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Painlevé property and exact solutions for a nonlinear wave equation with generalized power-law nonlinearities
Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
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Painlevé property and exact solutions for a nonlinear wave equation with generalized power-law nonlinearities Matthew Russo, Robert A. Van Gorder ⇑, S. Roy Choudhury Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 3 October 2012 Received in revised form 17 November 2012 Accepted 25 November 2012 Available online 3 December 2012 Keywords: Painlevé analysis Nonlinear hyperbolic PDE Power-law nonlinearity Exact solutions
By employing a variety of techniques, we investigate several classes of solutions of a family of nonlinear partial differential equations (NLPDEs) with generalized nonlinearities, special cases of which include the Klein–Gordon equation, the Landau–Ginzburg–Higgs equation, the u4 and u6 equations, the Rayleigh wave equation. The Painlevé property for our class of equations is studied first, showing that there are integrable families of such equations satisfying the strong Painlevé property (under a traveling wave assumption). From the truncated Laurent expansions, we introduce the auto-Bäcklund transformation for the two families shown to admit the strong Painlevé property. A multi-parameter family of exact solutions is then constructed from these auto-Bäcklund transformations for each of the cases, leading to travelling wave solutions. From here, assuming only travelling wave solutions, we then discuss more general methods of obtaining travelling wave solutions for those cases which do not satisfy the strong Painlevé property. Such solutions constitute rare exact solutions to a complicated nonlinear partial differential equation. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In the present paper, we shall be concerned with the nonlinear hyperbolic equation partial differential equation (PDE) q p X X utt uxx ¼ ai ut2i1 þ bk uk : i¼1
ð1Þ
k¼1
When the first sum in (1) is removed, we have a nonlinear (provided p P 2) Klein–Gordon equation. More particularly, two similar cases of this equation considered are known as the u42 equation and Landau–Ginzburg–Higgs equation [1], both of which occur when p = 3 and q ¼ 0. When p ¼ 5 and q ¼ 0, we recover the u6 equation [1]. Meanwhile, physically relevant solutions still occur when at least one ai – 0. For example, when q ¼ 2; p ¼ 0, we recover the Rayleigh wave equation. There are many methods which have been used over the past decades to find various types of exact analytic solutions of nonlinear partial differential equations (NLPDEs). Among them are (we included only some recent references): a. b. c. d. e.
the extended tanh–coth method [2,3]; Hirota’s method [4,5]; Painlevé series systematically truncated for ‘partial integrability’ [6,7]; invariant Painlevé analysis [6]; Bell Polynomial analyses [8];
⇑ Corresponding author. E-mail address: [email protected] (R.A. Van Gorder). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.11.019
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f. series solutions based on appropriate basis functions [9]; g. similarity reduction methods [10,11]. In this paper, we shall employ a potpourri of methods, Painlevé techniques and Bell Polynomial–Hirota-type exponential polynomials to analyze a large variety of solutions of (1). The paper consists of two main parts. In the first, we first show that the Painlevé property holds for certain subclasses of (1). In particular, two important subclasses satisfy the strong Painlevé property, whereas an infinite number of classes satisfy the weak Painlevé property. From the truncated Laurent expansions, we introduce the auto-Bäcklund transformation for the two families shown to admit the strong Painlevé property. A multiparameter family of exact solutions is then constructed from these auto-Bäcklund transformations for each of the two cases, leading to travelling wave solutions. In the second part of the paper, we consider a general travelling wave transformation, which maps our nonlinear PDE (1) into an ordinary differential equation (ODE) governing all possible travelling wave solutions, for arbitrary non-negative integer values of p and q. In order to recover the solutions which are rational functions of exponentials, an alternate form of this equation is obtained. The solutions obtained here through each of the two methods constitute rare exact solutions to a complicated nonlinear partial differential equation. 2. Painlevé property for several cases of p and q We wish to find the values of p and q for which (1) admits the Painlevé property. Recall that an equation has the strong Painlevé property if it has no movable critical points (for instance, branch points or essential singularities) and it is solution in a neighborhood of a singularity /0 scales as ð/ /0 Þa for positive integer a. For an overview of the method, see [12–14]; for a recent application, see [15]. Assume that / depends on both x and t, and scale / so that /0 ¼ 0. Applying the singular manifold method about the singular manifold
/ðx; tÞ ¼ 0;
ð2Þ
we use a leading order analysis with the substitution
u ¼ u0 /a
ð3Þ
and balance the highest order nonlinearities with the highest order derivatives. Substitution of (3) into (1) gives us two possible values for a: term domination. Case (i): u2p1 t If p 6 2q 1, the ut2q1 term dominates. Balancing utt ; uxx and aq u2q1 gives us t
a¼
3 2q ; 2 2q "
u0 ¼
3 2q
a2q1 ð2 2qÞ
ð4Þ
ða/t Þ12q /2t /2x 2
1 #2q2
:
ð5Þ
Furthermore, in any case other case where the ut2q1 term dominates the singularity analysis, we fall into this case. Case (ii): up term domination. It is necessary that p > 2q 1 for the up term to dominate the singularity analysis. Balancing utt ; uxx and bp up gives us
a¼
2 ; p1
"
2ð1 þ pÞ 2 /t /2x u0 ¼ 2 bp ð1 pÞ
ð6Þ 1 #p1
:
ð7Þ
From here it is clear that for case (i) we do not have integrability, since our value for a is both positive and not an integer. Thus we need only consider integrability for case (ii). We see immediately that a is only an integer for the cases where p is equal to 2 or 3. Now, since p > 2q 1 is a necessary yet not sufficient condition for the up term to dominate the singularity analysis, we need to ensure that the highest u2q1 term does not dominate. This means that we need t
pjaj > ð2q 1Þðjaj þ 1Þ ) 2p > ðp þ 1Þð2q 1Þ;
ð8Þ
so when p = 2, we can have q ¼ 0 or q ¼ 1, while when p = 3, we can have either q ¼ 0 or q ¼ 1 (note that, while p ¼ 3; q ¼ 2 satisfies p > 2q 1 we have from (8) 6 > 4ð3Þ ¼ 12, which is absurd). So, we have four possible integrable cases. In order to determine the integrability of the four cases, we proceed with the resonance analysis. Plugging u ¼ u0 /a þ m/rþa into the leading order terms and keeping the most singular and linear in m we have for case (ii):
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
h i m ðr þ aÞðr þ a 1Þð/2t /2x Þ bp up1 0 p ¼ 0;
ð9Þ
2ðp þ 1Þ : p1
) r ¼ 1;
1625
ð10Þ
The 1 resonance corresponds to the arbitrariness of the singular manifold whereas the other resonance corresponds the position in the series expansion of u where we should have arbitrary terms. For the remainder of the paper we will consider only case (ii) as case (i) is clearly never integrable. For case (ii) we determine whether (1) possesses the strong Painlevé Property for p = 2, 3. We consider u as an expansion of powers of /ðx; tÞ (following the notation of [18]): 1 X 2 /p1 ðx; tÞ u
j
j p1
j¼0
ðx; tÞ/p1 ðx; tÞ:
ð11Þ
Substituting (11) into (1) we have the following generalized recurrence relation
u
j 2p j 2p j 2p þ 1 2u j 1;t /t 2u j 1;x /x þ u j 1 ð/tt /xx Þ þ þ2 þ 1 ð/2t /2x Þu j p1 p1 p1 p1 p1 p1 p1 1km q 2Y i1 X X Xj 2p þ 4i 2 2 1 1 m ai ukj2pþ4i cm ukcmm u1k ukj2pþ4i g2i1 cm g2i1 g2i1 ;t t p 1 p 1 p1 p1 t m¼1 i¼1 k2K c2C
j 2;tt p1
u
i
i
p X
j2pþ2l p1
bl
l¼0
þ
j 2;xx p1
X
j2pþ2l ðb1 þþbl1 Þ p1
X
b1 ¼0
uj2pþ2lðb p1
bl ¼0
1 þbþbl1 Þ
ub1 ubl ;
where
K ¼ fk ¼ ðk1 ; . . . ; k2i Þ j km 2 f0; 1g for all m ¼ 1; 2; . . . ; 2ig;
C ¼ c ¼ ðc1 ; c2 ; . . . ; c2i1 Þ j 0 6 cm 6
gl k1 þ b þ k2i þ 2i þ
j 2p þ 4i gm1 for all m ¼ 1; 2; . . . ; 2i 1 ; p1
l X
cj :
j¼1
We need to verify the arbitrariness of the term at the resonance. Plugging in j ¼ 2ðp þ 1Þ gives us
" 2pðp þ 1Þ ðp 1Þ
ð/2t 2
#
/2x Þ
pbp up1 0
u2ðpþ1Þ ¼ 0: p1
Thus we have verified the arbitrariness at the resonance. Eq. (1) possesses the strong Painlevé property for the special cases p = 2 and p = 3, when p > 2ðq 1Þ. 3. Integrability and closed-form exact solutions of traveling wave type We now consider the two special cases where (1) satisfies the strong Painlevé property and find special solutions following a procedure used in [19]. Such a method was also employed in [16,17]. We begin by defining two new functions to be used extensively through the remainder of this paper:
C
/t ; /x
ð12Þ
V
/xx : /x
ð13Þ
We would like to insert (12) truncated at the constant term into (1) (to find our auto-Bäcklund transform equations) and match orders of /. Truncating the Laurent series at constant terms, we shall obtain exact solutions for our nonlinear hyperbolic equation (1) in each of these two interesting cases. For computational simplicity, we will place certain specific yet useful restrictions on C and V; under appropriate restrictions on C and V, and hence on /ðx; tÞ, we have integrable equations. 3.1. The case of p ¼ 2; q ¼ 1 When q ¼ 1 and p = 2, we have the nonlinear wave equation
