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Application Of A Generalized Bernoulli Sub-ODE Method
For Finding Traveling Solutions Of Some Nonlinear
Equations
Bin ZhengShandong University of Technology
School of ScienceZhangzhou Road 12, Zibo, 255049
Abstract: In this paper, a generalized Bernoulli sub-ODE method is proposed to construct exact travelingsolutions of nonlinear evolution equations. We apply the method to establish traveling solutions of thevariant Boussinseq equations, (2+1)-dimensional NNV equations and (2+1)-dimensional Boussinesq andKadomtsev-Petviashvili equations. As a result, some new exact traveling wave solutions are found.
KeyWords: Bernoulli sub-ODE method, traveling wave solutions, variant Boussinseq equations, NNVequations, Boussinesq and Kadomtsev-Petviashvili equations.
1 Introduction
It is well known that nonlinear evolution equa-tions (NLEEs) are widely used to describe manycomplex physical phenomena such as fluid me-chanics, plasma physics, optical fibers, biology,solid state physics, chemical kinematics, chemi-cal physics, and so on. So, the powerful and ef-ficient methods to find analytic solutions of non-linear equations have drawn a lot of interest bya diverse group of scientists. In the literature,there is a wide variety of approaches to nonlinearproblems for constructing traveling wave solution-s. Some of these approaches are the homogeneousbalance method [1, 2], the hyperbolic tangent ex-pansion method [3, 4], the trial function method[5], the tanh-method [6-8], the non-linear trans-form method [9], the inverse scattering transform[10], the Backlund transform [11, 12], the Hirotasbilinear method [13, 14], the generalized Riccatiequation method [15, 16], the Weierstrass ellipticfunction method [17], the theta function method[18-20], the sine-cosine method [21], the Jaco-bi elliptic function expansion [22, 23], the com-plex hyperbolic function method [24-26], the trun-cated Painleve expansion [27], the F-expansionmethod [28], the rank analysis method [29], theexp-function expansion method [30], the (G/G)-expansion method [31-40] and so on.
In [41], we proposed a new Bernoulli sub-ODEmethod to construct exact traveling wave solu-
tions for NLEEs. In this paper, we will applythe Bernoulli sub-ODE method to construct ex-act traveling wave solutions for some special non-linear equations. First, we reduce the nonlinearequations to ODEs by a traveling wave variabletransformation. Second, we suppose the solutioncan be expressed as an polynomial of single vari-able G, where G = G() satisfied the Bernoul-li equation. Then the degree of the polynomialcan be determined by the homogeneous balancemethod, and the coefficients can be obtained bysolving a set of algebraic equations.
The rest of the paper is organized as follows.In Section 2, we describe the Bernoulli sub-ODEmethod for finding traveling wave solutions ofnonlinear evolution equations, and give the mainsteps of the method. In the subsequent section-s, we will apply the method to find exact trav-eling wave solutions of the variant Boussinseq e-quation, (2+1)-dimensional NNV equations and(2+1)-dimensional Boussinesq and Kadomtsev-Petviashvili equations. In the last Section, someconclusions are presented.
2 Description Of The BernoulliSub-ODE Method
In this section we describe the Bernoulli Sub-ODEMethod.
First we present the solutions of the following
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ODE:G + G = G2, (1)
where = 0 , G = G().When = 0, Eq. (1) is the type of Bernoulli
equation, and we can obtain the solution as
G =1
+ de
, (2)
where d is an arbitrary constant.When = 0, the solution of Eq. (1) is given
byG = de, (3)
where d is an arbitrary constant.Suppose that a nonlinear equation, say in two
or three independent variables x, y, t, is given by
P (u, ut, ux, uy, , utt, uxt, uxx, uxy...) = 0, (4)
where u = u(x, y, t) is an unknown function, P is apolynomial in u = u(x, y, t) and its various partialderivatives, in which the highest order derivativesand nonlinear terms are involved. By using thesolutions of Eq. (1), we can construct a serials ofexact solutions of nonlinear equations:
Step 1. We suppose that
u(x, y, t) = u(), = (x, y, t). (5)
The traveling wave variable (5) permits us reduc-ing (4) to an ODE for u = u()
P (u, u, u, ...) = 0. (6)
Step 2. Suppose that the solution of (6) canbe expressed by a polynomial in G as follows:
u() = amGm + am1G
m1 + ...+ a0, (7)
where G = G() satisfies Eq. (1), and am, am1..., a0, are constants to be determined laterwith am = 0. The positive integer m can be de-termined by considering the homogeneous balancebetween the highest order derivatives and nonlin-ear terms appearing in (6).
