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Nonlinear self-adjointness, conservation laws, exact solutions of a system of dispersive evolution equations
Commun Nonlinear Sci Numer Simulat 19 (2014) 3036–3043
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Nonlinear self-adjointness, conservation laws, exact solutions of a system of dispersive evolution equations R. Tracinà b,⇑, M.S. Bruzón a, M.L. Gandarias a, M. Torrisi b a b
Departamento de Matemáticas, Universidad de Cádiz, PO Box 40, 11510 Puerto Real, Cádiz, Spain Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy
a r t i c l e
i n f o
Article history: Received 20 September 2013 Received in revised form 22 November 2013 Accepted 6 December 2013 Available online 3 February 2014
a b s t r a c t In this paper, we classify a ð1 þ 1Þ-dimensional nonlinear system of dispersive evolution equations both from point of view of symmetries and nonlinear self-adjointness. Then, conservation laws are established from the property of nonlinear self-adjointness and from these we obtain exact solutions. Ó 2014 Elsevier B.V. All rights reserved.
Keywords: Nonlinearly self-adjoint system Conservation laws Exact solutions
1. Introduction We consider the ð1 þ 1Þ-dimensional nonlinear system of dispersive evolution equations
F 1 ut ðuxxx þ a1 vv x Þx a2 v 2 ¼ 0; F 2 v t uxxx b1 u b2 v ¼ 0;
ð1Þ
where a1 ; a2 ; b1 ; b1 are constants and satisfy the condition ja1 j þ ja2 j – 0. This system has been studied in [16] in order to find exact solutions by using invariant subspace method. This system offers interesting features, it admits diverse invariant subspaces of solutions defined by 2nd or 3rd order linear differential equations. The invariant subspace method, proposed in [18,4,5], could be considered as a generalized separation of variables method for nonlinear differential equations. It is well known that evolution equation (1) do not possess an usual Lagrangian, that is, there is no function L such that the considered equations are the Euler–Lagrange arising from L. Hence the Noethers theorem cannot be directly applied to obtain conservation laws on the basis of the equations symmetries. This, however, can be overcome by applying the general concept of nonlinear self-adjointness, devised and developed by Ibragimov [8,10,11,13,14], and Gandarias [6], which enables one to establish the conservation laws for any differential equation. Recently, several papers have been devoted to search for self-adjoint equations. This procedure can be applied not only to classes of single differential equations of any order but to the systems where the number of equations is equal to the number of dependent variables (see e.g. [12,19]). It is well known that conservation laws are useful in many respects. As shown in [14,15], they can be used to construct exact solutions different from group invariant solutions. ⇑ Corresponding author. Tel.: +39 0957383088. E-mail addresses: [email protected] (R. Tracinà), [email protected] (M.S. Bruzón), [email protected] (M.L. Gandarias), [email protected] (M. Torrisi). http://dx.doi.org/10.1016/j.cnsns.2013.12.005 1007-5704/Ó 2014 Elsevier B.V. All rights reserved.
R. Tracinà et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 3036–3043
3037
There exist different approaches to the study of conservation laws for equations without Lagrangian. In [1,2] the interested reader can be find a general treatment of the direct construction method for conservation laws of partial differential equations. Our first step is to investigate and to prove the nonlinear self-adjointness of the evolution system (1) which is the most important step to apply the Ibragimov’s techniques. Then, once obtained the Lie point symmetries, the conservation laws for system (1) can be immediately established. Finally, we show that it is possible to use these results in order to obtain exact solutions. 2. Adjoint system and conditions for self-adjointness For system (1) we introduce the formal Lagrangian