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utt uxx a1 ut b1 u b2 u2 ¼ 0:
ð14Þ
Assuming a solution
uðx; tÞ ¼
u0 ðx; tÞ 2
/ðx; tÞ
þ
u1 ðx; tÞ þ u2 ðx; tÞ; /ðx; tÞ
ð15Þ
we find
u0 ¼
6 2 ð/ /2x Þ; b2 t
ð16Þ
u1 ¼
1
6 5/2t /tt þ 8/t /x /xt þ a1 /3t a1 /t /2x þ /tt /2x þ /xx /2t 5/xx /2x ; b2 ð/2t /2x Þ 5
ð17Þ
u2 ¼
1
/2x Þ3 a1 /3t /2x /tt
4a1 /4t /x /xt þ 64a1 /2t /3x /xt 32a1 /3t /2x /xx þ 58a1 /t /4x /tt þ 30a1 /t /4x /xx
50b2 ð/2t 88
600/3t /tt /x /xt þ 412/2t /2x /tt /xx 104/t /3x /tt /xt 104/3t /x /xx /xt 600/t /3x /xt /xx
þ a21 /6t þ 25b1 /6t 25b1 /6x þ 75/4t /2tt 49/4x /2tt 49/4t /2xx þ 75/4x /2xx þ 180/4x /2xt þ 60/5x /xtt 100/5x /xxx 100/5t /ttt þ 180/4t /2xt þ 60/5t /xxt 2a21 /4t /2x þ a21 /2t /4x 75b1 /4t /2x þ 75b1 /2t /4x þ 150/2t /2x /2tt 30/4t /tt /xx þ 344/2t /2x /2xt 30/4x /xx /tt þ 150/2t /2x /2xx þ 180/x /4t /xtt 60/x /4t /xxx 240/3x /2t /xtt 240/2x /3t /xxt þ 180/t /4x /xxt þ 160/3x /2t /xxx 60/t /4x /ttt þ 160/3t /2x /ttt þ 30a1 /5t /tt
þ 2a1 /5t /xx 60a1 /5x /xt :
ð18Þ
1
Due to the length of the equation for Oð/ Þ it will be omitted here. Following the procedure from the previous example we use (13) and (14) in (17)–(19) to get
u0 ¼
6/2x 2 ðC 1Þ; b2
ð19Þ
u1 ¼
h i 6/x ðb2 ðC 2 1ÞÞ1 5ðC 2 C t þ C 3 C x þ C 4 VÞ þ 9CC x þ 10C 2 V þ a1 C 3 a1 C þ C t 5V ; 5
ð20Þ
u2 ¼
h
1 2
3
50b2 ðC 1Þ
600ðC 3 C x V þ C 4 V 2 ÞðC t þ CC x þ C 2 VÞ þ 412ðC 2 C t V þ C 3 C x V þ C 4 V 2 Þ
104CðC x þ CVÞðC x þ CC x þ C 2 VÞ 104ðC 3 C x V þ C 4 V 2 Þ 600ðCC X V þ C 2 V 2 Þ 88a1 ðC 3 C t þ C 4 C x þ C 5 VÞ 4a1 ðC 4 C x þ C 5 VÞ þ 64a1 ðC 2 C x þ C 3 VÞ 32a1 C 3 V þ 58a1 ðCC t þ C 2 C x þ C 3 VÞ þ 30a1 CV þ a21 C 6 þ 25b1 C 6 25b1 þ 75C 4 ðC t þ CC x þ C 2 VÞ2 49ðC t þ CC x þ C 2 VÞ2 49C 4 V 2 þ 75V 2 þ 180ðC x þ CVÞ2 þ 60ðC xt þ C 2x þ 2CC x V þ C 2 V x þ C t V þ C 2 V 2 Þ 100ðV x þ V 2 Þ 100C 5 ðC tt þ 2C t C x þ 3CC t V þ CC xt þ C 2 V t þ C 2 C x V þ C 3 V 2 Þ þ 180C 4 ðC x þ CVÞ2 þ 60C 5 ðV t þ C x V þ CV 2 Þ 2a21 C 4 þ a21 C 2 75b1 C 4 þ 75b1 C 2 þ 150C 2 ðC t þ CC x þ C 2 VÞ2 30ðC 4 C t V þ C 5 C x V þ C 6 V 2 Þ þ 344C 2 ðC x þ CVÞ2 30ðC t V þ CC x V þ C 2 V 2 Þ þ 150C 2 V 2 þ 180ðC 4 C xt þ C 4 C t V þ C 5 V t þ C 4 C 2x þ 2C 5 C x V þ C 6 V 2 Þ 60ðC 4 V x þ C 4 V 2 Þ 240ðC 2 C xt þ C 2 C t V þ C 3 V t þ C 2 C 2x þ 2C 3 C x V þ C 4 V 2 Þ 240ðC 3 V t þ C 3 C x V þ C 4 V 2 Þ þ 180ðCV t þ CC x V þ C 2 V 2 Þ þ 160ðC 2 V x þ C 2 V 2 Þ 60ðCC tt þ 2CC t C x þ 3C 2 C t V þ C 2 C xt þ C 2 C 2x þ 2C 3 C x V þ C 3 V t þ C 4 V 2 Þ þ 160ðC 3 C tt þ 2C 3 C t C x þ 3C 4 C t V þ C 4 C xt
i þ C 4 C 2x þ 2C 5 C x V þ C 5 V t þ C 6 V 2 Þ þ 30a1 ðC 5 C t þ C 6 C x þ C 7 VÞ þ 2a1 C 5 V 60a1 ðC x þ CVÞ :
ð21Þ
Forcing C and V to be constants drastically reduces the complexity of Eqs. (21) and (22), as well as the complexity of the equation for Oð/1 Þ. Under such a restriction, we find
u1 ¼
u2 ¼
i 6/x h 5VðC 2 1Þ þ a1 C ; 5b2 1 2
50b2 ðC 1Þ
h
i 30a1 CVðC 2 1Þ 25V 2 ðC 2 1Þ2 þ a21 C 2 þ 25b1 ðC 2 1Þ ;
ð22Þ
ð23Þ
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
Oð/1 Þ ¼
h i 6/x ðC 2 1Þ4 a31 C 3 ðC 2 1Þ3 25a1 CV 2 ðC 2 1Þ5 ; 125b2
a1 C 1
V ¼
5ðC 21 1Þ
1627
ð24Þ
ð25Þ
;
( ) 5ðC 21 1Þ a1 C 1 ðx þ C 1 tÞ þ C 2 ; exp /ðx; tÞ ¼ a1 C 1 5ðC 21 1Þ
ð26Þ
n o n o 2a1 C 1 a1 C 1 6a1 C 1 exp 5ðC ðx þ C tÞ 6 exp ðx þ C tÞ ð 1Þa1 C 1 1 1 2 2 5ðC 1 1Þ 1 1Þ n o uðx; tÞ ¼ n o 2 þ 2 2 5ðC 1Þ a C 5ðC 1Þ 1 1 5b2 a11C 1 exp 5ðC12 1Þ ðx þ C 1 tÞ þ C 2 5b2 a11C1 exp 5ðCa12C1Þ ðx þ C 1 tÞ þ C 2 1
h
1 50b2 ðC 21
1Þ
1
a21 C 21
6
i a21 C 21 þ a21 C 21 þ 25b1 ðC 21 1Þ :
ð27Þ
3.2. The case of p ¼ 2; q ¼ 0 The case of p = 2 and q ¼ 0 proceeds similar to the p ¼ 2; q ¼ 1 case. The form of (15) is unchanged, and we find that u0 ; u1 and u2 are determined uniquely by the Oð/4 Þ; Oð/3 Þ and Oð/1 Þ conditions, respectively. Note that taking
/ðx; tÞ ¼ exp ðVðx þ C 1 tÞÞ þ C 2
ð28Þ 2
0
for constant V; C 1 and C 2 identically satisfies the Oð/ Þ condition. What is left is the Oð/ Þ condition, and this reduces to
V 2 ðC 21 1Þ b1 V 2 ðC 21 1Þ þ b1 ¼ 0 ) V 2 ðC 21 1Þ ¼ b1 :
ð29Þ
So, there are two possibilities. When jC 1 j < 1, we take the þb1 root, obtaining
V1 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 C 21 1
ð30Þ
:
Otherwise, when jC 1 j > 1, we take the b1 root, obtaining
V2 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 1 C 21
ð31Þ
:
Recalling the solution (15), when jC 1 j < 1 we have the exact solution
uðx; tÞ ¼
6b1 C 2 eV 1 ðxþC 1 tÞ b2 ðeV 1 ðxþC1 tÞ þ C 2 Þ
2
;
ð32Þ
while when jC 1 j > 1 we obtain the exact solution
6b1 C 2 eV 2 ðxþC1 tÞ
b1 : b2
ð33Þ
utt uxx b1 u b2 u2 b3 u3 ¼ 0:
ð34Þ
uðx; tÞ ¼
2
b2 ðeV 2 ðxþC 1 tÞ þ C 2 Þ
3.3. The case of p ¼ 3; q ¼ 0 When p = 3 and q ¼ 0 we have
Assuming a solution of the form
uðx; tÞ ¼
u0 ðx; tÞ þ u1 ðx; tÞ; /ðx; tÞ
ð35Þ
we have
u0 ¼
1 2 2 2 /t /2x ; b3
ð36Þ
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M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
pffiffiffi pffiffiffi pffiffiffi u1 ¼ 4 2/t /x /xt 3/2x /2t 2/xx 3/2t /2x 2/tt ;
ð37Þ
1 3 1 6b23 /2t /2x 2 ;
ð38Þ
3 1 2b2 b3 2 /2t /2x 2
Oð/1 Þ : u0;tt u0;xx b1 u0 2b2 u0 u1 3b3 u0 u21 :
ð39Þ
Using these conditions, and letting C and V be constant, we obtain
u0 ¼
u1 ¼
sffiffiffiffiffi 1 2 / ðC 2 1Þ2 ; b3 x
ð40Þ
pffiffiffi 32 1 32 1 pffiffiffi pffiffiffi 1 6b23 C 2 1 4 2C 2 V 2 3 C 2 V 2 3C 2 1 C 2 V 2b2 b3 2 C 2 1 ;
pffiffiffi /x 2 2 ffi ð6b1 b3 2b2 ÞðC 2 1Þ þ 3b3 ðC 2 1Þ2 V 2 : Oð/1 Þ : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 6 b3 ðC 1Þ
ð41Þ
ð42Þ
Once again we let C ¼ C 1 , where C 1 is a constant, and find for V
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u2b 6b1 b3 ; V ¼ t 2 2 3b3 ðC 1 1Þ
ð43Þ
which gives us
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 v 9 u < u = u2b2 6b1 b3 u 3b3 ðC 2 1Þ 2 1 t t exp ðx þ C 1 tÞ þ C 2 ; /ðx; tÞ ¼ 2 2 : ; 3b3 ðC 1 1Þ 2b2 6b1 b3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 v 9 u < u = u2b2 6b1 b3 u 3b3 ðC 2 1Þ 2 1 t t exp ðx þ C /ðx; tÞ ¼ tÞ þ C2; 1 2 : ; 3b3 ðC 21 1Þ 2b2 6b1 b3
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b2 6b1 b3 ðx þ C 1 tÞ 3b2 ðC 2 1Þ 3 1 uðx; tÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3b3 ðC 21 1Þ 2b22 6b1 b3 tÞ þ C2 exp ðx þ C 1 2 2 3b3 ðC 1 1Þ 2b2 6b1 b3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 u 2 u 32 u u b 3b1 b3 u b2 3b1 b3 u b2 3b1 b3 1 2 2 2 2 2 2 3 C1 t 2 2 2 3C 1 1 C 1 t 2 2 2b2 b3 2 C 1 1 A @8C 1 t 2 2 3b3 ðC 1 1Þ 3b3 ðC 1 1Þ 3b3 ðC 1 1Þ 32 1 1 : 6b23 C 21 1
ð44Þ
ð44Þ
qffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
2ðC 1 1Þ exp b3
ð45Þ
3.4. The case of p ¼ 3; q ¼ 1 The case of p = 3 and q ¼ 1 proceeds similar to the p ¼ 3; q ¼ 0 case. We take a solution of the form (35), and we find that u0 and u1 are uniquely determined by the Oð/3 Þ and Oð/2 Þ conditions, respectively. Upon taking
/ðx; tÞ ¼ exp ðVðx þ C 1 tÞÞ þ C 2
ð46Þ 1
for constant V; C 1 and C 2 identically satisfies the Oð/ Þ condition provided that 2
2
3b3 ðC 21 1Þ2 V 2 ½b1 b3 2b2 þ ð2b2 ð6b1 þ a21 Þb3 ÞC 21 ¼ 0;
ð47Þ
so we take
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ub1 b3 2b2 þ ð2b2 ð6b1 þ a2 Þb3 ÞC 2 2 2 1 1 : V ¼t 3b3 ðC 21 1Þ2 With this choice of V, we finally have from the Oð/0 Þ condition that
ð48Þ
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
pffiffiffiffiffi pffiffiffi 2 3 12 2 b3 b2 2 ðb2 4b1 b3 Þa1 b1 C 1 ðC 21 1Þ þ ð4b2 18b1 b3 ÞðC 21 1Þ3=2 2a31 b3 C 1 5 5 pffiffiffiffiffiffiffiffi 3=2 3 6b3 2 2 a1 b3 C 21 2ðb2 3b1 b3 ÞðC 21 1Þ 5 ¼ 0;
1629
ð49Þ
a nonlinear relation for C 1 . When a real-valued solution exists for C 1 , then we have a solution of the form (35). Note that this requirement is distinct from what we have seen for the previous cases considered above. Here, in order to have a solution (35), we need to restrict the wave speed to only values satisfying the condition (49). Let C 1 denote a solution to (49) (we omit a general solution form here, as it is too complicated for arbitrary parameter values). Then we find that the solution (35) becomes
pffiffiffi qffiffiffiffiffiffiffi pffiffiffi VðxþC 1 tÞ 2V C 2 b2 C2 2a1 C 1 1 e pffiffiffiffiffi uðx; tÞ ¼ pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ : VðxþC 1 tÞ 3b3 2 e þ C2 2 b 3 6 b3 C 1 1
ð50Þ
4. General method for travelling wave solutions In the previous section, we showed that one may obtain closed-form exact solutions by use of truncated Painlevé expansions. Let us now consider travelling wave solutions under a far more general framework. To this end, let us assume a travelling wave solution to (1) of the form uðx; tÞ ¼ wðzÞ, where x ct and c 2 R is the wave speed. Without making this assumption a priori, we still recovered solutions of the form in Section 3, lending some validity to the present approach. In fact, most known solutions of NLPDEs tend to be traveling waves, although some, such as ‘boomerons’, have non-constant wave speeds and this are not traveling waves. Under such a travelling wave assumption, (1) becomes q p X X 2i1 ð1 c2 Þw00 þ ai c2i1 w0 þ bj wj ¼ 0: i¼1
ð51Þ
j¼1