Step 3. Substituting (7) into (6) and using(1), collecting all terms with the same order ofG together, the left-hand side of (6) is convertedto another polynomial in G. Equating each coef-ficient of this polynomial to zero, yields a set ofalgebraic equations for am, am1..., k, c, and.
Step 4. Solving the algebraic equations sys-tem in Step 3, and by using the solutions of Eq.(1), we can construct the traveling wave solutionsof the nonlinear evolution equation (6).
In the following sections, we will apply themethod described above to some examples.
3 Application Of The Bernoul-li Sub-ODE Method For TheVariant Boussinseq Equations
In this section, we will consider the variantBoussinseq equations [42, 43]:
ut + uux + vx + uxxt = 0, (8)
vt + (uv)x + uxxx = 0, (9)
where and are arbitrary constants, > 0.Supposing that
= k(x ct), (10)
by (10), (8) and (9) are converted into ODEs
cu + uu + v k2cu = 0 (11)
cv + (uv) + k2u = 0. (12)Integrating (11) and (12) once, we have
cu+ 12u2 + v k2cu = g1, (13)
cv + uv + k2u = g2, (14)where g1 and g2 are the integration constants.
Suppose that the solution of (13) and (14) canbe expressed by a polynomial in G as follows:
u() =
mi=0
aiGi, (15)
v() =
ni=0
biGi, (16)
where ai, bi are constants, G = G() satisfies Eq.(1).
Balancing the order of u2 and v in Eq. (13),the order of u and uv in Eq. (14), then we canobtain 2m = n, n+2 = m+ n m = 1, n = 2,so Eqs. (15) and (16) can be rewritten as
u() = a2G2 + a1G+ a0, a2 = 0, (17)
v() = b2G2 + b1G+ b0, b2 = 0, (18)
where a1, a0, b2, b1, b0 are constants to be de-termined later.
Substituting (17) and (18) into (13) and (14)and collecting all the terms with the same powerof G together, equating each coefficient to zero,yields a set of simultaneous algebraic equationsas follows:
For Eq. (13):
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G0 : ca0 g1 + 12a20 + b0 = 0.
G1 : b1 + a0a1 ca1 k2ca12 = 0.
G2 : ca2 + b2 + 3k2ca1 + 12a21
4k2ca22 + a0a2 = 0.
G3 : a1a2 2k2ca12 + 10k2ca2 = 0.
G4 : 12a22 6k2ca22 = 0.
For Eq. (14):G0 : cb0 g2 + a0b0 = 0.
G1 : b1a0 + a1b0 cb1 + k2a12 = 0.
G2 : cb2+a1b13betak2a1+4k2a22+a2b0 + a0b2 = 0.
G3 : 2k2a12+a1b210k2a2+a2b1 = 0.
G4 : a2b2 + 6k2a2
2 = 0.
Solving the algebraic equations above yields:
a2 = 12k2c2,
a1 = 12k2c,
a0 =12(
+2c2+222k2c2
c ),
b2 = 6k22, b1 = 6k2,
b0 = 4 (+222k2c2
2c2),
g2 = 8 (2+444k4c4
3c3),
g1 =18(
4c42+32+444k4c42c2
).
(19)
Combining with (2) and (3), under the conditions = 0, we can obtain the traveling wave solutionsof the variant Boussinseq equations (8) and (9) asfollows:
u1() =1
2( + 2c2 + 222k2c2
c)
12k2c( 1 + de
)
+12k2c2(1
+ de
)2 (20)
v1() =
4( + 222k2c2
2c2)
+6k2(1
+ de
)
6k22( 1 + de
)2 (21)
Remark 1 When = 0, we obtain the trivialsolutions.
Remark 2 The exact traveling wave solutions(20)-(21) of the variant Boussinseq equations aredifferent from the results in [42, 43], and have notbeen reported by other authors to our best knowl-edge.
4 Application Of The BernoulliSub-ODE Method For (2+1)-dimensional NNV Equations
In this section, we consider the (2+1)-dimensionalNNV equations [4446]:
ut + auxxx + buyyy + cux + duy= 3a(uv)x + 3b(uw)y,
(22)
ux = vy, (23)
uy = wx. (24)
Suppose that
= kx+ ly + t. (25)
By (25), (22), (23) and (24) are converted intoODEs
u + ak3u + bl3u + cku + dlu
= 3ak(uv) + 3bl(uw),(26)
ku = lv, (27)
lu = kw. (28)
Integrating (26), (27) and (28) once times, wehave
u+ ak3u + bl3u + cku+ dlu= 3akuv + 3bluw + g1,
(29)
ku = lv + g2, (30)
lu = kw + g3, (31)
where g1, g2, g3 are the integration constants.Suppose that the solutions of (29), (30) and
(31) can be expressed by polynomials in G as fol-lows:
u() =mi=0
aiGi, (32)
v() =n
i=0
biGi, (33)
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w() =
si=0
ciGi, (34)
where ai, bi, ci are constants, G = G() satisfiesEq. (1).