ðut ðuxxx þ a1 vv x Þx a2 v 2 Þ þ v ðv t uxxx b1 u b2 v Þ; Lu
ð2Þ
and v are two new dependent variables. where u The adjoint system for the system of differential equations (1) is defined by
(
F 1 dL ¼ 0; du
ð3Þ
F 2 dL ¼ 0; dv where, in this case,
dL @L @L @L @L @L Dt Dx þ Dx Dx þ þ Dx Dx Dx Dx ; du @u @ut @ux @uxx @uxxxx dL @L @L @L @L @L Dt Dx þ Dx Dx þ þ Dx Dx Dx Dx dv @ v @v t @v x @ v xx @ v xxxx with Dt and Dx the total differentiations with respect to t and x respectively. Taking into account the Eq. (2), the adjoint system (3) for system (1) is
xxxx u t ¼ 0; F 1 v xxx b1 v u xx v 2a2 u v ¼ 0: F 2 v t b2 v a1 u
ð4Þ
According to [13], the system (1) will be nonlinearly self-adjoint if each equations F i (i ¼ 1; 2) of the adjoint system (4) coincides with ki1 F 1 þ ki2 F 2 (i ¼ 1; 2) after the following substitution
¼ /ðt; x; u; v Þ; u
v ¼ wðt; x; u; v Þ
ð5Þ
with /ðt; x; u; v Þ – 0 or wðt; x; u; v Þ – 0. In other words system (1) is said to be nonlinearly self-adjoint if the adjoint system (4) obeys the condition
(
F 1 u¼/ðt;x;u;v Þ; v ¼wðt;x;u;v Þ ¼ k11 F 1 þ k12 F 2 ; F 2 u¼/ðt;x;u;v Þ; v ¼wðt;x;u;v Þ ¼ k21 F 1 þ k22 F 2
ð6Þ
with regular undetermined coefficients kij ði; j ¼ 1; 2Þ. Using the differential consequences of (5), since / and w do not depend on the derivatives ut ; v t ; uxx ; . . ., Eqs. (6) split into the following equations for the coefficients kij ði; j ¼ 1; 2Þ
k11 ¼ /u ;
k12 ¼ /v ;
k21 ¼ wu ;
k22 ¼ wv
and into the system for the substitution (5)
/u ¼ /v ¼ wu ¼ wv ¼ 0; /xxxx /t þ wxxx b1 w ¼ 0; a1 /xx 2a2 / ¼ 0; wt b2 w ¼ 0: It is possible to verify that this system admits always solutions with /ðt; xÞ – 0 or wðt; xÞ – 0, then system (1) is nonlinearly self-adjoint for any value of the constants a1 ; a2 ; b1 and b2 . The value of these constants determines the expressions of the substitutions (5). In order to obtain the explicit forms of the substitutions (5) we distinguish the following cases, where we will use ci (i ¼ 1; 2; . . .) to indicate arbitrary constants.
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1. a1 ¼ 0. In this case
wðt; xÞ ¼ eb2 t f ðxÞ
/ðt; xÞ ¼ 0;
where f ðxÞ is solution of the equation
f 000 b1 f ¼ 0: Then we distinguish two cases (a) b1 ¼ 0
f ðxÞ ¼ c1 x2 þ c2 x þ c3 : (b) b1 – 0
f ðxÞ ¼ c1 e
p ffiffiffiffi 3
b1 x
þ e
p3ffiffiffiffi
b1 x 2
c2 cos
pffiffiffip pffiffiffip ffiffiffiffiffi ffiffiffiffiffi ! 3 3 b1 x 3 3 b1 x þ c3 sin : 2 2
2. a2 ¼ b2 ¼ 0. In this case
/ðt; xÞ ¼ ðc1 x þ c2 Þt þ c3 x þ c4 ;
wðt; xÞ ¼ f ðxÞ
where f ðxÞ must satisfies
f 0000 b1 f c1 x c2 ¼ 0: Then we have two possibilities (a) b1 ¼ 0
wðt; xÞ ¼
c1 4 c2 3 x þ x þ c 5 x2 þ c 6 x þ c 7 : 24 6
(b) b1 – 0
wðt; xÞ ¼ e
p ffiffiffiffi 3
b1 x=2
pffiffiffip pffiffiffip ffiffiffiffiffi ffiffiffiffiffi ! p ffiffiffiffi 3 3 b1 x 3 3 b1 x 3 c1 x þ c2 þ c 5 e b1 x c6 cos þ c7 sin : 2 2 b1
3. a2 ¼ 0; b2 – 0. In this case
/ðt; xÞ ¼ ðc4 x þ c2 Þeb2 t þ c3 x þ c1 ; while for w we must distinguish the following subcases: (a) b1 – 0
wðt; xÞ ¼ eb2 t e (b) b1 ¼ 0
wðt; xÞ ¼ eb2 t 4. a1 a2 – 0; b2 –
p ffiffiffiffi 3
b1 x=2
c6 cos
! pffiffiffip pffiffiffip ffiffiffiffiffi ffiffiffiffiffi ! p ffiffiffiffi 3 3 b1 x 3 3 b1 x 3 b ðxc4 þ c2 Þ þ c 5 e b1 x þ 2 : þ c7 sin 2 2 b1
2 c x c c5 x 4 2 þ c6 x þ c7 b2 x3 þ : 24 6 2
4a22
a21
.