We first consider two special cases before addressing the general case. 4.1. Degenerate cases When c ¼ 1, the equation becomes degenerate:
sgnðcÞ
q p X X 2i1 ai w0 þ bj wj ¼ 0: i¼1
ð52Þ
j¼1
This case is well-studied in the literature, however we make a few remarks since it is an exact reduction of the more general equation we are concerned with. Eq. (52) essentially defines a polynomial in w0 of degree 2q 1. Assume – 0, and let Pp 1 j Mk ða1 ; . . . ; a2q1 ; sgnðcÞ j¼1 bj w Þ denote the kth solution to the polynomial equation q X ai M2i1 þ i¼1
X 1 b wj ¼ 0: sgnðcÞ j¼1 j p
ð53Þ
Then, for (52) with real coefficients, there exists a real-valued solution satisfying the first order nonlinear differential equation
X 1 b wj þ kÞ sgnðcÞ j¼1 j p
w0k ¼ M k ða1 ; . . . ; a2q1 ;
ð54Þ
for k ¼ 1; 2; . . . ; 2q 1, provided M k is real-valued. For instance, in the case where q ¼ 1,
X 1 bj wj : a1 sgnðcÞ j¼1 p
w0 ¼
ð55Þ
Separating variables, we see that a solution to this nonlinear ODE then must satisfy the implicit integral relation
Z
w
wð0Þ
dn z ¼ : Pp j a sgnðcÞ 1 b n j j¼1
When p = 1, the exact solution is given in closed form by
ð56Þ
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wðzÞ ¼ wð0Þ exp
b1 z : a1 sgnðcÞ
ð57Þ
When p = 2, the exact solution is given by
wðzÞ ¼
wð0Þb1 : 1z ðwð0Þb2 þ b1 Þ exp a1 bsgnðcÞ wð0Þb2
ð58Þ
Note that p ¼ 1; 2 constitute the cases in which we find exact solutions in terms of exponential functions. When p P 3, this is possible in some special cases, but in general the implicit relation (56) will result in a linear combination of logarithm and inverse tangent functions, the inversion of which will strongly depend on the specific values of the bk ’s. For the general p = 3 case, we obtain
0 1 1 ln ðwÞ ln b1 þ b2 w þ b3 w2 b1 2b1
1
b2 z0 z B b2 þ b3 w C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA ¼ ; a1 sgnðcÞ 2 2 b1 4b1 b3 b2 4b1 b3 b2
ð59Þ
where z0 is a constant of integration. However, consider the case where b2 ¼ 0; we then may solve the relation for w to obtain the two solution branches
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2b1 ðz0 zÞ 1 ðz0 zÞ exp b1 exp 2b 1 b 3 a1 sgnðcÞ a1 sgnðcÞ wðzÞ ¼ : 2b1 ðz0 zÞ 1 b3 exp a1 sgnðcÞ
ð60Þ
More generally, let p P 3 be arbitrary, take b2 ¼ b3 ¼ ¼ bp1 ¼ 0, and let b1 and bp be arbitrary. Then, we have exact solutions
2
b1 exp
ðp1Þb1 ðz0 zÞ a1 sgnðcÞ
31=ðp1Þ
5 wðzÞ ¼ m‘ 4 1 ðz0 zÞ 1 bk exp ðp1Þb a1 sgnðcÞ
ð61Þ
;
where ‘ ¼ 1; 2; . . . ; p and m‘ is a pth root of unity. We should mention that in the case where ¼ 0, there are only constant solutions. Such solutions must satisfy the polynomial equation p X bj wj ¼ 0:
ð62Þ
j¼1
In the case where c = 0, we have stationary solutions. Such solutions necessarily satisfy the ODE
w00 þ
p X bj wj ¼ 0:
ð63Þ
j¼1
Again, such equations are well-studied in the literature. We shall say a few things about it, since it provides some motivation for later solutions to the more general case problem. When p = 1, the solution is trivial. When p = 2, one can find wðzÞ in terms of Weierstrass’s elliptic functions, while when p = 3 one can find wðzÞ in terms of Jacobi elliptic functions (under some appropriate restrictions on the bk ’s). 4.2. Constructing rational travelling wave solutions in exponentials For the integrable families of equations discovered in Section 2, we were able to provide exact solutions in terms of ratios of exponential functions in Section 3. As we have shown so far in Section 4, exact solutions may be found even in non-integrable cases. In the present section, we shall assume travelling wave solutions, and obtain said solutions in a manner different from that of Section 3. In particular, while we have obtained some particular solutions in Section 3, it is not clear that those were the only travelling wave solutions, since we made an assumption on the form of /ðx; tÞ. We shall focus on solutions which are functions of exponential factors of the form eaðzz0 Þ [2–8]. To simplify the process, let us make the change of independent variable Z ¼ eaðzz0 Þ where a and z0 are constants, and change of dependent variable WðZÞ ¼ wðzÞ. Then (51) becomes
a2 ð1 c2 ÞðZ 2 W 00 þ ZW 0 Þ þ
q p X X 2i1 ai c2i1 a2i1 Z 2i1 W 0 þ bj W j ¼ 0: i¼1
ð64Þ
j¼1
Then, in the new variable Z, solutions we seek take the form of rational functions in Z. To this end, let us assume a solution of the form
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
WðZÞ ¼
S0 þ S1 Z þ þ Sm Z m SðZÞ ¼ ; T 0 þ T 1 Z þ þ T n Z n TðZÞ
1631
ð65Þ
degðSÞ ¼ m P 0; degðTÞ ¼ n P 0. Eq. (64) then becomes
h
a2 ð1 c2 Þ Z 2 S00 T 2 2S0 TT 0 STT 00 þ 2ST 0 2 þ Z S0 T 2 STT 0
i
þ
q X 2i1 54i ai c2i1 a2i1 Z 2i1 S0 T ST 0 T i¼1
p X þ bj Sj T 3j ¼ 0:
ð66Þ
j¼1
We may always rearrange the factors of TðZÞ to put the left hand side of (66) into the form ðTðZÞÞv QðZÞ, where v P 0 is an integer power. As T is not identically zero by construction, we must then have that QðZÞ 0 if WðZÞ ¼ SðZÞ=TðZÞ is in fact a solution. Let Q denote the polynomial of minimal degree satisfying this condition. Since S and T are of finite degree, such a polynomial exists; let us then take degðQÞ ¼ jðm; nÞ. Note that the degree j depends on both m and n. Note that since degðSÞ ¼ m P 0; degðTÞ ¼ n P 0, and a was arbitrary yet non-zero, we have m þ n þ 3 free parameters. The coefficients of Q will depends on these m þ n þ 3 parameters. In particular,
QðZÞ ¼ Q0 ðS0 ; . . . Sm ; T 0 ; . . . ; T n ; aÞ þ þ Qj ðS0 ; . . . Sm ; T 0 ; . . . ; T n ; aÞZ j :
ð67Þ
jðm; nÞ 6 m þ n þ 2, then we have enough free parameters to force the coefficients of Q to be zero. Meanwhile, if jðm; nÞ > m þ n þ 2, we in general do not have enough free parameters to choose the coefficients of Q to be zero. An excepIf
tion to this latter case occurs is there are serendipitous cancellations leading to QðZÞ 0. Such cancellations would depend on fixing very specific values of the model parameters in the original equation (1), and as such much generality would be lost in making such assumptions. We can proceed no further with such generality, so we shall illustrate these ideas through a special case which holds the solutions of Section 3 as particular cases. 4.3. Possible rational solutions for the restriction p ¼ 3; q ¼ 1 In order to proceed further, let us take q ¼ 1 and p = 3. Under such an assumption, (66) becomes
a2 ð1 c2 ÞZ 2 S00 T 2 2S0 TT 0 STT 00 þ 2ST 0 2 þ a2 ð1 c2 Þ þ aa1 c Z S0 T 2 STT 0 þ b1 ST 2 þ b2 S2 T þ b3 S3 ¼ 0: 3
ð68Þ
2
We see that the highest order term will be either degðS Þ ¼ 3m or degðST Þ ¼ m þ 2n. Then,
jðm; nÞ ¼
3m
if m P n;
m þ 2n if m < n:
ð69Þ
Thus, by specifying the highest order nonlinearities p and q, we have determined jðm; nÞ. As previously discussed, we require jðm; nÞ 6 m þ n þ 2. We may then verify that when p = 3 and q ¼ 1, we have the possibilities ðm; nÞ 2 fð0; 1Þ; ð0; 2Þ; ð1; 0Þ; ð1; 1Þ; ð1; 2Þ; ð2; 2Þg. We then have six possible rational solutions. Mapping back to the original form of the solution (in variable z versus variable Z), we have the general rational solutions
wðzÞ ¼
S0 ; T 0 þ T 1 eaðzz0 Þ
ð70Þ
wðzÞ ¼
S0 ; T 0 þ T 1 eaðzz0 Þ þ T 2 e2aðzz0 Þ
ð71Þ
wðzÞ ¼ S0 þ S1 eaðzz0 Þ ;
ð72Þ
wðzÞ ¼
S0 þ S1 eaðzz0 Þ ; T 0 þ T 1 eaðzz0 Þ
ð73Þ
wðzÞ ¼
S0 þ S1 eaðzz0 Þ ; T 0 þ T 1 eaðzz0 Þ þ T 2 e2aðzz0 Þ
ð74Þ
wðzÞ ¼
S0 þ S1 eaðzz0 Þ þ S2 e2aðzz0 Þ ; T 0 þ T 1 eaðzz0 Þ þ T 2 e2aðzz0 Þ
ð75Þ
where the coefficients and the exponential growth/decay rate, a, depend on the original model parameters. Since the coefficients depend on the model parameters, there may exist model parameters for which some of the coefficients become singular, for one or more other six solutions. Thus, these are the admissible solutions, and for some specific values of the parameters a1 ; b1 ; b2 ; b3 ; c and , such solutions may break down. It is therefore important to test whether for given specific values of a1 ; b1 ; b2 ; b3 ; c and , an obtained function is really a solution. The primary danger is that, for certain parameter val-
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ues, solutions can become singular and hence non-physical. Thus, we must proceed with caution. See also the comments of Kudryashov [20,21] on such issues. Furthermore, these solutions are the only possible solutions; if either S or T is of degree three or greater, the rational function (65) fails to be a solution. 4.4. Explicit solutions for the case p ¼ 3; q ¼ 1 In order to demonstrate the method, let us pick a concrete example for which we may compute solutions exactly. Since we have already found solutions for the ðp; qÞ ¼ ð3; 0Þ and ðp; qÞ ¼ ð2; 1Þ cases in Section 3, let us consider the non-integrable case of ðp; qÞ ¼ ð3; 1Þ. To this end, we set SðZÞ ¼ S0 and TðZÞ ¼ T 0 þ T 1 Z þ T 2 Z 2 . From (68), we arrive at the following conditions:
Oð1Þ : b1 T 20 þ b2 S0 T 0 þ b3 S20 ¼ 0;
ð76Þ
OðZÞ : ð2b1 að1 c2 Þ aa1 cÞT 0 þ b2 S0 ¼ 0;
ð77Þ
OðZ 2 Þ : að1 c2 Þð2S0 T 21 2S0 T 0 T 2 Þ ðað1 c2 Þ þ aa1 cÞð2S0 T 0 T 2 þ S0 T 21 Þ þ b1 S0 ð2T 0 T 2 þ T 21 Þ þ b2 S20 T 2 ¼ 0;
ð78Þ
OðZ 3 Þ : 6ð1 c2 Þa2 þ 3ð1 c2 þ a1 cÞa 2b1 ¼ 0;
ð79Þ
OðZ 4 Þ : 6ð1 c2 Þa2 þ 2ð1 c2 þ a1 cÞa b1 ¼ 0:
ð80Þ
3
4
The OðZ Þ and OðZ Þ conditions imply the parameter constraints
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 and a ¼ : 6ðc2 1Þ
1 a1 ¼ c c
ð81Þ
The OðZÞ condition then implies 2b1 T 0 þ b2 S0 ¼ 0, i.e.