Balancing the order of u and uv in Eq. (29),the order of u and v in Eq. (30), the order of uand w in Eq. (31), then we can obtain m + 2 =m + n, m = n, m = s m = n = s = 2, soEq.(32), (33) and (34) can be rewritten as
u() = a2G2 + a1G+ a0, a2 = 0, (35)
v() = b2G2 + b1G+ b0, b2 = 0, (36)
w() = c2G2 + c1G+ c0, c2 = 0, (37)
where a2, a1, a0, b2, b1, b0, c2, c1, c0 are con-stants to be determined later.
Substituting (35), (36) and (37) into (29),(30) and (31) and collecting all the terms withthe same power of G together, equating each co-efficient to zero, yields a set of simultaneous alge-braic equations as follows:
For Eq. (29):
G0 : cka0 + dla0 3aka0b0 3bla0c0g0 + a0 = 0.
G1 : 3aka0b1 3bla0c1 + dla1 + ak3a12
+cka1 3aka1b0 + bl3a12 3bla1c0+a1 = 0.
G2 : 3ak3a1+ cka2 3aka0b2 3bla0c2+4ak3a2
2 3aka1b1 3bl3a1+ a2+4bl3a2
2 3bla1c1 + dla2 3aka2b03bla2c0 = 0.
G3 : 3aka1b2 + 2bl3a12 + 2ak3a12
3aka2b1 3bla1c2 10ak3a210bl3a2 3bla2c1 = 0.
G4 : 3aka2b2 3bla2c2 + 6ak3a22
+6bl3a22 = 0.
For Eq.(30):
G0 : ka0 lb0 g2 = 0.G1 : ka1 lb1 = 0.G2 : ka2 lb2 = 0.
For Eq. (31):
G0 : la0 kc0 g3 = 0.G1 : la1 kc1 = 0.G2 : la2 kc2 = 0.
Solving the algebraic equations above yields:
a2 = 2lk2, a1 = 2lk, a0 = a0,
b2 = 2k22, b1 = 2k2, b0 = b0,
c2 = 22l2, c1 = 2l2, k = k, l = l, = ,
c0 =1
3
3ak3a0 3bl3a0 + dl2k + ak4l2
bl2k
+1
3
ck2l 3ak2lb0 + bl42k + lkbl2k
,
g1 = a03ak3a0 3bl3a0 + ak4l2 + bl42k
lk,
g2 = ka0 lb0,
g3 = 1
3
6bl3a0 3ak3a0 + dl2k + ak4l2
l2b
13
ck2l 3ak2lb0 + bl42k + lkl2b
,
(38)
where k, l, , a0, b0 are arbitrary constants.
Under the condition = 0, combining with(2) and (3), we can obtain the traveling wave so-lutions of the (2+1)-dimensional NNV equations(22)-(24) as follows:
u() = 2lk2(1
+ de
)2 2lk( 1
+ de) + a0
(39)
v() = 2k22(1
+ de
)2 2k2( 1
+ de) + b0
(40)
w() = 22l2(1
+ de
)2 2l2( 1
+ de)
+1
3
3ak3a0 3bl3a0 + dl2k + ak4l2
bl2k
+1
3
ck2l 3ak2lb0 + bl42k + lkbl2k
(41)
where = kx+ ly + t.
Remark 3 Some authors have reported some ex-act solutions for the (2+1)-dimensional NNV e-quations in [44-46]. To our best knowledge, ourresults (39)-(41) have not been reported so far inthe literature.
Remark 4 When = 0 , we obtain the trivialsolutions.
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5 Application Of The BernoulliSub-ODE Method For (2+1)dimensional Boussinesq andKadomtsev-Petviashvili equa-tions
In this section we will consider the following(2+1) dimensional Boussinesq and Kadomtsev-Petviashvili equations [47] :
uy = qx, (42)
vx = qy, (43)
qt = qxxx + qyyy + 6(qu)x + 6(qv)y. (44)
In order to obtain the traveling wave solutions of(42)-(44), we suppose that
u(x, y, t) = u(),v(x, y, t) = v(),q(x, y, t) = q(), = ax+ dy ct,
(45)
where a, d, c are constants that to be determinedlater.