In this case we distinguish the following cases. (a) a1 a2 > 0
/ðt; xÞ ¼
A4 t
c1 e
þ
c2 eb2 t A4 b2
!
A4 t
sinðAxÞ þ c3 e
þ
c4 eb2 t A4 b2
! cosðAxÞ
ð7Þ
and wðt; xÞ ¼ f ðxÞeb2 t where f ðxÞ is solution of
f 000 b1 f ¼ c2 sinðAxÞ þ c4 cosðAxÞ; rffiffiffiffiffiffiffiffi ffi where A ¼ 2aa12 . (b) a1 a2 < 0
ð8Þ
R. Tracinà et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 3036–3043
/ðt; xÞ ¼
4
c1 eA t þ
c2 eb2 t A4 b2
!
c4 eb2 t
4
eAx þ c3 eA t þ
3039
!
A4 b2
eAx
ð9Þ
and wðt; xÞ ¼ f ðxÞeb2 t where f ðxÞ is solution of
f 000 b1 f ¼ c2 eAx þ c4 eAx : 4a22
5. a1 a2 – 0; b2 ¼
a21
ð10Þ
.
In this case we distinguish the following cases: (a) a1 a2 > 0
/ðt; xÞ ¼ ðc1 þ c2 tÞeb2 t sinðAxÞ þ ðc3 þ c4 tÞeb2 t cosðAxÞ and wðt; xÞ ¼ f ðxÞeb2 t where f ðxÞ is a solution of Eq. (8). (b) a1 a2 < 0
/ðt; xÞ ¼ ðc1 þ c2 tÞeb2 t eAx þ ðc3 þ c4 tÞeb2 t eAx and wðt; xÞ ¼ f ðxÞeb2 t where f ðxÞ is a solution of Eq. (10). 3. Lie symmetries In order to derive conservation laws for system (1) we perform the corresponding Lie symmetry analysis. The method for finding Lie point symmetries is well known, see for example [3,7,17]. The invariance of system (1) under a Lie group transformations with infinitesimal generator of the form
X ¼ n1
@ @ @ @ þ n2 þ g1 þ g2 @t @x @u @v
ð11Þ
yields a system of equations for the coordinates n1 ðt; x; u; v Þ; n2 ðt; x; u; v Þ; g1 ðt; x; u; v Þ and g2 ðt; x; u; v Þ. By solving this system we get that
n1 ¼ 4p1 t þ p3 ;
n2 ¼ p1 x þ p2 ;
g1 ¼ p4 u þ gðt; xÞ; g2 ¼ ðp1 þ p4 Þv ;
where the constants p1 ; p2 ; p3 ; p4 and the function gðt; xÞ must satisfy the following conditions
b1 p1 ¼ 0; b2 p1 ¼ 0;
a1 ð4p1 þ p4 Þ ¼ 0; a2 ð6p1 þ p4 Þ ¼ 0; g t g xxxxx ¼ 0; b1 g þ g xxx ¼ 0: We observe that, for each value of the constants a1 ; a2 ; b1 and b2 , system (1) admits the symmetry operator
X ¼ p3
@ @ þ p2 : @t @x
ð12Þ
Taking into account that ja1 j þ ja2 j – 0, we obtain the following expressions for the constants p1 ; p4 and for the function g depending on the values of the constants a1 ; a2 ; b1 and b2 . 1. If b1 – 0 we have
p1 ¼ p4 ¼ 0; 1=3
b1 ðxb1 tÞ
gðt; xÞ ¼ p5 e
þe
1=3 b 1 ðxb tÞ 1 2
! pffiffiffi pffiffiffi 3 b11=3 3 b1=3 1 p6 cos ðx b1 tÞ þ p7 sin ðx b1 tÞ ; 2 2
where p5 ; p6 and p7 are arbitrary constants. 2. If b1 ¼ 0 we obtain
gðt; xÞ ¼ p5 x2 þ p6 x þ p7 ; where p5 ; p6 and p7 are arbitrary constants, while for the constants p1 and p4 we must distinguish the following subcases
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(a) b2 – 0 or a1 a2 – 0 we have
p1 ¼ p4 ¼ 0: (b) b2 ¼ 0 and a1 ¼ 0 we have
p4 ¼ 6p1 : (c) b2 ¼ 0 and a2 ¼ 0 we have
p4 ¼ 4p1 : 4. Conservation laws In order to get conservation laws we use the following statement proved in [10,9]. Theorem 1. Any symmetry (Lie point, Lie–Bäcklund, nonlocal symmetry)
X ¼ ni ðx; u; uð1Þ ; . . .Þ
@ @ þ ga ðx; u; uð1Þ ; . . .Þ a @xi @u
of a system of m equations
F a ðx; u; uð1Þ ; . . . ; uðsÞ Þ ¼ 0;
a ¼ 1; . . . m
ð13Þ
with n independent variables x ¼ ðx1 ; . . . ; xn Þ , m dependent variables u ¼ ðu1 ; . . . ; um Þ and where uðsÞ denotes the set of the partial derivatives of s-th order of u, is inherited by the adjoint system. Specifically the operator
Y ¼ ni
@ @ @ þ ga a þ ga a @xi @u @v