S0 ¼
2b1 T 0: b2
ð82Þ 2
From here, the Oð1Þ condition implies 4b1 b3 b2 ¼ 0. Let us write this as 2
b3 ¼
b2 : 4b1
ð83Þ
Finally, the OðZ 2 Þ condition implies 2
b1 T 0 T 1 T 2 ¼ 0; b2
ð84Þ
yet T 0 – 0 and b1 – 0, so we need one of T 1 ¼ 0 or T 2 ¼ 0. If T 1 ¼ 0, we obtain the solution
WðZÞ ¼
2b1 T 0
ð85Þ
b2 ðT 0 þ T 2 Z 2 Þ
or, converting back to z,
2b1 wðzÞ ¼ b2 where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)1 b1 1 þ s2 exp 2 ; z 6ðc2 1Þ
(
ð86Þ
s2 ¼ T 2 =T 0 . Meanwhile, if T 2 ¼ 0, we obtain the solution WðZÞ ¼
2b1 T 0 b2 ðT 0 þ T 1 ZÞ
ð87Þ
which is equivalent to
2b1 wðzÞ ¼ b2 where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)1 b1 1 þ s1 exp ; z 6ðc2 1Þ
(
ð88Þ
s1 ¼ T 1 =T 0 . Hence, both (86) and (88) are solutions to the nonlinear PDE 2
utt uxx ¼ aut þ b1 þ b2 u2 þ
b2 3 u ; 4b1
ð89Þ
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
1633
where the wave speed c is related to a by a ¼ c 1c . (For all a 2 R, there exists c > 0 such that this relation holds, so it is completely arbitrary.) Each of these solutions has one free parameter, sk . (The parameter c and the bk ’s are fixed by the coefficients to the NLPDE.) Hence, we have constructed a one-parameter family of solutions to (89) subject to the constraint a ¼ c 1c . Next, let us consider the case SðZÞ ¼ S0 þ S1 Z; TðZÞ ¼ T 0 þ T 1 Z. We find that upon taking
S0 ¼
b2 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4b1 b3 2b3
T0;
b2
S1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4b1 b3 2b3
T1;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b2 4b1 b3 b2 b2 4b1 b3 1 a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 2 ð1 c2 Þb3 2b1 b3 b2 þ b2 4b1 b3
ð90Þ
ð91Þ
we have that (68) implies
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4b1 b3 T 0 T 1 Z b2 WðZÞ ¼ 2b3 T 0 þ T 1 Z 2b3
ð92Þ
is a solution to the NLPDE
utt uxx ¼ aut þ b1 u þ b2 u2 þ b3 u3 ;
ð93Þ
provided that a and the wave-speed c satisfy the nonlinear relation
1 1 a¼c þ c c
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 ð1 c2 Þb3 2b1 b3 b2 þ b2 4b1 b3 :
ð94Þ
In terms of z, we have the solution
wðzÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4b1 b3 T 0 T 1 eaðzz0 Þ T 0 þ T 1 eaðzz0 Þ
2b3
b2 : 2b3
ð95Þ
Note that T 0 and T 1 are free parameters. Hence, we have constructed a two-parameter family of solutions to the NLPDE (93) subject to the constraint (94). It is worth mentioning that, for fixed parameters a; b1 ; b2 ; b3 , we have that the wave number is determined by the parameters, due to (94). Indeed, for any set of parameter values, we must verify such a c exists. From (94), we find that the existence of c is equivalent to the existence of a real-valued solution to the quartic equation
0
2
c þ 2ac þ @2 a 4
3
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 b2 4b1 b3 2 2b1 b3 b2 þ b2 4b1 b3 Ac 2ac þ 1¼0 b3 b3
2b1 b3 b2 þ
ð96Þ
for c. Observe that, although we have used a different and more direct method in Section 4, the solution (95) is of the same form as the solution (50) obtained in Section 3. So, the two methods are consistent. 5. Conclusions In this paper, we first determine values of p and q which permit the strong Painlevé property of (1). We find that general forms of Eq. (1) corresponding to (i) (ii) (iii) (iv)
p ¼ 2; p ¼ 2; p ¼ 3; p ¼ 3;
q ¼ 1; q ¼ 0; q ¼ 0; q ¼ 1.
For each of the cases exhibiting the strong Painlevé property, we obtain exact solutions using the relevant auto-Bäcklund transformations and truncated Laurent series. These solutions are all travelling waves. In particular, we find that solutions are rational functions of exponentials. Using the appearance of travelling wave solutions for the integrable cases as a clue, we consider a general travelling wave transformation, which maps our nonlinear PDE (1) into an ODE (51) which governs travelling wave solutions for arbitrary p and q. We have three degenerate cases, when c ¼ 1; 0; 1. The two cases c ¼ 1 result in a reduction of order (we obtain a first order strongly nonlinear, as opposed to a second order weakly nonlinear, equation). For some special cases, solutions to
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M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
this degenerate case are provided. The second type of degeneracy, c ¼ 0, corresponds to stationary solutions. Solutions to the c ¼ 0 ODE are always first integrable. When p = 2 and p = 3, the stationary solutions take the form of elliptic functions. However, in order to recover the solutions which are rational functions of exponentials, an alternate form of the travelling wave ODE is useful. Under a change of variables Z ¼ exp ðaðz z0 ÞÞ (where z ¼ x ct is the wave variable) we transform the travelling wave ODE into a new ODE, the rational solutions of which yield rational exponential solutions of the original travelling wave ODE. We arrive at a condition on the order ½m; n of any rational solution of this type. Using the general equation with p = 3 and q ¼ 1 as an example, we obtain all possible rational exponential solutions to this equation. 5.1. Acknowledgments The authors appreciate the helpful comments of the reviewers, which have greatly improved the clarity of the paper. R.A.V. was supported in part by NSF Grant # 1144246. References [1] Dodd RK, Eilbeck JC, Gibbon JD, Morris HC. Solitons and nonlinear wave equations. London: Academic Press; 1982. [2] Wazwaz AM. The extended tanh method for new soliton solutions for many forms of the fifth-order KdV equations. Appl Math Comput 2007;184:1002–14. [3] Wazwaz AM. Hirota’s direct method and the tanh–coth method for multiple-soliton solutions of the Sawada–Kotera–Iot seventh-order equation. Appl Math Comput 2008;199:133–8. [4] Hirota R. The direct method in soliton theory. Cambridge: Cambridge University Press; 2001. [5] Caudrey PJ. Memories of Hirota’s method. Philos Trans R Soc A 2011;28:1215–27. [6] Conte R, Musette M. The Painlevé handbook. Berlin: Springer; 2008. [7] Conte R. Exact solutions of NLPDEs by singularity analysis. Lect Notes Phys 2003;632:1–83. [8] Tian SF, Zhang HQ. On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation. J Phys A Math Gen 2012;45:055203 (pp. 29). [9] Gao YT, Tian B. Some two-dimensional and non-traveling-wave observable effects of the shallow-water waves. Phys Lett A 2002;301:74–82. [10] Karaca MA, Hizel E. Similarity reductions of BBM equations. Appl Math Sci 2008;2:463–9. [11] Fang JP, Li JB, Zheng c L, Ren QB. Special conditional similarity reductions and exact solutions of the (2+1)-dimensional VCBKK system. Chaos Solitons Fractals 2008;35:530–5. [12] Weiss J, Tabor M, Carnevale G. The Painlevé property and singularity analysis of integrable and nonintegrable systems. J Math Phys 1983;24:522. [13] Weiss J. The Painlevé property for partial differential equations II: Bäcklund transformations, Lax pairs, and the Schwarzian derivative. J Math Phys 1983;24:1405. [14] Weiss J. On classes of integrable systems and the Painlevé property. J Math Phys 1984;25:13. [15] Hearns J, Van Gorder RA, Choudhury SR. Painlevé test, integrability, and exact solutions for density-dependent reaction–diffusion equations with polynomial reaction function. Appl Math Comput 2012;219:3055–64. [16] Kudryashov NA. Exact soliton solutions of the generalized evolution equation of wave dynamics. PMM J Appl Math Mech 1988;52:361–5. [17] Kudryashov NA. Exact solutions of the generalized Kuramoto–Sivashinsky equation. Phys Lett A 1990;147:287–91. [18] Gilson C, Pickering A. Factorization and Painleve analysis of a class of nonlinear third-order partial differential equations. J Phys A Math Gen 1995;28:2871–88. [19] Roy Choudhury S. A unified approach to integrable systems via Painlevé analysis. Contemp Math 2002;301:139–61. [20] Kudryashov NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simul 2009;14:3507–29. [21] Kudryashov NA. On new travelling wave solutions of the KdV and the KdV–Burgers equations. Commun Nonlinear Sci Numer Simul 2009;14:1891–900.
