolution equation
Fðu; ut ; ux ; uxt ; utt ; . . .Þ ¼ 0;
ð1Þ
we consider its traveling wave solutions u(x, t) = u(f), f = kx + kt + d then Eq. (1) is reduced to an nonlinear ordinary differential equation
Q ðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð2Þ
d : df
where a prime denotes Step2: For a given ansatz equation (for example, the ansatz equation is /0 = a/2 + b/ + c in this paper), the form of u is decided and the homogeneous balance method is used on Eq. (2) to find the coefficients of u. Step3: The homogeneous balance method is used to solve the ansatz equation. Step4: Finally, the traveling wave solutions of Eq. (1) are obtained by combining Steps 2 and 3. From the above procedure, it is easy to find that the homogeneous balance method is more effective and simple than other methods and a lot of solutions can be obtained in the same time. This method can be also applied to other nonlinear evolution equations. 3. The application of the HB method to a KP like equation coupled to a Schrödinger equation For the KP like equation coupled to a Schrödinger equation [21]
ðut uxxxxx 30u2 ux 20ux uxx 10uuxxx Þx 3uyy ¼ 0; iv y ¼ v xx þ uv ;
ð3Þ
which u = u(x, y, t) and v = v(x, y, t) are assumed to be real and complex functions, respectively. Long wave in shallow water is a subject of broad interests and has a long colorful history. Physically, it has a rich variety of phenomenological manifestation, especially the existence of wave permanent in form and robust in maintaining their entities through mutual interaction and collision as well as the remarkable property of exhibiting recurrences of initial data when circumstances should prevail. These characteristics are due to the intimate interplay between the roles of nonlinearity and dispersion. Let us consider the traveling wave solutions
uðx; y; tÞ ¼ uðfÞ;
v ðx; y; tÞ ¼ eih v ðfÞ;
f ¼ kx þ ky þ mt þ d;
h ¼ px þ qy þ lt þ n;
ð4Þ
where k, k, m, d, p, q, l and n are constants. From the system (3) we obtain the relation k = 2pk and coupled nonlinear ordinary differential equations 6
2
4
4
ðkm 3k2 Þu k u0000 10k u3 5k u02 10k uu00 ¼ 0; ðp2 qÞv k
2
v 00 uv ¼ 0:
ð5Þ
We now seek the solutions of Eq. (5) in the form
u¼
v¼
m X i¼0 m X
ai /i ;
ð6Þ
bi /i ;
ð7Þ
i¼0
where ai, bi are constants to be determined later and / satisfy the Riccati equation
/0 ¼ a/2 þ b/ þ c;
ð8Þ
where a, b and c are constants and / satisfy Eq. (8). It is easy to show that m = 2 when balancing u0000 with uu00 . Therefore use the ansatz
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u ¼ a0 þ a1 / þ a2 /2 ;
ð9Þ
v ¼ b0 þ b1 / þ b2 /2 :
ð10Þ
Substituting Eq. (9), (10) into Eq. (5) along with (8) and collecting all terms with the same power in /i (i = 0, 1, 2, 3, 4, 5) yields a set of algebraic system for a0, a1, a2, b0, b1, b2, k, m p q and l, namely 2
6
3
4
4
4
2
6
12p2 k a0 k a1 b c 10k a0 a1 bc 5k a21 c2 20k a0 a2 c2 10k a30 16k a2 ac3 6
2
6
2
8k a1 abc 14k a2 b c2 þ kma0 ¼ 0; 4
4
6
2
6
6
3
4
80k a0 a2 ac 30k a0 a1 ab 232k a2 ab c 136k a2 a2 c2 15k a1 ab 15k a21 b
4 30k a21 ac 6 60k a1 a2 bc
2 2
2
4
6
4
12p k a2 þ kma2 30k a0 a21 110k a1 ba2 c 16k a2 b 4
2
2
4 40k a22 c2
2
40k a0 a2 b 30k a20 a2 ¼ 0;
4
4
2
2
6
4
4
6
3
20k a21 bc 10k a0 a1 b 12p2 k a1 k a1 b 20k a0 a1 ac þ kma1 30k a2 b c 6
2
6
4
6
22k a1 ab c 120k a2 ac2 b 60k a0 a2 bc 16k a1 a2 c2
2 4 30k a20 a1 40k a1 c2 a2 ¼ 0; 4 2 6 4 6 4 2 60k a22 b 240k a2 a3 c 170k a1 aa2 b 60k a1 a3 b 60k a0 a2 a2 30k a0 a22 4 2 6 80k a22 a2 10k a32 120k a2 a4 ¼ 0; 4 2 4 6 6 100k a1 a2 a2 30k a1 a22 140k a22 ab 24k a1 a4 336k a2 a3 b ¼ 0; 4
2
4
4
4
2
ð11Þ
6
70k a1 b a2 40k a21 ab 100k a22 bc 140k a1 aa2 c 60k a0 a1 a2 440k a2 a2 bc 6
3
4
6
4
2
6
2
130k a2 ab 20k a0 a1 a2 40k a1 a3 c 100k a0 a2 ab 10k a31 50k a1 a2 b ¼ 0; 2
2
2k b2 c2 a0 b0 qb0 k b1 bc þ p2 b0 ¼ 0; 2
2
2
2
qb2 þ p2 b2 a2 b0 3k b1 ab 8k b2 ac a1 b1 4k b2 b a0 b2 ¼ 0; 2
2
2
2
6k b2 bc a1 b0 þ p2 b1 qb1 k b1 b 2k b1 ac a0 b1 ¼ 0; 2
6k b2 a2 a2 b2 ¼ 0; 2
2
2k b1 a2 a2 b1 a1 b2 10k b2 ab ¼ 0: For which, with the aid of Maple, we find 2
3
4
1 m2 b 1 m2 b 1 b m2 ; a1 ¼ ; a2 ¼ ; 96 q2 96 cq2 384 q2 c2 b2 c2 b2 c b0 ¼ 4 2 ; b1 ¼ 4 ; b2 ¼ b2 : b b
a0 ¼
ð12Þ
It is to be noted that the Riccati equation (8) can be solved using the homogeneous balance method as follows i 2 0 Case: I. Let / ¼ Rm i¼0 ci tanh f: Balancing / with / leads to
/ ¼ c0 þ c1 tanh f:
ð13Þ
Substituting Eq. (13) into (8), we have the following solution of Eq. (8)
/¼
1 ðb þ 2 tanh fÞ; 2a
2
ac ¼
b 1: 4
ð14Þ
Substituting Eqs. (12) and (13) into (9), (10) we have the following new traveling wave solution of KP like equation coupled to a Schrödinger equation (3)
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m2 b2 96q2 b2
2
b 1 4ac
!