with appropriately chosen coefficients ga , is admitted by the system of equations consisting of Eqs. (13) and adjoint equations
F a ðx; u; v ; . . . ; uðsÞ ; v ðsÞ Þ
dðv b F b Þ ¼ 0; dua
a ¼ 1; . . . m:
ð14Þ
Furthermore, the combined system (13), (14) has the conservation law Di ðC i Þ ¼ 0, where
"
! ! # " ! # @L @L @L @L @L a þ D þ D þ . . . D D ðW Þ D j j j k k x x x x x @uai @uaij @uaijk @uaij @uaijk " # @L þ Dxj Dxk ðW a Þ ... þ ; @uaijk
C i ¼ ni L þ W a
ð15Þ
with
W a ¼ ga ni uai ;
a ¼ 1; . . . m:
We apply this theorem to look for conservation laws of system (1). Due to the fact that system (1) is nonlinearly self-adjoint, we will provide conserved vectors for system (1) by substituting (5) into the vector (15). By considering the symmetry operator of system (1) in the form (11) and the formal Lagragian (2) for system (1), Eq. (15) become
@L @L þ W2 ; @ut @v t @L @L @L @L @L @L @L D3x Dx þ D2x þ W2 þ Dx ðW 1 Þ Dx þ Dx ðW 2 Þ C 2 ¼ n2 L þ W 1 D2x @uxxx @uxxxx @v x @ v xx @uxxx @uxxxx @ v xx @L @L @L þ D3x ðW 1 Þ ; Dx þD2x ðW 1 Þ @uxxx @uxxxx @uxxxx
C 1 ¼ n1 L þ W 1
ð16Þ
where
W 1 g1 n1 ut n2 ux ; W 2 g2 n1 v t n2 v x : We compute conservation laws for system (1) in the case 4 of the Section 2, i.e. a1 a2 – 0 and b2 – 1=3
xb1
For sake of simplicity, we consider in (7)–(10) c2 ¼ c3 ¼ c4 ¼ 0 and choose f ðxÞ ¼ c5 e become
4a22
a21
.
. Then the functions / and w
R. Tracinà et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 3036–3043
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4 /ðt; xÞ ¼ c1 eA t sinðAxÞ; 1=3
wðt; xÞ ¼ c5 exb1
b2 t
;
where we recall that A ¼
rffiffiffiffiffiffiffiffi ffi 2a2 a1 and the constants c1 and c5 must satisfy the condition
jc1 j þ jc5 j – 0: Considering the symmetry operator (12) with jp2 j þ jp3 j – 0 we obtain
W 1 ¼ p3 ut p2 ux ; W 2 ¼ p3 v t p2 v x : By setting
s1 ¼ c5 p2 b1=3 p3 b2 ; 1
s2 ¼ c1 p2 ;
s3 ¼ c1 p3
from (16), (1) and (2) we obtain the conserved vector
C ðC 1 ; C 2 Þ s1 C 11 þ s2 C 12 þ s3 C 13 ; s1 C 21 þ s2 C 22 þ s3 C 23 ; where 1=3
C 11 ¼ exb1 C 21
b2 t
1=3 xb1 b2 t
¼e
v;
2=3 b1=3 1 ux uxx b1 u
ð17Þ
and, for a1 a2 > 0, 4
C 12 ¼ AeA t cosðAxÞu; C 22 ¼
a2 A4 t 2 e 4a2 u 2a1uxx a21 v 2 sinðAxÞ þ 2a2 ux a1 uxxx a21 vv x cosðAxÞ ; 2 A a1 4
C 13 ¼ A4 eA t sinðAxÞu; C 23 ¼ 2
a22 A4 t e 2 a1 uxxx 2a2 ux þ a21 vv x sinðAxÞ A 2a1 uxx þ a21 v 2 4a2 u cosðAxÞ ; a31
while, for a1 a2 < 0,
C 12 ¼ AeA C 22 ¼ eA
4
4
tþAx
tþAx
u; a1 Avv x a2 v 2 Auxxx þ A2 uxx A3 ux þ A4 u ;
C 13 ¼ A3 C 12 ; C 23 ¼ A3 C 22 : 5. Exact solutions from conservation laws Given a conservation law
Dt ðC 1 Þ þ Dx ðC 2 Þ ¼ 0 it is possible to look for solutions for system (1) that satisfy the conditions
C 1 ¼ f ðxÞ;
C 2 ¼ gðtÞ;
with suitable functions f ðxÞ and gðtÞ. We use this procedure by considering the conserved vectors obtained previously. In particular we consider the conserved vector ðC 11 ; C 21 Þ given by (17). In this case the conservation law becomes
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1=3 1=3 2=3 Dt exb1 b2 t v þ Dx exb1 b2 t b1=3 ¼ 0: 1 ux uxx b1 u Then we look for solutions of system (1) that satisfy the conditions 1=3
exb1
1=3
exb1
b2 t
b2 t
v ¼ f ðxÞ;
ð18Þ
1=3 2=3 b1 ux uxx b1 u ¼ gðtÞ:
ð19Þ
From the condition (18) we obtain