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Painlevé property and exact solutions for a nonlinear wave equation with generalized power-law nonlinearities Matthew Russo, Robert A. Van Gorder ⇑, S. Roy Choudhury Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 3 October 2012 Received in revised form 17 November 2012 Accepted 25 November 2012 Available online 3 December 2012 Keywords: Painlevé analysis Nonlinear hyperbolic PDE Power-law nonlinearity Exact solutions
By employing a variety of techniques, we investigate several classes of solutions of a family of nonlinear partial differential equations (NLPDEs) with generalized nonlinearities, special cases of which include the Klein–Gordon equation, the Landau–Ginzburg–Higgs equation, the u4 and u6 equations, the Rayleigh wave equation. The Painlevé property for our class of equations is studied first, showing that there are integrable families of such equations satisfying the strong Painlevé property (under a traveling wave assumption). From the truncated Laurent expansions, we introduce the auto-Bäcklund transformation for the two families shown to admit the strong Painlevé property. A multi-parameter family of exact solutions is then constructed from these auto-Bäcklund transformations for each of the cases, leading to travelling wave solutions. From here, assuming only travelling wave solutions, we then discuss more general methods of obtaining travelling wave solutions for those cases which do not satisfy the strong Painlevé property. Such solutions constitute rare exact solutions to a complicated nonlinear partial differential equation. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In the present paper, we shall be concerned with the nonlinear hyperbolic equation partial differential equation (PDE) q p X X utt uxx ¼ ai ut2i1 þ bk uk : i¼1
ð1Þ
k¼1
When the first sum in (1) is removed, we have a nonlinear (provided p P 2) Klein–Gordon equation. More particularly, two similar cases of this equation considered are known as the u42 equation and Landau–Ginzburg–Higgs equation [1], both of which occur when p = 3 and q ¼ 0. When p ¼ 5 and q ¼ 0, we recover the u6 equation [1]. Meanwhile, physically relevant solutions still occur when at least one ai – 0. For example, when q ¼ 2; p ¼ 0, we recover the Rayleigh wave equation. There are many methods which have been used over the past decades to find various types of exact analytic solutions of nonlinear partial differential equations (NLPDEs). Among them are (we included only some recent references): a. b. c. d. e.
the extended tanh–coth method [2,3]; Hirota’s method [4,5]; Painlevé series systematically truncated for ‘partial integrability’ [6,7]; invariant Painlevé analysis [6]; Bell Polynomial analyses [8];
⇑ Corresponding author. E-mail address: [email protected] (R.A. Van Gorder). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.11.019
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f. series solutions based on appropriate basis functions [9]; g. similarity reduction methods [10,11]. In this paper, we shall employ a potpourri of methods, Painlevé techniques and Bell Polynomial–Hirota-type exponential polynomials to analyze a large variety of solutions of (1). The paper consists of two main parts. In the first, we first show that the Painlevé property holds for certain subclasses of (1). In particular, two important subclasses satisfy the strong Painlevé property, whereas an infinite number of classes satisfy the weak Painlevé property. From the truncated Laurent expansions, we introduce the auto-Bäcklund transformation for the two families shown to admit the strong Painlevé property. A multiparameter family of exact solutions is then constructed from these auto-Bäcklund transformations for each of the two cases, leading to travelling wave solutions. In the second part of the paper, we consider a general travelling wave transformation, which maps our nonlinear PDE (1) into an ordinary differential equation (ODE) governing all possible travelling wave solutions, for arbitrary non-negative integer values of p and q. In order to recover the solutions which are rational functions of exponentials, an alternate form of this equation is obtained. The solutions obtained here through each of the two methods constitute rare exact solutions to a complicated nonlinear partial differential equation. 2. Painlevé property for several cases of p and q We wish to find the values of p and q for which (1) admits the Painlevé property. Recall that an equation has the strong Painlevé property if it has no movable critical points (for instance, branch points or essential singularities) and it is solution in a neighborhood of a singularity /0 scales as ð/ /0 Þa for positive integer a. For an overview of the method, see [12–14]; for a recent application, see [15]. Assume that / depends on both x and t, and scale / so that /0 ¼ 0. Applying the singular manifold method about the singular manifold
/ðx; tÞ ¼ 0;
ð2Þ
we use a leading order analysis with the substitution
u ¼ u0 /a
ð3Þ
and balance the highest order nonlinearities with the highest order derivatives. Substitution of (3) into (1) gives us two possible values for a: term domination. Case (i): u2p1 t If p 6 2q 1, the ut2q1 term dominates. Balancing utt ; uxx and aq u2q1 gives us t
a¼
3 2q ; 2 2q "
u0 ¼
3 2q
a2q1 ð2 2qÞ
ð4Þ
ða/t Þ12q /2t /2x 2
1 #2q2
:
ð5Þ
Furthermore, in any case other case where the ut2q1 term dominates the singularity analysis, we fall into this case. Case (ii): up term domination. It is necessary that p > 2q 1 for the up term to dominate the singularity analysis. Balancing utt ; uxx and bp up gives us
a¼
2 ; p1
"
2ð1 þ pÞ 2 /t /2x u0 ¼ 2 bp ð1 pÞ
ð6Þ 1 #p1
:
ð7Þ
From here it is clear that for case (i) we do not have integrability, since our value for a is both positive and not an integer. Thus we need only consider integrability for case (ii). We see immediately that a is only an integer for the cases where p is equal to 2 or 3. Now, since p > 2q 1 is a necessary yet not sufficient condition for the up term to dominate the singularity analysis, we need to ensure that the highest u2q1 term does not dominate. This means that we need t
pjaj > ð2q 1Þðjaj þ 1Þ ) 2p > ðp þ 1Þð2q 1Þ;
ð8Þ
so when p = 2, we can have q ¼ 0 or q ¼ 1, while when p = 3, we can have either q ¼ 0 or q ¼ 1 (note that, while p ¼ 3; q ¼ 2 satisfies p > 2q 1 we have from (8) 6 > 4ð3Þ ¼ 12, which is absurd). So, we have four possible integrable cases. In order to determine the integrability of the four cases, we proceed with the resonance analysis. Plugging u ¼ u0 /a þ m/rþa into the leading order terms and keeping the most singular and linear in m we have for case (ii):
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
h i m ðr þ aÞðr þ a 1Þð/2t /2x Þ bp up1 0 p ¼ 0;
ð9Þ
2ðp þ 1Þ : p1
) r ¼ 1;
1625
ð10Þ
The 1 resonance corresponds to the arbitrariness of the singular manifold whereas the other resonance corresponds the position in the series expansion of u where we should have arbitrary terms. For the remainder of the paper we will consider only case (ii) as case (i) is clearly never integrable. For case (ii) we determine whether (1) possesses the strong Painlevé Property for p = 2, 3. We consider u as an expansion of powers of /ðx; tÞ (following the notation of [18]): 1 X 2 /p1 ðx; tÞ u
j
j p1
j¼0
ðx; tÞ/p1 ðx; tÞ:
ð11Þ
Substituting (11) into (1) we have the following generalized recurrence relation
u
j 2p j 2p j 2p þ 1 2u j 1;t /t 2u j 1;x /x þ u j 1 ð/tt /xx Þ þ þ2 þ 1 ð/2t /2x Þu j p1 p1 p1 p1 p1 p1 p1 1km q 2Y i1 X X Xj 2p þ 4i 2 2 1 1 m ai ukj2pþ4i cm ukcmm u1k ukj2pþ4i g2i1 cm g2i1 g2i1 ;t t p 1 p 1 p1 p1 t m¼1 i¼1 k2K c2C
j 2;tt p1
u
i
i
p X
j2pþ2l p1
bl
l¼0
þ
j 2;xx p1
X
j2pþ2l ðb1 þþbl1 Þ p1
X
b1 ¼0
uj2pþ2lðb p1
bl ¼0
1 þbþbl1 Þ
ub1 ubl ;
where
K ¼ fk ¼ ðk1 ; . . . ; k2i Þ j km 2 f0; 1g for all m ¼ 1; 2; . . . ; 2ig;
C ¼ c ¼ ðc1 ; c2 ; . . . ; c2i1 Þ j 0 6 cm 6
gl k1 þ b þ k2i þ 2i þ
j 2p þ 4i gm1 for all m ¼ 1; 2; . . . ; 2i 1 ; p1
l X
cj :
j¼1
We need to verify the arbitrariness of the term at the resonance. Plugging in j ¼ 2ðp þ 1Þ gives us
" 2pðp þ 1Þ ðp 1Þ
ð/2t 2
#
/2x Þ
pbp up1 0
u2ðpþ1Þ ¼ 0: p1
Thus we have verified the arbitrariness at the resonance. Eq. (1) possesses the strong Painlevé property for the special cases p = 2 and p = 3, when p > 2ðq 1Þ. 3. Integrability and closed-form exact solutions of traveling wave type We now consider the two special cases where (1) satisfies the strong Painlevé property and find special solutions following a procedure used in [19]. Such a method was also employed in [16,17]. We begin by defining two new functions to be used extensively through the remainder of this paper:
C
/t ; /x
ð12Þ
V
/xx : /x
ð13Þ
We would like to insert (12) truncated at the constant term into (1) (to find our auto-Bäcklund transform equations) and match orders of /. Truncating the Laurent series at constant terms, we shall obtain exact solutions for our nonlinear hyperbolic equation (1) in each of these two interesting cases. For computational simplicity, we will place certain specific yet useful restrictions on C and V; under appropriate restrictions on C and V, and hence on /ðx; tÞ, we have integrable equations. 3.1. The case of p ¼ 2; q ¼ 1 When q ¼ 1 and p = 2, we have the nonlinear wave equation