b tanhðfÞ 2ac
!2
;
2
2
4a2 b
ðð4ac b Þ 2b tanhðfÞÞ2 :
ð15Þ ð16Þ
This is a bell-shaped solution. i Similarly, let / ¼ Rm i¼0 ci coth f, then we obtain the following new traveling wave soliton solutions of KP like equation coupled to a Schrödinger Eq. (3)
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m2 b2 96q2 b2
2
b 1 4ac 2
2
4a2 b
!
!2 b cothðfÞ ; 2ac
ðð4ac b Þ 2b cothðfÞÞ2 :
ð17Þ ð18Þ
M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973
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where f = kx + ky + mt + d. Case: II. From [22], when a = 1, b = 0, the Riccati Eq. (8) has the following solutions
8 pffiffiffiffiffiffi pffiffiffiffiffiffi > < c tanhð cfÞ; c < 0; c ¼ 0; / ¼ 1f ; > pffiffiffi : pffiffiffi c tanð cfÞ; c > 0:
ð19Þ
It is seen that the tanh function in (6) is only a special function in (19), so we conjecture that Eq. (3) may admit other types of traveling wave solutions in Eq. (19) in addition to the tanh-type one. Moreover, we hope to construct them in a unified way. For this purpose, we shall use the Riccati equation (8) once again to generate an associated algebraic system, but not use one of the functions in (19). Another advantage of the Riccati Eq. (8) is that the sign of c can be used to exactly judge the type of the traveling wave solution for Eq. (3). For example, if c < 0, we are sure that Eq. (3) admits tanh-type and coth-type traveling wave solutions. Especially Eq. (3) will possess three types of traveling wave solutions if c is an arbitrary constant. In this way, we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods. From (9), (12) and (19), we have the following new traveling wave solutions of KP like equation coupled to a Schrödinger Eq. (3), which contain traveling wave solutions, periodic wave solutions and rational solutions as follows. When c < 0, we have
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 4c þ 4b c tanhð cfÞ þ b ðtanhð cfÞÞ2 ; 384cq2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi b c v ðx; y; tÞ ¼ 22 4c þ 4b c tanhð cfÞ þ b2 ðtanhð cfÞÞ2 : b uðx; y; tÞ ¼
m 2 b2
ð20Þ ð21Þ
This is a linear combinations of kink wave and bell-shaped wave solutions which is a new solution for (3). When c = 0, we have
! 2 4bc b ; þ f 384c2 q2 f2 ! 2 b 4bc b þ 2 : v ðx; y; tÞ ¼ 22 4c2 f f b
m2 b2
uðx; y; tÞ ¼
4c2
ð22Þ ð23Þ
When c > 0, we have
uðx; y; tÞ ¼
m 2 b2 384q2 c
pffiffiffi pffiffiffi 2 4c þ 4b c tanð cfÞ þ b ðtanðfÞÞ2 ;
ð24Þ
pffiffiffi pffiffiffi pffiffiffi b c v ðx; y; tÞ ¼ 22 4c þ 4 c tanð cfÞb þ ðtanð cfÞÞ2 b2 ; b
ð25Þ
which contain a periodic-like solutions. Case: III. We suppose that the Riccati equation (8) has the following solutions of the form
/ ¼ A0 þ
m X ðAi f i þ Bi f i1 gÞ;
ð26Þ
i¼1
with
f ¼
1 ; cosh f þ r
g¼
sinh f ; cosh f þ r
which satisfy
f 0 ðfÞ ¼ f ðfÞgðfÞ;
g 0 ðfÞ ¼ 1 g 2 ðfÞ rf ðfÞ;
2
g ðfÞ ¼ 1 2rf ðfÞ þ ðr2 1Þf 2 ðfÞ: Balancing /0 with /2 leads to
/ ¼ A0 þ A1 f þ B1 g:
ð27Þ i
j
Substituting Eq. (27) into (8), collecting the coefficient of the same power f (f)g (f)(i = 0, 1, 2; j = 0, 1) and setting each of the obtained coefficients to zero yield the following set of algebra equations
aA20 þ aB21 þ bA0 þ c ¼ 0; 2aA0 A1 2arB21 rB1 þ bA1 ¼ 0; aA21 þ aðr 2 1ÞB21 þ ðr 2 1ÞB1 ¼ 0; 2aA0 B1 þ bB1 ¼ 0; 2aA1 B1 þ A1 ¼ 0;
ð28Þ
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which have solutions
A0 ¼
b ; 2a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr 2 1Þ ; 4a2
B1 ¼
A1 ¼
1 ; 2a
2
c¼
b 1 : 4a
ð29Þ
From Eqs. (27)–(29), we have
/¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sinh f ðr 2 1Þ 1 : bþ 2a cosh f þ r
ð30Þ
From Eqs. (9), (12) and (30), we obtain the new wave solutions of KP like equation coupled to a Schrödinger Eq. (3)
! pffiffiffiffiffiffiffiffiffiffiffiffiffi!!2 2 b b sinhðfÞ r 2 1 ; 1 þ 4ac 96q2 4ac coshðfÞ þ r pffiffiffiffiffiffiffiffiffiffiffiffiffi !!2 b2 bðsinhðfÞ r2 1Þ 2 v ðx; y; tÞ ¼ 2 2 ðb 4acÞ þ : coshðfÞ þ r 4a b uðx; y; tÞ ¼
m2 b2
ð31Þ
ð32Þ
Case: IV. We take / in the Riccati Eq. (8) being of the form
/ ¼ ep1 f qðzÞ þ p4 ðfÞ;
ð33Þ
where
z ¼ ep2 f þ p3 ; where p1, p2 and p3 are constants to be determined. 2 2 P 1 þb Substituting (33) into (8) we find that when c ¼ 4a ; we have
/¼
p1 ep1 f p b þ 1 : aðep1 f þ p3 Þ 2a
ð34Þ
If p3 = 1 in (34), we have
/¼
p1 1 b p1 f : tanh 2 2a 2a
ð35Þ
If p3 = 1 in (34), we have
/¼
p1 1 b p1 f : coth 2 2a 2a
ð36Þ
From (9), (12) and (34), we obtain the following new wave solutions of KP like equation coupled to a Schrödinger Eq. (3)
2 b p1 b p ep1 f p 1f 1þ ; ac 2 e 1 þ p3 2 b 2bp ep1 f v ðx; y; tÞ ¼ 22 2 ð4ac bðp1 bÞÞ þ p1 f 1 ; e þ p3 4a b uðx; y; tÞ ¼
m2 b2
96q2
ð37Þ ð38Þ
where f = kx + ky + mt + d. When p3 = 1, we have from (35)
!