v ðt; xÞ ¼ f ðxÞexb
1=3 þb2 t 1
ð20Þ
;
while, by supposing b1 – 0, from (19) we have xb
uðt; xÞ ¼ e
! !! pffiffiffi pffiffiffi 1=3 3 1=3 3 1=3 exb1 þtb2 g 1 ðtÞ cos x b1 b1 : þ g 2 ðtÞ sin x gðtÞ 2=3 2 2 3b1
1=3 1 2
ð21Þ
Now we determine the unknown functions g 1 ðtÞ; g 2 ðtÞ; gðtÞ and f ðxÞ requiring that u and v given by (21), (20) must satisfy system (1). We observe that the second equation of system (1) is satisfied, while the first reads
1=3 2=3 2 2 6b12=3 a1 4b1=3 b2 g t a2 f 2 þ 2exb1 tb2 g b4=3 1 ffx 2b1 f ffxx fx 1 pffiffiffi 1=3 ! pffiffiffi 1=3 !! 1=3 5xb 4tb2 pffiffiffi pffiffiffi 3b1 3b1 1 4=3 4=3 2 þ g 1 þ 2b1 g 1t þ 3g 2 cos x g 2 þ 2b1 g 2t 3g 1 sin x þ 3b1 e 2 2 2 ¼ 0:
ð22Þ
In order to find new solutions of system (1) we substitute the functions f ; g; g 1 and g 2 that satisfy Eq. (22) into (21) and (20). For example, setting
g 1 ðtÞ ¼ et g 2 ðtÞ ¼ et
pffiffiffi 4=3 pffiffiffi 4=3 ! t 3b1 t 3b1 ; c1 sin 2 2 ! pffiffiffi pffiffiffi t 3b4=3 t 3b14=3 1 ; c1 cos þ c2 sin 2 2
4=3 b 1 2
c2 cos
4=3 b 1 2
ð23Þ
with c1 and c2 arbitrary constants, (22) becomes
1=3 2=3 2 2 3b12=3 a1 4b1=3 a2 f 2 þ exb1 tb2 g b14=3 b2 g t ¼ 0: 1 ffx 2b1 f ffxx fx
ð24Þ
If b4=3 2b2 – 0, a particular solution of this equation is given by 1 xb
f ðxÞ ¼ c4 e
1=3 1 2
4=3
ðb1 b2 Þt
gðtÞ ¼ c3 e
;
2=3 2 2=3 b2 t 3 c4 b1 2a2 þ a1 b1 e þ ; 2 b4=3 2b2 1
ð25Þ
with c3 and c4 arbitrary constants. Taking into account (23) and these forms of f and g, from (21) and (20) we obtain the following solution for system (1) b
u¼e
1=3 1 2
1=3 1=3 ! !# pffiffiffi pffiffiffi 2c3 b4=3 2b2 eb1 r þ 3c24 b12=3 2a2 þ a1 b2=3 e2tb2 xb1 1 1 3 3 1=3 1=3 r þ c2 cos b r þ c1 sin b r ; 2 1 2 1 6b2=3 2b b4=3 "
1
v ¼ c4 etb x 2
1=3 b 1 2
2
1
;
where r ¼ x b1 t. If a1 a2 < 0, another particular solution of Eq. (24) is given by 1=3
f ðxÞ ¼ c4 eðb1
þA2Þx
;
4=3
gðtÞ ¼ c3 eðb1
b2 Þt
;
with c3 and c4 arbitrary constants. Together with (23) from (21) and (20) we obtain the following solution for system (1) b
u¼e
1=3 1 2
" r
! !# pffiffiffi pffiffiffi 1=3 c3 3 1=3 3 1=3 c2 cos b r þ c1 sin b r 2=3 eb1 r ; 2 1 2 1 3b1 A 2
v ¼ c4 eb tþx : 2
R. Tracinà et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 3036–3043
3043
6. Conclusions We verified that the nonlinear system of dispersive evolution equation (1) is nonlinearly self-adjoint. By using the conservation theorem proposed by Ibragimov, this property was used to obtain conservation laws from the classical symmetries admitted by this system even if the evolution equations (1) do not possess an usual Lagrangian. The knowledge of conservation laws was used to obtain exact solutions of system (1). Acknowledgments M.S.B. and M.L.G. acknowledge the financial support from Junta de Andalucia group FQM-201. M.T. and R.T. thank the support from University of Catania through PRA and from MIUR through PRIN ‘‘Kinetic and macroscopic models for particle transport in gases and semiconductors: analytical and computational aspects’’. M.T. also would like to thank GNFM (Gruppo Nazionale per Fisica-Matematica) for its support. We would like to thank the anonymous referee for his useful comments and interesting suggestions. References [1] Anco SC, Bluman G. Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications. Eur J Appl Math 2002;13:545–66. [2] Anco SC, Bluman G. Direct construction method for conservation laws for partial differential equations. Part II: General treatment. Eur J Appl Math 2002;41:567–85. [3] Bluman GW, Kumei S. Symmetries and differential equations. Applied mathematical sciences. Berlin: Springer; 1989. [4] Galaktionov VA. Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc R Soc Edinburgh Sec A Math 1995;125:225–46. [5] Galaktionov VA, Svirshchevskii SR. Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. Chapman & Hall/CRC; 2007 (Applied mathematics & nonlinear science, Taylor & Francis). [6] Gandarias ML. Weak self-adjoint differential equations. J Phys A Math Theor 2011;44(26):262001. [7] Ibragimov NH. Elementary Lie group analysis and ordinary differential equations. New York: Wiley; 1999. [8] Ibragimov NH. Integrating factors, adjoint equations and Lagrangians. J Math Anal Appl 2006;318(2):742–57. [9] Ibragimov NH. The answer to the question put to me by L.V. Ovsiannikov 33 years ago. Arch ALGA 2006;3:53–80. [10] Ibragimov NH. Quasi-self-adjoint differential equations. Arch ALGA 2007;4:55–60. [11] Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333(1):311–28. [12] Ibragimov NH, Torrisi M, Tracinà R. Self-adjointness and conservation laws of a generalized Burgers equation. J Phys A Math Theor 2011;44(14):145201. [13] Ibragimov NH. Nonlinear self-adjointness and conservation laws. J Phys A Math Theor 2011;44(43):432002. [14] Ibragimov NH. Nonlinear self-adjointness in constructing conservation laws. Arch ALGA 2011;7(8):1–99. [15] Ibragimov NH. Method of conservation laws for constructing solutions to systems of PDEs. Discontinuity, Nonlinearity Complexity 2012;1(4):353–65. [16] Ma WX, Liu Y. Invariant subspaces and exact solutions of a class of dispersive evolution equations. Commun Nonlinear Sci Numer Simul 2012;17(10):3795–801. [17] Olver PJ. Applications of Lie groups to differential equations. Berlin: Springer; 1986. [18] Titov SS. In: Ivanova TP, editor. Aerodynamics. Saratov: Saratov University; 1988. p. 104–9. [19] Tracinà R, Torrisi M. Quasi self-adjoint reaction diffusion systems, in: ICNAAM 2011 International Conference on Numerical and Applied Mathematics, AIP-Conference Proceedings, vol. 1389; 2011. p. 1386–9.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Nonlinear self-adjointness, conservation laws, exact solutions of a system of dispersive evolution equations R. Tracinà b,⇑, M.S. Bruzón a, M.L. Gandarias a, M. Torrisi b a b
Departamento de Matemáticas, Universidad de Cádiz, PO Box 40, 11510 Puerto Real, Cádiz, Spain Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy
a r t i c l e
i n f o
Article history: Received 20 September 2013 Received in revised form 22 November 2013 Accepted 6 December 2013 Available online 3 February 2014
a b s t r a c t In this paper, we classify a ð1 þ 1Þ-dimensional nonlinear system of dispersive evolution equations both from point of view of symmetries and nonlinear self-adjointness. Then, conservation laws are established from the property of nonlinear self-adjointness and from these we obtain exact solutions. Ó 2014 Elsevier B.V. All rights reserved.