1626
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
utt uxx a1 ut b1 u b2 u2 ¼ 0:
ð14Þ
Assuming a solution
uðx; tÞ ¼
u0 ðx; tÞ 2
/ðx; tÞ
þ
u1 ðx; tÞ þ u2 ðx; tÞ; /ðx; tÞ
ð15Þ
we find
u0 ¼
6 2 ð/ /2x Þ; b2 t
ð16Þ
u1 ¼
1
6 5/2t /tt þ 8/t /x /xt þ a1 /3t a1 /t /2x þ /tt /2x þ /xx /2t 5/xx /2x ; b2 ð/2t /2x Þ 5
ð17Þ
u2 ¼
1
/2x Þ3 a1 /3t /2x /tt
4a1 /4t /x /xt þ 64a1 /2t /3x /xt 32a1 /3t /2x /xx þ 58a1 /t /4x /tt þ 30a1 /t /4x /xx
50b2 ð/2t 88
600/3t /tt /x /xt þ 412/2t /2x /tt /xx 104/t /3x /tt /xt 104/3t /x /xx /xt 600/t /3x /xt /xx
þ a21 /6t þ 25b1 /6t 25b1 /6x þ 75/4t /2tt 49/4x /2tt 49/4t /2xx þ 75/4x /2xx þ 180/4x /2xt þ 60/5x /xtt 100/5x /xxx 100/5t /ttt þ 180/4t /2xt þ 60/5t /xxt 2a21 /4t /2x þ a21 /2t /4x 75b1 /4t /2x þ 75b1 /2t /4x þ 150/2t /2x /2tt 30/4t /tt /xx þ 344/2t /2x /2xt 30/4x /xx /tt þ 150/2t /2x /2xx þ 180/x /4t /xtt 60/x /4t /xxx 240/3x /2t /xtt 240/2x /3t /xxt þ 180/t /4x /xxt þ 160/3x /2t /xxx 60/t /4x /ttt þ 160/3t /2x /ttt þ 30a1 /5t /tt
þ 2a1 /5t /xx 60a1 /5x /xt :
ð18Þ
1
Due to the length of the equation for Oð/ Þ it will be omitted here. Following the procedure from the previous example we use (13) and (14) in (17)–(19) to get
u0 ¼
6/2x 2 ðC 1Þ; b2
ð19Þ
u1 ¼
h i 6/x ðb2 ðC 2 1ÞÞ1 5ðC 2 C t þ C 3 C x þ C 4 VÞ þ 9CC x þ 10C 2 V þ a1 C 3 a1 C þ C t 5V ; 5
ð20Þ
u2 ¼
h
1 2
3
50b2 ðC 1Þ
600ðC 3 C x V þ C 4 V 2 ÞðC t þ CC x þ C 2 VÞ þ 412ðC 2 C t V þ C 3 C x V þ C 4 V 2 Þ
104CðC x þ CVÞðC x þ CC x þ C 2 VÞ 104ðC 3 C x V þ C 4 V 2 Þ 600ðCC X V þ C 2 V 2 Þ 88a1 ðC 3 C t þ C 4 C x þ C 5 VÞ 4a1 ðC 4 C x þ C 5 VÞ þ 64a1 ðC 2 C x þ C 3 VÞ 32a1 C 3 V þ 58a1 ðCC t þ C 2 C x þ C 3 VÞ þ 30a1 CV þ a21 C 6 þ 25b1 C 6 25b1 þ 75C 4 ðC t þ CC x þ C 2 VÞ2 49ðC t þ CC x þ C 2 VÞ2 49C 4 V 2 þ 75V 2 þ 180ðC x þ CVÞ2 þ 60ðC xt þ C 2x þ 2CC x V þ C 2 V x þ C t V þ C 2 V 2 Þ 100ðV x þ V 2 Þ 100C 5 ðC tt þ 2C t C x þ 3CC t V þ CC xt þ C 2 V t þ C 2 C x V þ C 3 V 2 Þ þ 180C 4 ðC x þ CVÞ2 þ 60C 5 ðV t þ C x V þ CV 2 Þ 2a21 C 4 þ a21 C 2 75b1 C 4 þ 75b1 C 2 þ 150C 2 ðC t þ CC x þ C 2 VÞ2 30ðC 4 C t V þ C 5 C x V þ C 6 V 2 Þ þ 344C 2 ðC x þ CVÞ2 30ðC t V þ CC x V þ C 2 V 2 Þ þ 150C 2 V 2 þ 180ðC 4 C xt þ C 4 C t V þ C 5 V t þ C 4 C 2x þ 2C 5 C x V þ C 6 V 2 Þ 60ðC 4 V x þ C 4 V 2 Þ 240ðC 2 C xt þ C 2 C t V þ C 3 V t þ C 2 C 2x þ 2C 3 C x V þ C 4 V 2 Þ 240ðC 3 V t þ C 3 C x V þ C 4 V 2 Þ þ 180ðCV t þ CC x V þ C 2 V 2 Þ þ 160ðC 2 V x þ C 2 V 2 Þ 60ðCC tt þ 2CC t C x þ 3C 2 C t V þ C 2 C xt þ C 2 C 2x þ 2C 3 C x V þ C 3 V t þ C 4 V 2 Þ þ 160ðC 3 C tt þ 2C 3 C t C x þ 3C 4 C t V þ C 4 C xt
i þ C 4 C 2x þ 2C 5 C x V þ C 5 V t þ C 6 V 2 Þ þ 30a1 ðC 5 C t þ C 6 C x þ C 7 VÞ þ 2a1 C 5 V 60a1 ðC x þ CVÞ :
ð21Þ
Forcing C and V to be constants drastically reduces the complexity of Eqs. (21) and (22), as well as the complexity of the equation for Oð/1 Þ. Under such a restriction, we find
u1 ¼
u2 ¼
i 6/x h 5VðC 2 1Þ þ a1 C ; 5b2 1 2
50b2 ðC 1Þ
h
i 30a1 CVðC 2 1Þ 25V 2 ðC 2 1Þ2 þ a21 C 2 þ 25b1 ðC 2 1Þ ;
ð22Þ
ð23Þ
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
Oð/1 Þ ¼
h i 6/x ðC 2 1Þ4 a31 C 3 ðC 2 1Þ3 25a1 CV 2 ðC 2 1Þ5 ; 125b2
a1 C 1
V ¼
5ðC 21 1Þ
1627
ð24Þ
ð25Þ
;
( ) 5ðC 21 1Þ a1 C 1 ðx þ C 1 tÞ þ C 2 ; exp /ðx; tÞ ¼ a1 C 1 5ðC 21 1Þ
ð26Þ
n o n o 2a1 C 1 a1 C 1 6a1 C 1 exp 5ðC ðx þ C tÞ 6 exp ðx þ C tÞ ð 1Þa1 C 1 1 1 2 2 5ðC 1 1Þ 1 1Þ n o uðx; tÞ ¼ n o 2 þ 2 2 5ðC 1Þ a C 5ðC 1Þ 1 1 5b2 a11C 1 exp 5ðC12 1Þ ðx þ C 1 tÞ þ C 2 5b2 a11C1 exp 5ðCa12C1Þ ðx þ C 1 tÞ þ C 2 1
h
1 50b2 ðC 21
1Þ
1
a21 C 21
6
i a21 C 21 þ a21 C 21 þ 25b1 ðC 21 1Þ :
ð27Þ
3.2. The case of p ¼ 2; q ¼ 0 The case of p = 2 and q ¼ 0 proceeds similar to the p ¼ 2; q ¼ 1 case. The form of (15) is unchanged, and we find that u0 ; u1 and u2 are determined uniquely by the Oð/4 Þ; Oð/3 Þ and Oð/1 Þ conditions, respectively. Note that taking
/ðx; tÞ ¼ exp ðVðx þ C 1 tÞÞ þ C 2
ð28Þ 2
0
for constant V; C 1 and C 2 identically satisfies the Oð/ Þ condition. What is left is the Oð/ Þ condition, and this reduces to
V 2 ðC 21 1Þ b1 V 2 ðC 21 1Þ þ b1 ¼ 0 ) V 2 ðC 21 1Þ ¼ b1 :
ð29Þ
So, there are two possibilities. When jC 1 j < 1, we take the þb1 root, obtaining
V1 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 C 21 1
ð30Þ
:
Otherwise, when jC 1 j > 1, we take the b1 root, obtaining
V2 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 1 C 21
ð31Þ
:
Recalling the solution (15), when jC 1 j < 1 we have the exact solution
uðx; tÞ ¼
6b1 C 2 eV 1 ðxþC 1 tÞ b2 ðeV 1 ðxþC1 tÞ þ C 2 Þ
2
;
ð32Þ
while when jC 1 j > 1 we obtain the exact solution
6b1 C 2 eV 2 ðxþC1 tÞ
b1 : b2
ð33Þ
utt uxx b1 u b2 u2 b3 u3 ¼ 0:
ð34Þ
uðx; tÞ ¼
2
b2 ðeV 2 ðxþC 1 tÞ þ C 2 Þ
3.3. The case of p ¼ 3; q ¼ 0 When p = 3 and q ¼ 0 we have
Assuming a solution of the form
uðx; tÞ ¼
u0 ðx; tÞ þ u1 ðx; tÞ; /ðx; tÞ
ð35Þ
we have
u0 ¼
1 2 2 2 /t /2x ; b3
ð36Þ
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M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
pffiffiffi pffiffiffi pffiffiffi u1 ¼ 4 2/t /x /xt 3/2x /2t 2/xx 3/2t /2x 2/tt ;
ð37Þ
1 3 1 6b23 /2t /2x 2 ;
ð38Þ
3 1 2b2 b3 2 /2t /2x 2
Oð/1 Þ : u0;tt u0;xx b1 u0 2b2 u0 u1 3b3 u0 u21 :
ð39Þ
Using these conditions, and letting C and V be constant, we obtain
u0 ¼
u1 ¼
sffiffiffiffiffi 1 2 / ðC 2 1Þ2 ; b3 x
ð40Þ
pffiffiffi 32 1 32 1 pffiffiffi pffiffiffi 1 6b23 C 2 1 4 2C 2 V 2 3 C 2 V 2 3C 2 1 C 2 V 2b2 b3 2 C 2 1 ;
pffiffiffi /x 2 2 ffi ð6b1 b3 2b2 ÞðC 2 1Þ þ 3b3 ðC 2 1Þ2 V 2 : Oð/1 Þ : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 6 b3 ðC 1Þ
ð41Þ
ð42Þ
Once again we let C ¼ C 1 , where C 1 is a constant, and find for V
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u2b 6b1 b3 ; V ¼ t 2 2 3b3 ðC 1 1Þ
ð43Þ
which gives us
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 v 9 u < u = u2b2 6b1 b3 u 3b3 ðC 2 1Þ 2 1 t t exp ðx þ C 1 tÞ þ C 2 ; /ðx; tÞ ¼ 2 2 : ; 3b3 ðC 1 1Þ 2b2 6b1 b3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 v 9 u < u = u2b2 6b1 b3 u 3b3 ðC 2 1Þ 2 1 t t exp ðx þ C /ðx; tÞ ¼ tÞ þ C2; 1 2 : ; 3b3 ðC 21 1Þ 2b2 6b1 b3
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b2 6b1 b3 ðx þ C 1 tÞ 3b2 ðC 2 1Þ 3 1 uðx; tÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3b3 ðC 21 1Þ 2b22 6b1 b3 tÞ þ C2 exp ðx þ C 1 2 2 3b3 ðC 1 1Þ 2b2 6b1 b3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 u 2 u 32 u u b 3b1 b3 u b2 3b1 b3 u b2 3b1 b3 1 2 2 2 2 2 2 3 C1 t 2 2 2 3C 1 1 C 1 t 2 2 2b2 b3 2 C 1 1 A @8C 1 t 2 2 3b3 ðC 1 1Þ 3b3 ðC 1 1Þ 3b3 ðC 1 1Þ 32 1 1 : 6b23 C 21 1
ð44Þ
ð44Þ
qffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
2ðC 1 1Þ exp b3
ð45Þ
3.4. The case of p ¼ 3; q ¼ 1 The case of p = 3 and q ¼ 1 proceeds similar to the p ¼ 3; q ¼ 0 case. We take a solution of the form (35), and we find that u0 and u1 are uniquely determined by the Oð/3 Þ and Oð/2 Þ conditions, respectively. Upon taking
/ðx; tÞ ¼ exp ðVðx þ C 1 tÞÞ þ C 2
ð46Þ 1
for constant V; C 1 and C 2 identically satisfies the Oð/ Þ condition provided that 2
2
3b3 ðC 21 1Þ2 V 2 ½b1 b3 2b2 þ ð2b2 ð6b1 þ a21 Þb3 ÞC 21 ¼ 0;
ð47Þ
so we take
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ub1 b3 2b2 þ ð2b2 ð6b1 þ a2 Þb3 ÞC 2 2 2 1 1 : V ¼t 3b3 ðC 21 1Þ2 With this choice of V, we finally have from the Oð/0 Þ condition that
ð48Þ
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
pffiffiffiffiffi pffiffiffi 2 3 12 2 b3 b2 2 ðb2 4b1 b3 Þa1 b1 C 1 ðC 21 1Þ þ ð4b2 18b1 b3 ÞðC 21 1Þ3=2 2a31 b3 C 1 5 5 pffiffiffiffiffiffiffiffi 3=2 3 6b3 2 2 a1 b3 C 21 2ðb2 3b1 b3 ÞðC 21 1Þ 5 ¼ 0;
1629
ð49Þ
a nonlinear relation for C 1 . When a real-valued solution exists for C 1 , then we have a solution of the form (35). Note that this requirement is distinct from what we have seen for the previous cases considered above. Here, in order to have a solution (35), we need to restrict the wave speed to only values satisfying the condition (49). Let C 1 denote a solution to (49) (we omit a general solution form here, as it is too complicated for arbitrary parameter values). Then we find that the solution (35) becomes
pffiffiffi qffiffiffiffiffiffiffi pffiffiffi VðxþC 1 tÞ 2V C 2 b2 C2 2a1 C 1 1 e pffiffiffiffiffi uðx; tÞ ¼ pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ : VðxþC 1 tÞ 3b3 2 e þ C2 2 b 3 6 b3 C 1 1
ð50Þ
4. General method for travelling wave solutions In the previous section, we showed that one may obtain closed-form exact solutions by use of truncated Painlevé expansions. Let us now consider travelling wave solutions under a far more general framework. To this end, let us assume a travelling wave solution to (1) of the form uðx; tÞ ¼ wðzÞ, where x ct and c 2 R is the wave speed. Without making this assumption a priori, we still recovered solutions of the form in Section 3, lending some validity to the present approach. In fact, most known solutions of NLPDEs tend to be traveling waves, although some, such as ‘boomerons’, have non-constant wave speeds and this are not traveling waves. Under such a travelling wave assumption, (1) becomes q p X X 2i1 ð1 c2 Þw00 þ ai c2i1 w0 þ bj wj ¼ 0: i¼1
ð51Þ
j¼1
We first consider two special cases before addressing the general case. 4.1. Degenerate cases When c ¼ 1, the equation becomes degenerate:
sgnðcÞ
q p X X 2i1 ai w0 þ bj wj ¼ 0: i¼1
ð52Þ
j¼1