!2 bp1 1 p f ; tanh 2 1 96q2 4ac 2 b 1 v ðx; y; tÞ ¼ 22 2 ð4ac b2 Þ bp1 tanh p1 f : 2 4a b uðx; y; tÞ ¼
m2 b2
2
1
b 4ac
ð39Þ ð40Þ
When p3 = 1, we have from (41)
!
!2 bp1 1 uðx; y; tÞ ¼ ; p f coth 2 1 96q2 4ac 2 b 1 v ðx; y; tÞ ¼ 22 2 ð4ac b2 Þ bp1 coth p1 f : 2 4a b
m2 b2
2
b 1 4ac
Clearly, (15)–(18) is the special case of (39)–(42) with p1 = 2. Case: V. We suppose that the Riccati equation (8) have the following solutions of the form
ð41Þ ð42Þ
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/ ¼ A0 þ
m X
i1
sinh
ðAi sinh x þ Bi cosh xÞ;
i¼1
where dx=df ¼ sinh x or dx=df ¼ cosh x. It is easy to find that m = 1 by balancing /0 and /2. So we choose
/ ¼ A0 þ A1 sinh x þ B1 cosh x;
ð43Þ i
j
when dx=df ¼ sinh x, we substitute (43) and dx=df ¼ sinh x, into (8) and set the coefficient of sinh x cosh x (i = 0, 1, 2; j = 0, 1) to zero. A set of algebraic equations is obtained as follows
aA20 þ aB21 þ bA0 þ c ¼ 0; 2aA0 A1 þ bA1 ¼ 0; aA21 þ aB21 ¼ B1 ; 2aA0 B1 þ bB1 ¼ 0;
ð44Þ
2aA1 B1 þ A1 ¼ 0 for which, we have the following solutions
A0 ¼
b ; 2a
A1 ¼ 0;
B1 ¼
1 ; a
ð45Þ
2
where c ¼ b 4a4 and
A0 ¼
b ; 2a
rffiffiffiffiffiffi 1 ; 2a
B1 ¼
A1 ¼
1 ; 2a
ð46Þ
2
where c ¼ b 4a1. To dx=df ¼ sinh x, we have
sinh x ¼ csch f;
cosh x ¼ coth f:
ð47Þ
From (46) and (47), we obtain
/¼
b þ 2 coth f ; 2a
ð48Þ
2
where c ¼ b 4a4 and
/¼
b csch f þ coth f ; 2a
ð49Þ
2
where c ¼ b 4a1. Clearly (48) is the special case of (36) with p1 = 2. From (8), (9), (12), (48) and (49), we get the new traveling wave solutions of Eq. (3) in the following form
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m 2 b2 96q2 b2
2
b 1 4ac
!
b cothðfÞ 2ac
!2 ;
2
2
4a2 b
ðð4ac b Þ 2b cothðfÞÞ2 ;
ð50Þ ð51Þ
2
where c ¼ b 4a4 and
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m 2 b2 96q2 b2
2
1
b 4ac
!
!2 b ðcothðfÞ cschðfÞÞ ; 4ac
2
2
4a2 b
ðð4ac b Þ bðcothðfÞ cschðfÞÞÞ2 ;
ð52Þ ð53Þ
2
where c ¼ b 4a1. Clearly (50), (51) is the special case of (41), (42) with p1 = 2. Similarly, when dx=df ¼ cosh x, we obtain the following exact traveling wave (periodic-like) solutions of KP like equation coupled to a Schrödinger Eq. (3)
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m 2 b2 96q2 b2
2
1
b 4ac 2
2
4a2 b
!
b cotðfÞ 2ac
ðð4ac b Þ 2b cotðfÞÞ2 :
!2 ;
ð54Þ ð55Þ
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M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973 2
where c ¼ b 4a4 and
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m2 b2 96q2 b2
2
b 1 4ac 2
2
4a2 b
!
!2 b ðcotðfÞ cscðfÞÞ ; 4ac
ðð4ac b Þ bðcotðfÞ cscðfÞÞÞ2 :
ð56Þ ð57Þ
4. Conclusion HB method provides us a solutions polynomials in two elementary bell-shaped and kink-shaped functions, this covers the large majority of physically interesting solitary waves Eqs. (15), (16), (39), (40), in addition we also obtain a periodic-like wave solutions Eqs. (24), (25), (54), (55), (56), (57) which play an important interesting in physics. Remark. The majority of solutions for (8) in [23], contain solitary wave, periodic wave, rational solutions, etc. but in this paper we study the solutions of (8) which contains kink-type, bell-shaped, periodic-like, rational solutions which are wellknown solutions and also new solutions for instance in cases III, V are obtained, i.e. we recovered by HB method the wellknown solutions and adding some new solutions.