Keywords: Nonlinearly self-adjoint system Conservation laws Exact solutions
1. Introduction We consider the ð1 þ 1Þ-dimensional nonlinear system of dispersive evolution equations
F 1 ut ðuxxx þ a1 vv x Þx a2 v 2 ¼ 0; F 2 v t uxxx b1 u b2 v ¼ 0;
ð1Þ
where a1 ; a2 ; b1 ; b1 are constants and satisfy the condition ja1 j þ ja2 j – 0. This system has been studied in [16] in order to find exact solutions by using invariant subspace method. This system offers interesting features, it admits diverse invariant subspaces of solutions defined by 2nd or 3rd order linear differential equations. The invariant subspace method, proposed in [18,4,5], could be considered as a generalized separation of variables method for nonlinear differential equations. It is well known that evolution equation (1) do not possess an usual Lagrangian, that is, there is no function L such that the considered equations are the Euler–Lagrange arising from L. Hence the Noethers theorem cannot be directly applied to obtain conservation laws on the basis of the equations symmetries. This, however, can be overcome by applying the general concept of nonlinear self-adjointness, devised and developed by Ibragimov [8,10,11,13,14], and Gandarias [6], which enables one to establish the conservation laws for any differential equation. Recently, several papers have been devoted to search for self-adjoint equations. This procedure can be applied not only to classes of single differential equations of any order but to the systems where the number of equations is equal to the number of dependent variables (see e.g. [12,19]). It is well known that conservation laws are useful in many respects. As shown in [14,15], they can be used to construct exact solutions different from group invariant solutions. ⇑ Corresponding author. Tel.: +39 0957383088. E-mail addresses: [email protected] (R. Tracinà), [email protected] (M.S. Bruzón), [email protected] (M.L. Gandarias), [email protected] (M. Torrisi). http://dx.doi.org/10.1016/j.cnsns.2013.12.005 1007-5704/Ó 2014 Elsevier B.V. All rights reserved.
R. Tracinà et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 3036–3043
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There exist different approaches to the study of conservation laws for equations without Lagrangian. In [1,2] the interested reader can be find a general treatment of the direct construction method for conservation laws of partial differential equations. Our first step is to investigate and to prove the nonlinear self-adjointness of the evolution system (1) which is the most important step to apply the Ibragimov’s techniques. Then, once obtained the Lie point symmetries, the conservation laws for system (1) can be immediately established. Finally, we show that it is possible to use these results in order to obtain exact solutions. 2. Adjoint system and conditions for self-adjointness For system (1) we introduce the formal Lagrangian
ðut ðuxxx þ a1 vv x Þx a2 v 2 Þ þ v ðv t uxxx b1 u b2 v Þ; Lu
ð2Þ