This case is well-studied in the literature, however we make a few remarks since it is an exact reduction of the more general equation we are concerned with. Eq. (52) essentially defines a polynomial in w0 of degree 2q 1. Assume – 0, and let Pp 1 j Mk ða1 ; . . . ; a2q1 ; sgnðcÞ j¼1 bj w Þ denote the kth solution to the polynomial equation q X ai M2i1 þ i¼1
X 1 b wj ¼ 0: sgnðcÞ j¼1 j p
ð53Þ
Then, for (52) with real coefficients, there exists a real-valued solution satisfying the first order nonlinear differential equation
X 1 b wj þ kÞ sgnðcÞ j¼1 j p
w0k ¼ M k ða1 ; . . . ; a2q1 ;
ð54Þ
for k ¼ 1; 2; . . . ; 2q 1, provided M k is real-valued. For instance, in the case where q ¼ 1,
X 1 bj wj : a1 sgnðcÞ j¼1 p
w0 ¼
ð55Þ
Separating variables, we see that a solution to this nonlinear ODE then must satisfy the implicit integral relation
Z
w
wð0Þ
dn z ¼ : Pp j a sgnðcÞ 1 b n j j¼1
When p = 1, the exact solution is given in closed form by
ð56Þ
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M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
wðzÞ ¼ wð0Þ exp
b1 z : a1 sgnðcÞ
ð57Þ
When p = 2, the exact solution is given by
wðzÞ ¼
wð0Þb1 : 1z ðwð0Þb2 þ b1 Þ exp a1 bsgnðcÞ wð0Þb2
ð58Þ
Note that p ¼ 1; 2 constitute the cases in which we find exact solutions in terms of exponential functions. When p P 3, this is possible in some special cases, but in general the implicit relation (56) will result in a linear combination of logarithm and inverse tangent functions, the inversion of which will strongly depend on the specific values of the bk ’s. For the general p = 3 case, we obtain
0 1 1 ln ðwÞ ln b1 þ b2 w þ b3 w2 b1 2b1
1
b2 z0 z B b2 þ b3 w C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA ¼ ; a1 sgnðcÞ 2 2 b1 4b1 b3 b2 4b1 b3 b2
ð59Þ
where z0 is a constant of integration. However, consider the case where b2 ¼ 0; we then may solve the relation for w to obtain the two solution branches
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2b1 ðz0 zÞ 1 ðz0 zÞ exp b1 exp 2b 1 b 3 a1 sgnðcÞ a1 sgnðcÞ wðzÞ ¼ : 2b1 ðz0 zÞ 1 b3 exp a1 sgnðcÞ
ð60Þ
More generally, let p P 3 be arbitrary, take b2 ¼ b3 ¼ ¼ bp1 ¼ 0, and let b1 and bp be arbitrary. Then, we have exact solutions
2
b1 exp
ðp1Þb1 ðz0 zÞ a1 sgnðcÞ
31=ðp1Þ
5 wðzÞ ¼ m‘ 4 1 ðz0 zÞ 1 bk exp ðp1Þb a1 sgnðcÞ
ð61Þ
;
where ‘ ¼ 1; 2; . . . ; p and m‘ is a pth root of unity. We should mention that in the case where ¼ 0, there are only constant solutions. Such solutions must satisfy the polynomial equation p X bj wj ¼ 0:
ð62Þ
j¼1
In the case where c = 0, we have stationary solutions. Such solutions necessarily satisfy the ODE
w00 þ
p X bj wj ¼ 0:
ð63Þ
j¼1
Again, such equations are well-studied in the literature. We shall say a few things about it, since it provides some motivation for later solutions to the more general case problem. When p = 1, the solution is trivial. When p = 2, one can find wðzÞ in terms of Weierstrass’s elliptic functions, while when p = 3 one can find wðzÞ in terms of Jacobi elliptic functions (under some appropriate restrictions on the bk ’s). 4.2. Constructing rational travelling wave solutions in exponentials For the integrable families of equations discovered in Section 2, we were able to provide exact solutions in terms of ratios of exponential functions in Section 3. As we have shown so far in Section 4, exact solutions may be found even in non-integrable cases. In the present section, we shall assume travelling wave solutions, and obtain said solutions in a manner different from that of Section 3. In particular, while we have obtained some particular solutions in Section 3, it is not clear that those were the only travelling wave solutions, since we made an assumption on the form of /ðx; tÞ. We shall focus on solutions which are functions of exponential factors of the form eaðzz0 Þ [2–8]. To simplify the process, let us make the change of independent variable Z ¼ eaðzz0 Þ where a and z0 are constants, and change of dependent variable WðZÞ ¼ wðzÞ. Then (51) becomes
a2 ð1 c2 ÞðZ 2 W 00 þ ZW 0 Þ þ
q p X X 2i1 ai c2i1 a2i1 Z 2i1 W 0 þ bj W j ¼ 0: i¼1
ð64Þ
j¼1
Then, in the new variable Z, solutions we seek take the form of rational functions in Z. To this end, let us assume a solution of the form
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
WðZÞ ¼
S0 þ S1 Z þ þ Sm Z m SðZÞ ¼ ; T 0 þ T 1 Z þ þ T n Z n TðZÞ
1631
ð65Þ
degðSÞ ¼ m P 0; degðTÞ ¼ n P 0. Eq. (64) then becomes
h
a2 ð1 c2 Þ Z 2 S00 T 2 2S0 TT 0 STT 00 þ 2ST 0 2 þ Z S0 T 2 STT 0
i
þ
q X 2i1 54i ai c2i1 a2i1 Z 2i1 S0 T ST 0 T i¼1
p X þ bj Sj T 3j ¼ 0:
ð66Þ
j¼1
We may always rearrange the factors of TðZÞ to put the left hand side of (66) into the form ðTðZÞÞv QðZÞ, where v P 0 is an integer power. As T is not identically zero by construction, we must then have that QðZÞ 0 if WðZÞ ¼ SðZÞ=TðZÞ is in fact a solution. Let Q denote the polynomial of minimal degree satisfying this condition. Since S and T are of finite degree, such a polynomial exists; let us then take degðQÞ ¼ jðm; nÞ. Note that the degree j depends on both m and n. Note that since degðSÞ ¼ m P 0; degðTÞ ¼ n P 0, and a was arbitrary yet non-zero, we have m þ n þ 3 free parameters. The coefficients of Q will depends on these m þ n þ 3 parameters. In particular,
QðZÞ ¼ Q0 ðS0 ; . . . Sm ; T 0 ; . . . ; T n ; aÞ þ þ Qj ðS0 ; . . . Sm ; T 0 ; . . . ; T n ; aÞZ j :
ð67Þ
jðm; nÞ 6 m þ n þ 2, then we have enough free parameters to force the coefficients of Q to be zero. Meanwhile, if jðm; nÞ > m þ n þ 2, we in general do not have enough free parameters to choose the coefficients of Q to be zero. An excepIf
tion to this latter case occurs is there are serendipitous cancellations leading to QðZÞ 0. Such cancellations would depend on fixing very specific values of the model parameters in the original equation (1), and as such much generality would be lost in making such assumptions. We can proceed no further with such generality, so we shall illustrate these ideas through a special case which holds the solutions of Section 3 as particular cases. 4.3. Possible rational solutions for the restriction p ¼ 3; q ¼ 1 In order to proceed further, let us take q ¼ 1 and p = 3. Under such an assumption, (66) becomes
a2 ð1 c2 ÞZ 2 S00 T 2 2S0 TT 0 STT 00 þ 2ST 0 2 þ a2 ð1 c2 Þ þ aa1 c Z S0 T 2 STT 0 þ b1 ST 2 þ b2 S2 T þ b3 S3 ¼ 0: 3
ð68Þ
2
We see that the highest order term will be either degðS Þ ¼ 3m or degðST Þ ¼ m þ 2n. Then,
jðm; nÞ ¼
3m
if m P n;
m þ 2n if m < n:
ð69Þ
Thus, by specifying the highest order nonlinearities p and q, we have determined jðm; nÞ. As previously discussed, we require jðm; nÞ 6 m þ n þ 2. We may then verify that when p = 3 and q ¼ 1, we have the possibilities ðm; nÞ 2 fð0; 1Þ; ð0; 2Þ; ð1; 0Þ; ð1; 1Þ; ð1; 2Þ; ð2; 2Þg. We then have six possible rational solutions. Mapping back to the original form of the solution (in variable z versus variable Z), we have the general rational solutions
wðzÞ ¼
S0 ; T 0 þ T 1 eaðzz0 Þ
ð70Þ
wðzÞ ¼
S0 ; T 0 þ T 1 eaðzz0 Þ þ T 2 e2aðzz0 Þ
ð71Þ
wðzÞ ¼ S0 þ S1 eaðzz0 Þ ;
ð72Þ
wðzÞ ¼
S0 þ S1 eaðzz0 Þ ; T 0 þ T 1 eaðzz0 Þ
ð73Þ
wðzÞ ¼
S0 þ S1 eaðzz0 Þ ; T 0 þ T 1 eaðzz0 Þ þ T 2 e2aðzz0 Þ
ð74Þ
wðzÞ ¼
S0 þ S1 eaðzz0 Þ þ S2 e2aðzz0 Þ ; T 0 þ T 1 eaðzz0 Þ þ T 2 e2aðzz0 Þ
ð75Þ
where the coefficients and the exponential growth/decay rate, a, depend on the original model parameters. Since the coefficients depend on the model parameters, there may exist model parameters for which some of the coefficients become singular, for one or more other six solutions. Thus, these are the admissible solutions, and for some specific values of the parameters a1 ; b1 ; b2 ; b3 ; c and , such solutions may break down. It is therefore important to test whether for given specific values of a1 ; b1 ; b2 ; b3 ; c and , an obtained function is really a solution. The primary danger is that, for certain parameter val-
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M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
ues, solutions can become singular and hence non-physical. Thus, we must proceed with caution. See also the comments of Kudryashov [20,21] on such issues. Furthermore, these solutions are the only possible solutions; if either S or T is of degree three or greater, the rational function (65) fails to be a solution. 4.4. Explicit solutions for the case p ¼ 3; q ¼ 1 In order to demonstrate the method, let us pick a concrete example for which we may compute solutions exactly. Since we have already found solutions for the ðp; qÞ ¼ ð3; 0Þ and ðp; qÞ ¼ ð2; 1Þ cases in Section 3, let us consider the non-integrable case of ðp; qÞ ¼ ð3; 1Þ. To this end, we set SðZÞ ¼ S0 and TðZÞ ¼ T 0 þ T 1 Z þ T 2 Z 2 . From (68), we arrive at the following conditions:
Oð1Þ : b1 T 20 þ b2 S0 T 0 þ b3 S20 ¼ 0;
ð76Þ
OðZÞ : ð2b1 að1 c2 Þ aa1 cÞT 0 þ b2 S0 ¼ 0;
ð77Þ
OðZ 2 Þ : að1 c2 Þð2S0 T 21 2S0 T 0 T 2 Þ ðað1 c2 Þ þ aa1 cÞð2S0 T 0 T 2 þ S0 T 21 Þ þ b1 S0 ð2T 0 T 2 þ T 21 Þ þ b2 S20 T 2 ¼ 0;
ð78Þ
OðZ 3 Þ : 6ð1 c2 Þa2 þ 3ð1 c2 þ a1 cÞa 2b1 ¼ 0;
ð79Þ
OðZ 4 Þ : 6ð1 c2 Þa2 þ 2ð1 c2 þ a1 cÞa b1 ¼ 0:
ð80Þ
3
4
The OðZ Þ and OðZ Þ conditions imply the parameter constraints
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 and a ¼ : 6ðc2 1Þ
1 a1 ¼ c c
ð81Þ
The OðZÞ condition then implies 2b1 T 0 þ b2 S0 ¼ 0, i.e.