In summary we have used the extended homogeneous balance method to obtain many traveling wave solutions of KP like equation coupled to a Schrödinger equation. In Eq. (8), if we set b = 0, a = l, c = l HB method is reduced to tanh method [24] but HB method is more general than tanh method, extended tanh method [25] because this method is readily applicable to a large variety of nonlinear PDEs in contrast to the tanh method. Some merits are available for HB method. First, all the nonlinear PDEs which can be solved by tanh function method can be solved easily by HB method.and we have more multiple soliton solutions and triangular periodic solution (including rational solutions). Second, we used only the special solutions of Eq. (3), we can obtain more traveling wave solutions. Third, not only HB method contains the hyperbolic tangent expansion method, but also it is a computerizable method, which allow us to perform complicated and tedious differential calculation on computer. Therefore, the HB method is a generalized tanh function method for many nonlinear PDEs. One can construct the multisoliton solutions for the Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation using the algorithm in [26,27]. References [1] Mariana Antonova, Anjan Biswas, Adiabatic parameter dynamics of perturbed solitary waves, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 734– 748. [2] E.G. Fan, H.Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998) 403–406. [3] E.G. Fan, Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations, Phys. Lett. A 294 (2002) 26–30. [4] M.L. Wang, Y.M. Wang, A new Bäcklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients, Phys. Lett. A 287 (2001) 211–216. [5] M.L. Wang, Solitary wave solution for variant Boussinesq equations, Phys. Lett. A 199 (1995) 169–172. [6] M.L. Wang, Application of homogeneous balance method to exact solutions of nonlinear equation in mathematical physics, Phys. Lett. A 216 (1996) 67– 75. [7] M.L. Wang, Y.B. Zhou, Z.B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [8] Mohammed Khalfallah, New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 1169–1175. [9] Mohammed Khalfallah, Exact traveling wave solutions of the Boussinesq–Burgers equation, Math. Comput. Model. 49 (2009) 666–671. [10] A.S. Abdel Rady, A.H. Khater, E.S. Osman, Mohammed Khalfallah, New periodic wave and soliton solutions for system of coupled Korteweg–de Vries equations, Appl. Math. Comput. 207 (2009) 406414. [11] A.S. Abdel Rady, E.S. Osman, Mohammed Khalfallah, ‘‘Multi soliton solution for the system of coupled Korteweg–de Vries equations’’, Appl. Math. Comput. 210 (2009) 177–181. [12] A.S. Abdel Rady, E.S. Osman, Mohammed Khalfallah, On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 264–274. [13] A.S. Abdel Rady, Mohammed Khalfallah, On soliton solutions for Boussinesq–Burgers equations, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 886–894. [14] A.S. Abdel Rady, E.S. Osman, Mohammed Khalfallah, Multi soliton solution, rational solution of the Boussinesq–Burgers equations, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 1172–1176. [15] E.G. Fan, Two new applications of the homogeneous balance method, Phys. Lett. A 265 (2000) 353–357. [16] W.X. Ma, B. Fuchssteiner, Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation, Int. J. Nonlinear Mech. 31 (1996) 329–338. [17] C.L. Bai, Exact solutions for nonlinear partial differential equation: a new approach, Phys. Lett. A 288 (2001) 191–195. [18] C.L. Bai, Hong Zhao, Complex hyperbolic-function method and its applications to nonlinear equations, Phys. Lett. A 355 (2006) 32–38. [19] C.L. Bai, A new generalization of variable coefficients algebraic method for solving nonlinear evolution equations, Chaos Soliton Fract. 34 (2007) 1114– 1129. [20] Hong Zhao, Applications of the generalized algebraic method to special-type nonlinear equations, Chaos Soliton Fract. 36 (2008) 359–369. [21] X.B. Hu, The higher-order KdV equation with a source and nonlinear superposition formula, Chaos Soliton Fract. 7 (1996) 211–215. [22] X.Q. Zhao, D.B. Tang, A new note on a homogeneous balance method, Phys. Lett. A 297 (2002) 59–67.
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[23] Shun-dong Zhu, The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional BoitiLeonPempinelle equation, Chaos Soliton Fract. 37 (2008) 1335–1342. [24] E.G. Fan, Soliton solutions for a generalized Hirota Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A 282 (2001) 18–22. [25] C.L. Bai, H. Zhuo, Generalized extended tanh-function method and its application, Chaos Soliton Fract. 27 (2006) 1026–1035. [26] C.L. Bai, Multiple soliton solutions of the high order Broer–Kaup equations, Commun. Theor. Phys. 34 (2000) 729–732. [27] C.L. Bai, Abundant multi-soliton structure of the (3 + 1)-dimensional Nizhnik–Novikov–Veselov equations, Commun. Theor. Phys. 41 (2004) 15–20.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
New soliton solutions for a Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation Mohammed K. Elboree Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt
a r t i c l e
i n f o
Keywords: Homogeneous balance method Traveling wave solutions Soliton solutions Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation Riccati equation
a b s t r a c t The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Many phenomena in physics and other fields were described by nonlinear evolution equations. When we want to understand the physical mechanism of phenomena in nature, described by nonlinear evolution equations, we have to explored exact traveling wave solutions. The investigation of the exact traveling wave solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, elastic media, optical fibers, etc. In recent years, the homogeneous balance (HB) method has been widely applied to derive the nonlinear transformation and exact solutions (especially the solitary wave solutions) [1–3] and auto Bäcklund transformations [3,4,15] as well as the similarity reductions [3,15] of nonlinear partial differential equations (PDEs) in mathematical physics. Wang et al. [5–7] and Mohammed Khalfallah and co workers [8–14] applied the (HB) method to obtain the new exact traveling wave solutions of a given nonlinear partial differential equation. Fan [15] showed that there is