Application Of A Generalized Bernoulli Sub-ODE … · Application Of A Generalized Bernoulli Sub-ODE Method For Finding Traveling Solutions Of Some Nonlinear Equations Bin Zheng Shandong

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

[email protected]

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.

Key–Words: 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 Hirota’sbilinear 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(ξ) 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

WSEAS TRANSACTIONS on MATHEMATICS Bin Zheng

E-ISSN: 2224-2880 618 Issue 7, Volume 11, July 2012

ODE:G′ + λG = µG2, (1)

where λ ̸= 0 , G = G(ξ).When µ ̸= 0, Eq. (1) is the type of Bernoulli

equation, and we can obtain the solution as

G =1

µ

λ+ deλξ

, (2)

where d is an arbitrary constant.When µ = 0, the solution of Eq. (1) is given

byG = de−λξ, (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(ξ), ξ = ξ(x, y, t). (5)

The traveling wave variable (5) permits us reduc-ing (4) to an ODE for u = u(ξ)

P (u, u′, u′′, ...) = 0. (6)

Step 2. Suppose that the solution of (6) canbe expressed by a polynomial in G as follows:

u(ξ) = amGm + am−1Gm−1 + ...+ a0, (7)

where G = G(ξ) satisfies Eq. (1), and am, am−1

..., a0, µ 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, am−1..., k, c, λ andµ.

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 + αuxxt = 0, (8)

vt + (uv)x + βuxxx = 0, (9)

where α and β are arbitrary constants, β > 0.Supposing that

ξ = k(x− ct), (10)

by (10), (8) and (9) are converted into ODEs

−cu′ + uu′ + v′ − αk2cu′′′ = 0 (11)

−cv′ + (uv)′ + βk2u′′′ = 0. (12)

Integrating (11) and (12) once, we have

−cu+1

2u2 + v − αk2cu′′ = g1, (13)

−cv + uv + βk2u′′ = g2, (14)

where g1 and g2 are the integration constants.Suppose that the solution of (13) and (14) can

be expressed by a polynomial in G as follows:

u(ξ) =

m∑i=0

aiGi, (15)

v(ξ) =

n∑i=0

biGi, (16)

where ai, bi are constants, G = G(ξ) 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(ξ) = a2G2 + a1G+ a0, a2 ̸= 0, (17)

v(ξ) = 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):



G0 : −ca0 − g1 +12a

20 + b0 = 0.

G1 : b1 + a0a1 − ca1 − αk2ca1λ2 = 0.

G2 : −ca2 + b2 + 3αk2ca1µλ + 12a

21 −

4αk2ca2λ2 + a0a2 = 0.

G3 : a1a2 − 2αk2ca1µ2 + 10αk2ca2µλ = 0.

G4 : 12a

22 − 6αk2ca2µ

2 = 0.

For Eq. (14):G0 : −cb0 − g2 + a0b0 = 0.

G1 : b1a0 + a1b0 − cb1 + βk2a1λ2 = 0.

G2 : −cb2+a1b1−3betak2a1µλ+4βk2a2λ2+

a2b0 + a0b2 = 0.

G3 : 2βk2a1µ2+a1b2−10βk2a2µλ+a2b1 = 0.

G4 : a2b2 + 6βk2a2µ2 = 0.

Solving the algebraic equations above yields:

a2 = 12αk2cµ2,a1 = −12αk2cµλ,

a0 =12(

β+2αc2+2λ2α2k2c2

αc ),

b2 = −6βk2µ2, b1 = 6βk2µλ,

b0 = −β4 (

−β+2λ2α2k2c2

α2c2),

g2 = −β8 (

−β2+4λ4α4k4c4

α3c3),

g1 =18(

−4c4α2+3β2+4λ4α4k4c4

α2c2).

(19)

Combining with (2) and (3), under the conditionsµ ̸= 0, we can obtain the traveling wave solutionsof the variant Boussinseq equations (8) and (9) asfollows:

u1(ξ) =1

2(β + 2αc2 + 2λ2α2k2c2

αc)

−12αk2cµλ( 1µλ + deλξ

)

+12αk2cµ2(1

µλ + deλξ

)2 (20)

v1(ξ) = −β

4(−β + 2λ2α2k2c2

α2c2)

+6βk2µλ(1

µλ + deλξ

)

−6βk2µ2(1

µλ + deλξ

)2 (21)

Remark 1 When µ = 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 [44–46]:

ut + auxxx + buyyy + cux + duy= 3a(uv)x + 3b(uw)y,

(22)

ux = vy, (23)

uy = wx. (24)

Suppose that

ξ = kx+ ly + ωt. (25)

By (25), (22), (23) and (24) are converted intoODEs

ωu′ + ak3u′′′ + bl3u′′′ + cku′ + dlu′

= 3ak(uv)′ + 3bl(uw)′,(26)

ku′ = lv′, (27)

lu′ = kw′. (28)

Integrating (26), (27) and (28) once times, wehave

ωu+ 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(ξ) =m∑i=0

aiGi, (32)

v(ξ) =n∑

i=0

biGi, (33)



w(ξ) =

s∑i=0

ciGi, (34)

where ai, bi, ci are constants, G = G(ξ) 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(ξ) = a2G2 + a1G+ a0, a2 ̸= 0, (35)

v(ξ) = b2G2 + b1G+ b0, b2 ̸= 0, (36)

w(ξ) = 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 − 3bla0c0

−g0 + ωa0 = 0.