S0 ¼
2b1 T 0: b2
ð82Þ 2
From here, the Oð1Þ condition implies 4b1 b3 b2 ¼ 0. Let us write this as 2
b3 ¼
b2 : 4b1
ð83Þ
Finally, the OðZ 2 Þ condition implies 2
b1 T 0 T 1 T 2 ¼ 0; b2
ð84Þ
yet T 0 – 0 and b1 – 0, so we need one of T 1 ¼ 0 or T 2 ¼ 0. If T 1 ¼ 0, we obtain the solution
WðZÞ ¼
2b1 T 0
ð85Þ
b2 ðT 0 þ T 2 Z 2 Þ
or, converting back to z,
2b1 wðzÞ ¼ b2 where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)1 b1 1 þ s2 exp 2 ; z 6ðc2 1Þ
(
ð86Þ
s2 ¼ T 2 =T 0 . Meanwhile, if T 2 ¼ 0, we obtain the solution WðZÞ ¼
2b1 T 0 b2 ðT 0 þ T 1 ZÞ
ð87Þ
which is equivalent to
2b1 wðzÞ ¼ b2 where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)1 b1 1 þ s1 exp ; z 6ðc2 1Þ
(
ð88Þ
s1 ¼ T 1 =T 0 . Hence, both (86) and (88) are solutions to the nonlinear PDE 2
utt uxx ¼ aut þ b1 þ b2 u2 þ
b2 3 u ; 4b1
ð89Þ
M. Russo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1623–1634
1633
where the wave speed c is related to a by a ¼ c 1c . (For all a 2 R, there exists c > 0 such that this relation holds, so it is completely arbitrary.) Each of these solutions has one free parameter, sk . (The parameter c and the bk ’s are fixed by the coefficients to the NLPDE.) Hence, we have constructed a one-parameter family of solutions to (89) subject to the constraint a ¼ c 1c . Next, let us consider the case SðZÞ ¼ S0 þ S1 Z; TðZÞ ¼ T 0 þ T 1 Z. We find that upon taking
S0 ¼
b2 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4b1 b3 2b3
T0;
b2
S1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4b1 b3 2b3
T1;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b2 4b1 b3 b2 b2 4b1 b3 1 a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 2 ð1 c2 Þb3 2b1 b3 b2 þ b2 4b1 b3
ð90Þ
ð91Þ
we have that (68) implies
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4b1 b3 T 0 T 1 Z b2 WðZÞ ¼ 2b3 T 0 þ T 1 Z 2b3
ð92Þ
is a solution to the NLPDE
utt uxx ¼ aut þ b1 u þ b2 u2 þ b3 u3 ;
ð93Þ
provided that a and the wave-speed c satisfy the nonlinear relation
1 1 a¼c þ c c
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 ð1 c2 Þb3 2b1 b3 b2 þ b2 4b1 b3 :
ð94Þ
In terms of z, we have the solution
wðzÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4b1 b3 T 0 T 1 eaðzz0 Þ T 0 þ T 1 eaðzz0 Þ
2b3
b2 : 2b3
ð95Þ
Note that T 0 and T 1 are free parameters. Hence, we have constructed a two-parameter family of solutions to the NLPDE (93) subject to the constraint (94). It is worth mentioning that, for fixed parameters a; b1 ; b2 ; b3 , we have that the wave number is determined by the parameters, due to (94). Indeed, for any set of parameter values, we must verify such a c exists. From (94), we find that the existence of c is equivalent to the existence of a real-valued solution to the quartic equation
0
2
c þ 2ac þ @2 a 4
3
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 b2 4b1 b3 2 2b1 b3 b2 þ b2 4b1 b3 Ac 2ac þ 1¼0 b3 b3
2b1 b3 b2 þ
ð96Þ
for c. Observe that, although we have used a different and more direct method in Section 4, the solution (95) is of the same form as the solution (50) obtained in Section 3. So, the two methods are consistent. 5. Conclusions In this paper, we first determine values of p and q which permit the strong Painlevé property of (1). We find that general forms of Eq. (1) corresponding to (i) (ii) (iii) (iv)
p ¼ 2; p ¼ 2; p ¼ 3; p ¼ 3;
q ¼ 1; q ¼ 0; q ¼ 0; q ¼ 1.
For each of the cases exhibiting the strong Painlevé property, we obtain exact solutions using the relevant auto-Bäcklund transformations and truncated Laurent series. These solutions are all travelling waves. In particular, we find that solutions are rational functions of exponentials. Using the appearance of travelling wave solutions for the integrable cases as a clue, we consider a general travelling wave transformation, which maps our nonlinear PDE (1) into an ODE (51) which governs travelling wave solutions for arbitrary p and q. We have three degenerate cases, when c ¼ 1; 0; 1. The two cases c ¼ 1 result in a reduction of order (we obtain a first order strongly nonlinear, as opposed to a second order weakly nonlinear, equation). For some special cases, solutions to
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this degenerate case are provided. The second type of degeneracy, c ¼ 0, corresponds to stationary solutions. Solutions to the c ¼ 0 ODE are always first integrable. When p = 2 and p = 3, the stationary solutions take the form of elliptic functions. However, in order to recover the solutions which are rational functions of exponentials, an alternate form of the travelling wave ODE is useful. Under a change of variables Z ¼ exp ðaðz z0 ÞÞ (where z ¼ x ct is the wave variable) we transform the travelling wave ODE into a new ODE, the rational solutions of which yield rational exponential solutions of the original travelling wave ODE. We arrive at a condition on the order ½m; n of any rational solution of this type. Using the general equation with p = 3 and q ¼ 1 as an example, we obtain all possible rational exponential solutions to this equation. 5.1. Acknowledgments The authors appreciate the helpful comments of the reviewers, which have greatly improved the clarity of the paper. R.A.V. was supported in part by NSF Grant # 1144246. References [1] Dodd RK, Eilbeck JC, Gibbon JD, Morris HC. Solitons and nonlinear wave equations. London: Academic Press; 1982. [2] Wazwaz AM. The extended tanh method for new soliton solutions for many forms of the fifth-order KdV equations. Appl Math Comput 2007;184:1002–14. [3] Wazwaz AM. Hirota’s direct method and the tanh–coth method for multiple-soliton solutions of the Sawada–Kotera–Iot seventh-order equation. Appl Math Comput 2008;199:133–8. [4] Hirota R. The direct method in soliton theory. Cambridge: Cambridge University Press; 2001. [5] Caudrey PJ. Memories of Hirota’s method. Philos Trans R Soc A 2011;28:1215–27. [6] Conte R, Musette M. The Painlevé handbook. Berlin: Springer; 2008. [7] Conte R. Exact solutions of NLPDEs by singularity analysis. Lect Notes Phys 2003;632:1–83. [8] Tian SF, Zhang HQ. On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation. J Phys A Math Gen 2012;45:055203 (pp. 29). [9] Gao YT, Tian B. Some two-dimensional and non-traveling-wave observable effects of the shallow-water waves. Phys Lett A 2002;301:74–82. [10] Karaca MA, Hizel E. Similarity reductions of BBM equations. Appl Math Sci 2008;2:463–9. [11] Fang JP, Li JB, Zheng c L, Ren QB. Special conditional similarity reductions and exact solutions of the (2+1)-dimensional VCBKK system. Chaos Solitons Fractals 2008;35:530–5. [12] Weiss J, Tabor M, Carnevale G. The Painlevé property and singularity analysis of integrable and nonintegrable systems. J Math Phys 1983;24:522. [13] Weiss J. The Painlevé property for partial differential equations II: Bäcklund transformations, Lax pairs, and the Schwarzian derivative. J Math Phys 1983;24:1405. [14] Weiss J. On classes of integrable systems and the Painlevé property. J Math Phys 1984;25:13. [15] Hearns J, Van Gorder RA, Choudhury SR. Painlevé test, integrability, and exact solutions for density-dependent reaction–diffusion equations with polynomial reaction function. Appl Math Comput 2012;219:3055–64. [16] Kudryashov NA. Exact soliton solutions of the generalized evolution equation of wave dynamics. PMM J Appl Math Mech 1988;52:361–5. [17] Kudryashov NA. Exact solutions of the generalized Kuramoto–Sivashinsky equation. Phys Lett A 1990;147:287–91. [18] Gilson C, Pickering A. Factorization and Painleve analysis of a class of nonlinear third-order partial differential equations. J Phys A Math Gen 1995;28:2871–88. [19] Roy Choudhury S. A unified approach to integrable systems via Painlevé analysis. Contemp Math 2002;301:139–61. [20] Kudryashov NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simul 2009;14:3507–29. [21] Kudryashov NA. On new travelling wave solutions of the KdV and the KdV–Burgers equations. Commun Nonlinear Sci Numer Simul 2009;14:1891–900.