a close connection among the HB method, Wiess, Tabor, Carnevale (WTC) method and Clarkson, Kruskal (CK) method. The tanh method, developed for years, is one of most direct and effective algebraic method for finding exact solutions of nonlinear equations. Recently, much work has been concentrated on the various extensions and applications of the method [16–20]. As a mathematical model of complex physical phenomena, nonlinear evolution equations are involved in many fields from physics to biology, chemistry, engineering, plasma physics, optical fibers and solid state physics etc. Many methods were developed for finding the exact traveling wave solutions of nonlinear evolution equations, such as Hirota’s method, Bäcklund and Darboux transformation, Painlevé expansions, Homogeneous balance method, Jacobi elliptic function, Extended tanh-function method, F-expansion method and extended F-expansion method. In this paper, we use the HB method to solve the Riccati equation /0 = a/2 + b/ + c and the reduced nonlinear ordinary differential equation for the KP like equation coupled to a Schrödinger equation, respectively. It makes the HB method use more extensively. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.074
M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973
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We aim to in this paper to obtain for many new exact traveling wave solutions which contain rational and periodic-like solutions. 2. Summary of the method This method consists of the following steps. Step1: For a given nonlinear evolution equation
Fðu; ut ; ux ; uxt ; utt ; . . .Þ ¼ 0;
ð1Þ
we consider its traveling wave solutions u(x, t) = u(f), f = kx + kt + d then Eq. (1) is reduced to an nonlinear ordinary differential equation
Q ðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð2Þ
d : df
where a prime denotes Step2: For a given ansatz equation (for example, the ansatz equation is /0 = a/2 + b/ + c in this paper), the form of u is decided and the homogeneous balance method is used on Eq. (2) to find the coefficients of u. Step3: The homogeneous balance method is used to solve the ansatz equation. Step4: Finally, the traveling wave solutions of Eq. (1) are obtained by combining Steps 2 and 3. From the above procedure, it is easy to find that the homogeneous balance method is more effective and simple than other methods and a lot of solutions can be obtained in the same time. This method can be also applied to other nonlinear evolution equations. 3. The application of the HB method to a KP like equation coupled to a Schrödinger equation For the KP like equation coupled to a Schrödinger equation [21]
ðut uxxxxx 30u2 ux 20ux uxx 10uuxxx Þx 3uyy ¼ 0; iv y ¼ v xx þ uv ;
ð3Þ
which u = u(x, y, t) and v = v(x, y, t) are assumed to be real and complex functions, respectively. Long wave in shallow water is a subject of broad interests and has a long colorful history. Physically, it has a rich variety of phenomenological manifestation, especially the existence of wave permanent in form and robust in maintaining their entities through mutual interaction and collision as well as the remarkable property of exhibiting recurrences of initial data when circumstances should prevail. These characteristics are due to the intimate interplay between the roles of nonlinearity and dispersion. Let us consider the traveling wave solutions
uðx; y; tÞ ¼ uðfÞ;
v ðx; y; tÞ ¼ eih v ðfÞ;
f ¼ kx þ ky þ mt þ d;
h ¼ px þ qy þ lt þ n;
ð4Þ
where k, k, m, d, p, q, l and n are constants. From the system (3) we obtain the relation k = 2pk and coupled nonlinear ordinary differential equations 6
2
4
4
ðkm 3k2 Þu k u0000 10k u3 5k u02 10k uu00 ¼ 0; ðp2 qÞv k
2
v 00 uv ¼ 0:
ð5Þ
We now seek the solutions of Eq. (5) in the form
u¼
v¼
m X i¼0 m X
ai /i ;
ð6Þ
bi /i ;
ð7Þ
i¼0
where ai, bi are constants to be determined later and / satisfy the Riccati equation
/0 ¼ a/2 þ b/ þ c;
ð8Þ
where a, b and c are constants and / satisfy Eq. (8). It is easy to show that m = 2 when balancing u0000 with uu00 . Therefore use the ansatz
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M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973
u ¼ a0 þ a1 / þ a2 /2 ;
ð9Þ
v ¼ b0 þ b1 / þ b2 /2 :
ð10Þ
Substituting Eq. (9), (10) into Eq. (5) along with (8) and collecting all terms with the same power in /i (i = 0, 1, 2, 3, 4, 5) yields a set of algebraic system for a0, a1, a2, b0, b1, b2, k, m p q and l, namely 2
6
3
4
4
4
2
6
12p2 k a0 k a1 b c 10k a0 a1 bc 5k a21 c2 20k a0 a2 c2 10k a30 16k a2 ac3 6
2
6
2
8k a1 abc 14k a2 b c2 þ kma0 ¼ 0; 4
4
6
2
6
6
3
4
80k a0 a2 ac 30k a0 a1 ab 232k a2 ab c 136k a2 a2 c2 15k a1 ab 15k a21 b
4 30k a21 ac 6 60k a1 a2 bc
2 2
2
4
6
4
12p k a2 þ kma2 30k a0 a21 110k a1 ba2 c 16k a2 b 4
2
2
4 40k a22 c2
2
40k a0 a2 b 30k a20 a2 ¼ 0;
4
4
2
2
6
4
4
6
3
20k a21 bc 10k a0 a1 b 12p2 k a1 k a1 b 20k a0 a1 ac þ kma1 30k a2 b c 6
2
6
4
6
22k a1 ab c 120k a2 ac2 b 60k a0 a2 bc 16k a1 a2 c2
2 4 30k a20 a1 40k a1 c2 a2 ¼ 0; 4 2 6 4 6 4 2 60k a22 b 240k a2 a3 c 170k a1 aa2 b 60k a1 a3 b 60k a0 a2 a2 30k a0 a22 4 2 6 80k a22 a2 10k a32 120k a2 a4 ¼ 0; 4 2 4 6 6 100k a1 a2 a2 30k a1 a22 140k a22 ab 24k a1 a4 336k a2 a3 b ¼ 0; 4
2
4
4
4
2
ð11Þ
6
70k a1 b a2 40k a21 ab 100k a22 bc 140k a1 aa2 c 60k a0 a1 a2 440k a2 a2 bc 6
3
4
6
4
2
6
2
130k a2 ab 20k a0 a1 a2 40k a1 a3 c 100k a0 a2 ab 10k a31 50k a1 a2 b ¼ 0; 2
2
2k b2 c2 a0 b0 qb0 k b1 bc þ p2 b0 ¼ 0; 2
2
2
2
qb2 þ p2 b2 a2 b0 3k b1 ab 8k b2 ac a1 b1 4k b2 b a0 b2 ¼ 0; 2
2
2
2
6k b2 bc a1 b0 þ p2 b1 qb1 k b1 b 2k b1 ac a0 b1 ¼ 0; 2
6k b2 a2 a2 b2 ¼ 0; 2
2
2k b1 a2 a2 b1 a1 b2 10k b2 ab ¼ 0: For which, with the aid of Maple, we find 2
3
4
1 m2 b 1 m2 b 1 b m2 ; a1 ¼ ; a2 ¼ ; 96 q2 96 cq2 384 q2 c2 b2 c2 b2 c b0 ¼ 4 2 ; b1 ¼ 4 ; b2 ¼ b2 : b b
a0 ¼
ð12Þ
It is to be noted that the Riccati equation (8) can be solved using the homogeneous balance method as follows i 2 0 Case: I. Let / ¼ Rm i¼0 ci tanh f: Balancing / with / leads to
/ ¼ c0 þ c1 tanh f:
ð13Þ
Substituting Eq. (13) into (8), we have the following solution of Eq. (8)
/¼
1 ðb þ 2 tanh fÞ; 2a
2
ac ¼
b 1: 4
ð14Þ
Substituting Eqs. (12) and (13) into (9), (10) we have the following new traveling wave solution of KP like equation coupled to a Schrödinger equation (3)
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m2 b2 96q2 b2
2
b 1 4ac
!