G1 : −3aka0b1 − 3bla0c1 + dla1 + ak3a1λ2

+cka1 − 3aka1b0 + bl3a1λ2 − 3bla1c0

+ωa1 = 0.

G2 : −3ak3a1µλ+ cka2 − 3aka0b2 − 3bla0c2

+4ak3a2λ2 − 3aka1b1 − 3bl3a1µλ+ ωa2

+4bl3a2λ2 − 3bla1c1 + dla2 − 3aka2b0

−3bla2c0 = 0.

G3 : −3aka1b2 + 2bl3a1µ2 + 2ak3a1µ

2

−3aka2b1 − 3bla1c2 − 10ak3a2µλ

−10bl3a2µλ− 3bla2c1 = 0.

G4 : −3aka2b2 − 3bla2c2 + 6ak3a2µ2

+6bl3a2µ2 = 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 = 2lkµ2, a1 = −2lµλk, a0 = a0,

b2 = 2k2µ2, b1 = −2µk2λ, b0 = b0,

c2 = 2µ2l2, c1 = −2µl2λ, k = k, l = l, ω = ω,

c0 =1

3

−3ak3a0 − 3bl3a0 + dl2k + ak4lλ2

bl2k

+1

3

ck2l − 3ak2lb0 + bl4λ2k + ωlk

bl2k,

g1 = −a0−3ak3a0 − 3bl3a0 + ak4lλ2 + bl4λ2k

lk,

g2 = ka0 − lb0,

g3 = −1

3

−6bl3a0 − 3ak3a0 + dl2k + ak4lλ2

l2b

−1

3


l2b,

(38)

where k, l, ω, a0, b0 are arbitrary constants.

Under the condition µ ̸= 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(ξ) = 2lkµ2(1

µλ + deλξ

)2 − 2lµλk(1

µλ + deλξ

) + a0

(39)

v(ξ) = 2k2µ2(1

µλ + deλξ

)2 − 2µk2λ(1

µλ + deλξ

) + b0

(40)

w(ξ) = 2µ2l2(1

µλ + deλξ

)2 − 2µl2λ(1

µλ + deλξ

)

+1

3

−3ak3a0 − 3bl3a0 + dl2k + ak4lλ2

bl2k

+1

3


bl2k(41)

where ξ = kx+ ly + ωt.

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 µ = 0 , we obtain the trivialsolutions.



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(ξ),v(x, y, t) = v(ξ),q(x, y, t) = q(ξ),ξ = 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′ − 6auq′

−6aqu′ − 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(ξ) =l∑

i=0

aiGi, (52)

v(ξ) =m∑i=0

biGi, (53)

q(ξ) =

n∑i=0

ciGi, (54)

where ai, bi, ci are constants, and G = G(ξ) 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(ξ) = a2G2 + a1G+ a0, a2 ̸= 0, (55)

v(ξ) = b2G2 + b1G+ b0, b2 ̸= 0, (56)

u(ξ) = 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 + a3c1λ2 − 6db0c1

−6aa0c1 + d3c1λ2 − cc1 = 0.

G2 : 4a3c2λ2 − 6aa0c2 − 6db1c1 − 6aa1c1

−6aa2c0 + 4d3c2λ2 − cc2 − 3d3c1µλ

−6db0c2 − 3a3c1µλ− 6db2c0 = 0.

G3 : −6aa2c1 − 6db2c1 + 2a3c1µ2 − 10d3c2µλ

−6aa1c2 − 10a3c2µλ− 6db1c2

+2d3c1µ2 = 0.

G4 : 6d3c2µ2−6aa2c2+6a3c2µ

2−6db2c2 = 0.

Solving the algebraic equations above yield-s:



Case 1:

a0 = a0, a1 = −µλa2, a2 = a2µ2,

b0 = b0, b1 = −d2µλ, b2 = d2µ2, a = a,

c0 = c0, c1 = −dµλa, c2 = dµ2a,

g2 = ab0 − dc0, g1 = da0 − ac0, d = d,

c =−6a3c0 − 6d3c0 + a4dλ2 − 6d2b0a− 6a2a0d+ d4λ2a

ad,

g3 = −c0−6a3c0 − 6d3c0 + a4dλ2 + d4λ2a

ad,

(58)

where a0, b0, c0, a, d are arbitrary constants.