Using the wave variable (45), (42)-(44) can beconverted into ODEs
du aq = 0, (46)av dq = 0, (47)(a3 + d3)q cq 6auq6aqu 6dvq 6dqv = 0. (48)
Integrating the ODEs above, we obtain
du aq = g1, (49)av dq = g2, (50)(a3 + d3)q cq 6auq 6dvq = g3.(51)
Supposing that the solutions of the ODEs abovecan be expressed by a polynomial in G as follows:
u() =l
i=0
aiGi, (52)
v() =mi=0
biGi, (53)
q() =
ni=0
ciGi, (54)
where ai, bi, ci are constants, and G = G() satis-fies Eq. (1).
Balancing the order of u and q in Eq. (52),the order of v and q in Eq. (53) and the order of
q and vq in Eq. (54), we have l+1 = n+1, m+1 = n+1, n+3 = m+n+1 l = m = n = 2.So Eq.(52)-(54) can be rewritten as
u() = a2G2 + a1G+ a0, a2 = 0, (55)
v() = b2G2 + b1G+ b0, b2 = 0, (56)
u() = c2G2 + c1G+ c0, c2 = 0, (57)
where ai, bi, ci are constants to be determinedlater.
Substituting (55)-(57) into the ODEs (49)-(51), collecting all terms with the same powerof G together, equating each coefficient to zero,yields a set of simultaneous algebraic equationsas follows:
For Eq. (49):G0 : a0d ac0 g1 = 0.
G1 : a1d ac1 = 0.
G2 : a2d ac2 = 0.
For Eq. (50):G0 : ab0 g2 dc0 = 0.
G1 : ab1 dc1 = 0.
G2 : dc2 + ab2 = 0.
For Eq. (51):
G0 : g3 cc0 6db0c0 6aa0c0 = 0.
G1 : 6aa1c0 6db1c0 + a3c12 6db0c1
6aa0c1 + d3c12 cc1 = 0.
G2 : 4a3c22 6aa0c2 6db1c1 6aa1c1
6aa2c0 + 4d3c22 cc2 3d3c1
6db0c2 3a3c1 6db2c0 = 0.
G3 : 6aa2c1 6db2c1 + 2a3c12 10d3c2
6aa1c2 10a3c2 6db1c2
+2d3c12 = 0.
G4 : 6d3c226aa2c2+6a3c226db2c2 = 0.
Solving the algebraic equations above yield-s:
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Case 1:
a0 = a0, a1 = a2, a2 = a22,
b0 = b0, b1 = d2, b2 = d22, a = a,
c0 = c0, c1 = da, c2 = d2a,
g2 = ab0 dc0, g1 = da0 ac0, d = d,
c =6a3c0 6d3c0 + a4d2 6d2b0a 6a2a0d+ d42a
ad,
g3 = c06a3c0 6d3c0 + a4d2 + d42a
ad,
(58)
where a0, b0, c0, a, d are arbitrary constants.
Assume = 0, then substituting the resultsabove into (55)-(57), combining with (2) we canobtain the traveling wave solution of (2+1) di-mensional BKP equation as follows:
u1() = a22(
1 + de
)2 a2( 1
+ de) + a0,
(59)
v1() = d22(
1 + de
)2 d2( 1
+ de) + b0,
(60)
q1() = d2a(
1 + de
)2 da( 1
+ de) + c0,
(61)where
= ax+ dy 6a3c0 6d3c0 + a4d2
adt
6d2b0a 6a2a0d+ d42a
adt. (62)
Case 2:
a0 = a0, a1 = a1, a2 = d22,
b0 = b0, b1 = a1, b2 = d22,
d = d, c = 6d(b0 + a0),c0 = c0, c1 = a1, c2 = d22,g2 = db0 dc0, g1 = da0 + dc0,a = d, g3 = 0,
(63)
where a0, b0, c0, a1, d are arbitrary constants.
Similarly, under the condition = 0, we canobtain traveling wave solutions of (2+1) dimen-sional Boussinesq and Kadomtsev-Petviashvili e-quations as follows:
u2() = d22(
1 + de
)2 + a1(
1 + de
) + a0,
(64)
v2() = d22(
1 + de
)2 + a1(
1 + de
) + b0,
(65)
q2() = d22(1
+ de
)2 a1(
1 + de
) + c0,
(66)where
= dx+ dy 6d(b0 + a0)t. (67)
Case 3:
a0 = a0, a1 =12d
2(12 12
3i),
a2 = d22(12
12
3i),
b0 = b0, b1 =12d
2, b2 = d22,
c0 = c0, c1 =12d
2(12 12
3i),
c2 = d22(12
12
3i),
d = d, a = (12 12
3i)d,
c = 6da0(12 12
3i) 6db0,
g1 = da0 dc0(12 12
3i),
g2 = db0(12
12
3i) dc0,
g3 = 0,
(68)
where a0, b0, c0, d are arbitrary constants.