b tanhðfÞ 2ac
!2
;
2
2
4a2 b
ðð4ac b Þ 2b tanhðfÞÞ2 :
ð15Þ ð16Þ
This is a bell-shaped solution. i Similarly, let / ¼ Rm i¼0 ci coth f, then we obtain the following new traveling wave soliton solutions of KP like equation coupled to a Schrödinger Eq. (3)
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m2 b2 96q2 b2
2
b 1 4ac 2
2
4a2 b
!
!2 b cothðfÞ ; 2ac
ðð4ac b Þ 2b cothðfÞÞ2 :
ð17Þ ð18Þ
M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973
5969
where f = kx + ky + mt + d. Case: II. From [22], when a = 1, b = 0, the Riccati Eq. (8) has the following solutions
8 pffiffiffiffiffiffi pffiffiffiffiffiffi > < c tanhð cfÞ; c < 0; c ¼ 0; / ¼ 1f ; > pffiffiffi : pffiffiffi c tanð cfÞ; c > 0:
ð19Þ
It is seen that the tanh function in (6) is only a special function in (19), so we conjecture that Eq. (3) may admit other types of traveling wave solutions in Eq. (19) in addition to the tanh-type one. Moreover, we hope to construct them in a unified way. For this purpose, we shall use the Riccati equation (8) once again to generate an associated algebraic system, but not use one of the functions in (19). Another advantage of the Riccati Eq. (8) is that the sign of c can be used to exactly judge the type of the traveling wave solution for Eq. (3). For example, if c < 0, we are sure that Eq. (3) admits tanh-type and coth-type traveling wave solutions. Especially Eq. (3) will possess three types of traveling wave solutions if c is an arbitrary constant. In this way, we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods. From (9), (12) and (19), we have the following new traveling wave solutions of KP like equation coupled to a Schrödinger Eq. (3), which contain traveling wave solutions, periodic wave solutions and rational solutions as follows. When c < 0, we have
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 4c þ 4b c tanhð cfÞ þ b ðtanhð cfÞÞ2 ; 384cq2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi b c v ðx; y; tÞ ¼ 22 4c þ 4b c tanhð cfÞ þ b2 ðtanhð cfÞÞ2 : b uðx; y; tÞ ¼
m 2 b2
ð20Þ ð21Þ
This is a linear combinations of kink wave and bell-shaped wave solutions which is a new solution for (3). When c = 0, we have
! 2 4bc b ; þ f 384c2 q2 f2 ! 2 b 4bc b þ 2 : v ðx; y; tÞ ¼ 22 4c2 f f b
m2 b2
uðx; y; tÞ ¼
4c2
ð22Þ ð23Þ
When c > 0, we have
uðx; y; tÞ ¼
m 2 b2 384q2 c
pffiffiffi pffiffiffi 2 4c þ 4b c tanð cfÞ þ b ðtanðfÞÞ2 ;
ð24Þ
pffiffiffi pffiffiffi pffiffiffi b c v ðx; y; tÞ ¼ 22 4c þ 4 c tanð cfÞb þ ðtanð cfÞÞ2 b2 ; b
ð25Þ
which contain a periodic-like solutions. Case: III. We suppose that the Riccati equation (8) has the following solutions of the form
/ ¼ A0 þ
m X ðAi f i þ Bi f i1 gÞ;
ð26Þ
i¼1
with
f ¼
1 ; cosh f þ r
g¼
sinh f ; cosh f þ r
which satisfy
f 0 ðfÞ ¼ f ðfÞgðfÞ;
g 0 ðfÞ ¼ 1 g 2 ðfÞ rf ðfÞ;
2
g ðfÞ ¼ 1 2rf ðfÞ þ ðr2 1Þf 2 ðfÞ: Balancing /0 with /2 leads to
/ ¼ A0 þ A1 f þ B1 g:
ð27Þ i
j
Substituting Eq. (27) into (8), collecting the coefficient of the same power f (f)g (f)(i = 0, 1, 2; j = 0, 1) and setting each of the obtained coefficients to zero yield the following set of algebra equations
aA20 þ aB21 þ bA0 þ c ¼ 0; 2aA0 A1 2arB21 rB1 þ bA1 ¼ 0; aA21 þ aðr 2 1ÞB21 þ ðr 2 1ÞB1 ¼ 0; 2aA0 B1 þ bB1 ¼ 0; 2aA1 B1 þ A1 ¼ 0;
ð28Þ
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M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973
which have solutions
A0 ¼
b ; 2a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr 2 1Þ ; 4a2
B1 ¼
A1 ¼
1 ; 2a
2
c¼
b 1 : 4a
ð29Þ
From Eqs. (27)–(29), we have
/¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sinh f ðr 2 1Þ 1 : bþ 2a cosh f þ r
ð30Þ