Assume µ ̸= 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(ξ) = a2µ2(1

µλ + deλξ

)2 − µλa2(1

µλ + deλξ

) + a0,

(59)

v1(ξ) = d2µ2(1

µλ + deλξ

)2 − d2µλ(1

µλ + deλξ

) + b0,

(60)

q1(ξ) = dµ2a(1

µλ + deλξ

)2 − dµλa(1

µλ + deλξ

) + c0,

(61)where

ξ = ax+ dy − −6a3c0 − 6d3c0 + a4dλ2

adt

−−6d2b0a− 6a2a0d+ d4λ2a

adt. (62)

Case 2:

a0 = a0, a1 = a1, a2 = d2µ2,b0 = b0, b1 = a1, b2 = d2µ2,d = d, c = 6d(−b0 + a0),c0 = c0, c1 = −a1, c2 = −d2µ2,g2 = −db0 − dc0, g1 = da0 + dc0,a = −d, g3 = 0,

(63)

where a0, b0, c0, a1, d are arbitrary constants.

Similarly, under the condition µ ̸= 0, we canobtain traveling wave solutions of (2+1) dimen-sional Boussinesq and Kadomtsev-Petviashvili e-quations as follows:

u2(ξ) = d2µ2(1

µλ + deλξ

)2 + a1(1

µλ + deλξ

) + a0,

(64)

v2(ξ) = d2µ2(1

µλ + deλξ

)2 + a1(1

µλ + deλξ

) + b0,

(65)

q2(ξ) = −d2µ2(1

µλ + deλξ

)2 − a1(1

µλ + deλξ

) + c0,

(66)where

ξ = −dx+ dy − 6d(−b0 + a0)t. (67)

Case 3:

a0 = a0, a1 =12d

2µλ(−12 ±

12

√3i),

a2 = d2µ2(−12 ±

12

√3i),

b0 = b0, b1 =12µλd

2, b2 = d2µ2,

c0 = c0, c1 =12d

2µλ(12 ±12

√3i),

c2 = d2µ2(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(ξ) = d2µ2(−12 ±

12

√3i)( 1

µλ+deλξ

)2

+12d

2µλ(−12 ±

12

√3i)( 1

µλ+deλξ

) + a0,(69)

v3(ξ) = d2µ2( 1µλ+deλξ

)2 + 12µλd

2( 1µλ+deλξ

) + b0,

(70)

q3(ξ) = d2µ2(12 ±12

√3i)( 1

µλ+deλξ

)2

+12d

2µλ(12 ±12

√3i)( 1

µλ+deλξ

) + c0,(71)

ξ = (1

2± 1

2

√3i)dx+dy+[6da0(

1

2± 1

2

√3i)+6db0]t,

(72)where µ ̸= 0.



Case 4:

a0 = a0, a1 =12d

2µλ(−12 ±

12

√3i),

a2 = d2µ2(−12 ±

12

√3i)

b0 = b0, b1 = −a1(12 ±12

√3i),

b2 = d2µ2

c0 = c0, c1 = −a1(−12 ±

12

√3i),

c2 = d2µ2(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(ξ) = d2µ2(−12 ±

12

√3i)( 1

µλ+deλξ

)2

+12d

2µλ(−12 ±

12

√3i)( 1

µλ+deλξ

) + a0,(74)

v4(ξ) = d2µ2( 1µλ+deλξ

)2

−a1(12 ±12

√3i)( 1

µλ+deλξ

) + b0,(75)

q4(ξ) = d2µ2(12 ±12

√3i)(12 ±

12

√3i)( 1

µλ+deλξ

)2

−a1(−12 ±

12

√3i)( 1

µλ+deλξ

) + c0,

(76)

ξ = (1

2± 1

2

√3i)dx+dy+[6da0(

1

2± 1

2

√3i)+6db0]t,

(77)where µ ̸= 0.

Remark 5 When µ = 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 µ1, λ1 represent µ, λ in [46] re-spectively. Then in (39)-(41), considering

d, µ, λ, a0, b0, l, k, ω are arbitrary constants,if we take

d =A+B√λ21 − 4µ1

, µ = B −A, λ =√

λ21 − 4µ1,

a0 = α0 − 2µ1, l = 1, k = 1, b0 = −2µ1,

ω = 3(a+b)(α0−2µ1)−A+B√λ21 − 4µ1

−(a+b)(λ21−4µ1)

−c− 6aµ1 − 3b(γ0 −1

2λ21),

where A, B, α0, γ0 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, λ1 > 0, µ1 = 0, (39)-(41)reduce the solitary solutions in [46, (3.41)-(3.43)].If we take

d =iA+B

i√

4µ1 − λ21

, µ = iA−B, λ = i√

4µ1 − λ21,

a0 = α0 − 2µ1, l = 1, k = 1, b0 = −2µ1,

ω = 3(a+b)(α0−2µ1)−iA+B

i√

λ21 − 4µ1

−(a+b)(λ21−4µ1)

−c− 6aµ1 − 3b(γ0 −1

2λ21),

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(ξ) 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



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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Application Of A Generalized Bernoulli Sub-ODE … · Application Of A Generalized Bernoulli Sub-ODE Method For Finding Traveling Solutions Of Some Nonlinear Equations Bin Zheng Shandong