Thus
u3() = d22(12
12
3i)( 1
+de
)2
+12d2(12
12
3i)( 1
+de
) + a0,(69)
v3() = d22( 1
+de
)2 + 12d2( 1
+de
) + b0,
(70)
q3() = d22(12
12
3i)( 1
+de
)2
+12d2(12
12
3i)( 1
+de
) + c0,(71)
= (1
2 12
3i)dx+dy+[6da0(
1
2 12
3i)+6db0]t,
(72)where = 0.
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Case 4:
a0 = a0, a1 =12d
2(12 12
3i),
a2 = d22(12
12
3i)
b0 = b0, b1 = a1(12 12
3i),
b2 = d22
c0 = c0, c1 = a1(12 12
3i),
c2 = d22(12
12
3i)
d = d, a = (12 12
3i)d,
c = 6da0(12 12
3i) 6db0
g1 = da0 dc0(12 12
3i),
g2 = db0(12
12
3i) dc0,
g3 = 0,
(73)
where a0, b0, c0, d are arbitrary constants.
Then
u4() = d22(12
12
3i)( 1
+de
)2
+12d2(12
12
3i)( 1
+de
) + a0,(74)
v4() = d22( 1
+de
)2
a1(12 12
3i)( 1
+de
) + b0,(75)
q4() = d22(12
12
3i)(12
12
3i)( 1
+de
)2
a1(12 12
3i)( 1
+de
) + c0,
(76)
= (1
2 12
3i)dx+dy+[6da0(
1
2 12
3i)+6db0]t,
(77)where = 0.
Remark 5 When = 0, we obtain the triv-ial solutions. The traveling wave solution-s established for (2+1)-dimensional Boussinesqand Kadomtsev-Petviashvili equations (59)-(61),(64)-(66), (69)-(73), (74)-(76) have not been re-ported by other authors to our best knowledge.
6 Comparison with Zayed re-sults
In [46,43], Zayed solved the (2+1)-dimensional N-NV equations and the variant Boussinseq equa-tions by using the (G/G) expansion method re-spectively. In this section, we will present somecomparisons between the established results inSection 3-4 and Zayed results.
Let 1, 1 represent , in [46] re-spectively. Then in (39)-(41), considering
d, , , a0, b0, l, k, are arbitrary constants,if we take
d =A+B21 41
, = B A, =
21 41,
a0 = 0 21, l = 1, k = 1, b0 = 21,
= 3(a+b)(021)A+B21 41
(a+b)(2141)
c 6a1 3b(0 1
221),
where A, B, 0, 0 are defined in [46], our solu-tions (39)-(41) reduce to the solutions derived in[46, (3.32 )-(3.34)]. Furthermore, under the con-dition A = 0, B = 0, 1 > 0, 1 = 0, (39)-(41)reduce the solitary solutions in [46, (3.41)-(3.43)].If we take
d =iA+B
i
41 21, = iAB, = i
41 21,
a0 = 0 21, l = 1, k = 1, b0 = 21,
= 3(a+b)(021)iA+B
i
21 41(a+b)(2141)
c 6a1 3b(0 1
221),
then our solutions (39)-(41) reduce to the solu-tions derived in [46, (3.35)-(3.37)]. So in this way,our results (39)-(41) extend Zayed results for the(2+1)-dimensional NNV equations in [46] .
For the variant Boussinseq equations, we notethat our solutions (20)-(21) are different solutionsfrom the results in [43, (33)-(38)].
7 Conclusions
In this paper we have seen that some new trav-eling wave solutions of the variant Boussinseq e-quations, (2+1)-dimensional NNV equations and(2+1)-dimensional Boussinesq and Kadomtsev-Petviashvili equations are successfully found byusing the Bernoulli sub-ODE method. The mainpoints of the method are that assuming the solu-tion of the ODE reduced by using the travelingwave variable as well as integrating can be ex-pressed by anm-th degree polynomial in G, whereG = G() is the general solutions of a Bernoullisub-ODE equation. The positive integerm can bedetermined by the general homogeneous balancemethod, and the coefficients of the polynomial can
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be obtained by solving a set of simultaneous alge-braic equations.
Compared to the methods used before, onecan see that this method is concise and effective.Also this method can be applied to other nonlin-ear problems.
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