From Eqs. (9), (12) and (30), we obtain the new wave solutions of KP like equation coupled to a Schrödinger Eq. (3)
! pffiffiffiffiffiffiffiffiffiffiffiffiffi!!2 2 b b sinhðfÞ r 2 1 ; 1 þ 4ac 96q2 4ac coshðfÞ þ r pffiffiffiffiffiffiffiffiffiffiffiffiffi !!2 b2 bðsinhðfÞ r2 1Þ 2 v ðx; y; tÞ ¼ 2 2 ðb 4acÞ þ : coshðfÞ þ r 4a b uðx; y; tÞ ¼
m2 b2
ð31Þ
ð32Þ
Case: IV. We take / in the Riccati Eq. (8) being of the form
/ ¼ ep1 f qðzÞ þ p4 ðfÞ;
ð33Þ
where
z ¼ ep2 f þ p3 ; where p1, p2 and p3 are constants to be determined. 2 2 P 1 þb Substituting (33) into (8) we find that when c ¼ 4a ; we have
/¼
p1 ep1 f p b þ 1 : aðep1 f þ p3 Þ 2a
ð34Þ
If p3 = 1 in (34), we have
/¼
p1 1 b p1 f : tanh 2 2a 2a
ð35Þ
If p3 = 1 in (34), we have
/¼
p1 1 b p1 f : coth 2 2a 2a
ð36Þ
From (9), (12) and (34), we obtain the following new wave solutions of KP like equation coupled to a Schrödinger Eq. (3)
2 b p1 b p ep1 f p 1f 1þ ; ac 2 e 1 þ p3 2 b 2bp ep1 f v ðx; y; tÞ ¼ 22 2 ð4ac bðp1 bÞÞ þ p1 f 1 ; e þ p3 4a b uðx; y; tÞ ¼
m2 b2
96q2
ð37Þ ð38Þ
where f = kx + ky + mt + d. When p3 = 1, we have from (35)
!
!2 bp1 1 p f ; tanh 2 1 96q2 4ac 2 b 1 v ðx; y; tÞ ¼ 22 2 ð4ac b2 Þ bp1 tanh p1 f : 2 4a b uðx; y; tÞ ¼
m2 b2
2
1
b 4ac
ð39Þ ð40Þ
When p3 = 1, we have from (41)
!
!2 bp1 1 uðx; y; tÞ ¼ ; p f coth 2 1 96q2 4ac 2 b 1 v ðx; y; tÞ ¼ 22 2 ð4ac b2 Þ bp1 coth p1 f : 2 4a b
m2 b2
2
b 1 4ac
Clearly, (15)–(18) is the special case of (39)–(42) with p1 = 2. Case: V. We suppose that the Riccati equation (8) have the following solutions of the form
ð41Þ ð42Þ
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M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973
/ ¼ A0 þ
m X
i1
sinh
ðAi sinh x þ Bi cosh xÞ;
i¼1
where dx=df ¼ sinh x or dx=df ¼ cosh x. It is easy to find that m = 1 by balancing /0 and /2. So we choose
/ ¼ A0 þ A1 sinh x þ B1 cosh x;
ð43Þ i
j
when dx=df ¼ sinh x, we substitute (43) and dx=df ¼ sinh x, into (8) and set the coefficient of sinh x cosh x (i = 0, 1, 2; j = 0, 1) to zero. A set of algebraic equations is obtained as follows
aA20 þ aB21 þ bA0 þ c ¼ 0; 2aA0 A1 þ bA1 ¼ 0; aA21 þ aB21 ¼ B1 ; 2aA0 B1 þ bB1 ¼ 0;
ð44Þ
2aA1 B1 þ A1 ¼ 0 for which, we have the following solutions
A0 ¼
b ; 2a
A1 ¼ 0;
B1 ¼
1 ; a
ð45Þ
2
where c ¼ b 4a4 and
A0 ¼
b ; 2a
rffiffiffiffiffiffi 1 ; 2a
B1 ¼
A1 ¼
1 ; 2a
ð46Þ
2
where c ¼ b 4a1. To dx=df ¼ sinh x, we have
sinh x ¼ csch f;
cosh x ¼ coth f:
ð47Þ
From (46) and (47), we obtain
/¼
b þ 2 coth f ; 2a
ð48Þ
2
where c ¼ b 4a4 and
/¼
b csch f þ coth f ; 2a
ð49Þ
2
where c ¼ b 4a1. Clearly (48) is the special case of (36) with p1 = 2. From (8), (9), (12), (48) and (49), we get the new traveling wave solutions of Eq. (3) in the following form
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m 2 b2 96q2 b2
2
b 1 4ac
!
b cothðfÞ 2ac
!2 ;
2
2
4a2 b
ðð4ac b Þ 2b cothðfÞÞ2 ;
ð50Þ ð51Þ
2
where c ¼ b 4a4 and
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m 2 b2 96q2 b2
2
1
b 4ac
!
!2 b ðcothðfÞ cschðfÞÞ ; 4ac
2
2
4a2 b
ðð4ac b Þ bðcothðfÞ cschðfÞÞÞ2 ;
ð52Þ ð53Þ
2
where c ¼ b 4a1. Clearly (50), (51) is the special case of (41), (42) with p1 = 2. Similarly, when dx=df ¼ cosh x, we obtain the following exact traveling wave (periodic-like) solutions of KP like equation coupled to a Schrödinger Eq. (3)
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m 2 b2 96q2 b2
2
1
b 4ac 2
2
4a2 b
!
b cotðfÞ 2ac
ðð4ac b Þ 2b cotðfÞÞ2 :
!2 ;
ð54Þ ð55Þ
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M.K. Elboree / Applied Mathematics and Computation 218 (2012) 5966–5973 2
where c ¼ b 4a4 and
uðx; y; tÞ ¼
v ðx; y; tÞ ¼
m2 b2 96q2 b2
2
b 1 4ac 2
2
4a2 b
!
!2 b ðcotðfÞ cscðfÞÞ ; 4ac
ðð4ac b Þ bðcotðfÞ cscðfÞÞÞ2 :
ð56Þ ð57Þ
4. Conclusion HB method provides us a solutions polynomials in two elementary bell-shaped and kink-shaped functions, this covers the large majority of physically interesting solitary waves Eqs. (15), (16), (39), (40), in addition we also obtain a periodic-like wave solutions Eqs. (24), (25), (54), (55), (56), (57) which play an important interesting in physics. Remark. The majority of solutions for (8) in [23], contain solitary wave, periodic wave, rational solutions, etc. but in this paper we study the solutions of (8) which contains kink-type, bell-shaped, periodic-like, rational solutions which are wellknown solutions and also new solutions for instance in cases III, V are obtained, i.e. we recovered by HB method the wellknown solutions and adding some new solutions.
In summary we have used the extended homogeneous balance method to obtain many traveling wave solutions of KP like equation coupled to a Schrödinger equation. In Eq. (8), if we set b = 0, a = l, c = l HB method is reduced to tanh method [24] but HB method is more general than tanh method, extended tanh method [25] because this method is readily applicable to a large variety of nonlinear PDEs in contrast to the tanh method. Some merits are available for HB method. First, all the nonlinear PDEs which can be solved by tanh function method can be solved easily by HB method.and we have more multiple soliton solutions and triangular periodic solution (including rational solutions). Second, we used only the special solutions of Eq. (3), we can obtain more traveling wave solutions. Third, not only HB method contains the hyperbolic tangent expansion method, but also it is a computerizable method, which allow us to perform complicated and tedious differential calculation on computer. Therefore, the HB method is a generalized tanh function method for many nonlinear PDEs. One can construct the multisoliton solutions for the Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation using the algorithm in [26,27]. References [1] Mariana Antonova, Anjan Biswas, Adiabatic parameter dynamics of perturbed solitary waves, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 734– 748. [2] E.G. Fan, H.Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998) 403–406. [3] E.G. Fan, Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations, Phys. Lett. A 294 (2002) 26–30. [4] M.L. Wang, Y.M. Wang, A new Bäcklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients, Phys. Lett. A 287 (2001) 211–216. [5] M.L. Wang, Solitary wave solution for variant Boussinesq equations, Phys. Lett. A 199 (1995) 169–172. [6] M.L. Wang, Application of homogeneous balance method to exact solutions of nonlinear equation in mathematical physics, Phys. Lett. A 216 (1996) 67– 75. [7] M.L. Wang, Y.B. Zhou, Z.B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [8] Mohammed Khalfallah, New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 1169–1175. [9] Mohammed Khalfallah, Exact traveling wave solutions of the Boussinesq–Burgers equation, Math. Comput. Model. 49 (2009) 666–671. [10] A.S. Abdel Rady, A.H. Khater, E.S. Osman, Mohammed Khalfallah, New periodic wave and soliton solutions for system of coupled Korteweg–de Vries equations, Appl. Math. Comput. 207 (2009) 406414. [11] A.S. Abdel Rady, E.S. Osman, Mohammed Khalfallah, ‘‘Multi soliton solution for the system of coupled Korteweg–de Vries equations’’, Appl. Math. Comput. 210 (2009) 177–181. [12] A.S. Abdel Rady, E.S. Osman, Mohammed Khalfallah, On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 264–274. [13] A.S. Abdel Rady, Mohammed Khalfallah, On soliton solutions for Boussinesq–Burgers equations, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 886–894. [14] A.S. Abdel Rady, E.S. Osman, Mohammed Khalfallah, Multi soliton solution, rational solution of the Boussinesq–Burgers equations, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 1172–1176. [15] E.G. Fan, Two new applications of the homogeneous balance method, Phys. Lett. A 265 (2000) 353–357. [16] W.X. Ma, B. Fuchssteiner, Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation, Int. J. Nonlinear Mech. 31 (1996) 329–338. [17] C.L. Bai, Exact solutions for nonlinear partial differential equation: a new approach, Phys. Lett. A 288 (2001) 191–195. [18] C.L. Bai, Hong Zhao, Complex hyperbolic-function method and its applications to nonlinear equations, Phys. Lett. A 355 (2006) 32–38. [19] C.L. Bai, A new generalization of variable coefficients algebraic method for solving nonlinear evolution equations, Chaos Soliton Fract. 34 (2007) 1114– 1129. [20] Hong Zhao, Applications of the generalized algebraic method to special-type nonlinear equations, Chaos Soliton Fract. 36 (2008) 359–369. [21] X.B. Hu, The higher-order KdV equation with a source and nonlinear superposition formula, Chaos Soliton Fract. 7 (1996) 211–215. [22] X.Q. Zhao, D.B. Tang, A new note on a homogeneous balance method, Phys. Lett. A 297 (2002) 59–67.
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[23] Shun-dong Zhu, The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional BoitiLeonPempinelle equation, Chaos Soliton Fract. 37 (2008) 1335–1342. [24] E.G. Fan, Soliton solutions for a generalized Hirota Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A 282 (2001) 18–22. [25] C.L. Bai, H. Zhuo, Generalized extended tanh-function method and its application, Chaos Soliton Fract. 27 (2006) 1026–1035. [26] C.L. Bai, Multiple soliton solutions of the high order Broer–Kaup equations, Commun. Theor. Phys. 34 (2000) 729–732. [27] C.L. Bai, Abundant multi-soliton structure of the (3 + 1)-dimensional Nizhnik–Novikov–Veselov equations, Commun. Theor. Phys. 41 (2004) 15–20.