and v are two new dependent variables. where u The adjoint system for the system of differential equations (1) is defined by
(
F 1 dL ¼ 0; du
ð3Þ
F 2 dL ¼ 0; dv where, in this case,
dL @L @L @L @L @L Dt Dx þ Dx Dx þ þ Dx Dx Dx Dx ; du @u @ut @ux @uxx @uxxxx dL @L @L @L @L @L Dt Dx þ Dx Dx þ þ Dx Dx Dx Dx dv @ v @v t @v x @ v xx @ v xxxx with Dt and Dx the total differentiations with respect to t and x respectively. Taking into account the Eq. (2), the adjoint system (3) for system (1) is
xxxx u t ¼ 0; F 1 v xxx b1 v u xx v 2a2 u v ¼ 0: F 2 v t b2 v a1 u
ð4Þ
According to [13], the system (1) will be nonlinearly self-adjoint if each equations F i (i ¼ 1; 2) of the adjoint system (4) coincides with ki1 F 1 þ ki2 F 2 (i ¼ 1; 2) after the following substitution
¼ /ðt; x; u; v Þ; u
v ¼ wðt; x; u; v Þ
ð5Þ
with /ðt; x; u; v Þ – 0 or wðt; x; u; v Þ – 0. In other words system (1) is said to be nonlinearly self-adjoint if the adjoint system (4) obeys the condition
(
F 1 u¼/ðt;x;u;v Þ; v ¼wðt;x;u;v Þ ¼ k11 F 1 þ k12 F 2 ; F 2 u¼/ðt;x;u;v Þ; v ¼wðt;x;u;v Þ ¼ k21 F 1 þ k22 F 2
ð6Þ
with regular undetermined coefficients kij ði; j ¼ 1; 2Þ. Using the differential consequences of (5), since / and w do not depend on the derivatives ut ; v t ; uxx ; . . ., Eqs. (6) split into the following equations for the coefficients kij ði; j ¼ 1; 2Þ
k11 ¼ /u ;
k12 ¼ /v ;
k21 ¼ wu ;
k22 ¼ wv
and into the system for the substitution (5)
/u ¼ /v ¼ wu ¼ wv ¼ 0; /xxxx /t þ wxxx b1 w ¼ 0; a1 /xx 2a2 / ¼ 0; wt b2 w ¼ 0: It is possible to verify that this system admits always solutions with /ðt; xÞ – 0 or wðt; xÞ – 0, then system (1) is nonlinearly self-adjoint for any value of the constants a1 ; a2 ; b1 and b2 . The value of these constants determines the expressions of the substitutions (5). In order to obtain the explicit forms of the substitutions (5) we distinguish the following cases, where we will use ci (i ¼ 1; 2; . . .) to indicate arbitrary constants.
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1. a1 ¼ 0. In this case
wðt; xÞ ¼ eb2 t f ðxÞ
/ðt; xÞ ¼ 0;
where f ðxÞ is solution of the equation
f 000 b1 f ¼ 0: Then we distinguish two cases (a) b1 ¼ 0
f ðxÞ ¼ c1 x2 þ c2 x þ c3 : (b) b1 – 0
f ðxÞ ¼ c1 e
p ffiffiffiffi 3
b1 x
þ e
p3ffiffiffiffi
b1 x 2
c2 cos
pffiffiffip pffiffiffip ffiffiffiffiffi ffiffiffiffiffi ! 3 3 b1 x 3 3 b1 x þ c3 sin : 2 2
2. a2 ¼ b2 ¼ 0. In this case
/ðt; xÞ ¼ ðc1 x þ c2 Þt þ c3 x þ c4 ;
wðt; xÞ ¼ f ðxÞ
where f ðxÞ must satisfies
f 0000 b1 f c1 x c2 ¼ 0: Then we have two possibilities (a) b1 ¼ 0
wðt; xÞ ¼
c1 4 c2 3 x þ x þ c 5 x2 þ c 6 x þ c 7 : 24 6
(b) b1 – 0
wðt; xÞ ¼ e
p ffiffiffiffi 3
b1 x=2
pffiffiffip pffiffiffip ffiffiffiffiffi ffiffiffiffiffi ! p ffiffiffiffi 3 3 b1 x 3 3 b1 x 3 c1 x þ c2 þ c 5 e b1 x c6 cos þ c7 sin : 2 2 b1
3. a2 ¼ 0; b2 – 0. In this case
/ðt; xÞ ¼ ðc4 x þ c2 Þeb2 t þ c3 x þ c1 ; while for w we must distinguish the following subcases: (a) b1 – 0
wðt; xÞ ¼ eb2 t e (b) b1 ¼ 0
wðt; xÞ ¼ eb2 t 4. a1 a2 – 0; b2 –
p ffiffiffiffi 3
b1 x=2
c6 cos
! pffiffiffip pffiffiffip ffiffiffiffiffi ffiffiffiffiffi ! p ffiffiffiffi 3 3 b1 x 3 3 b1 x 3 b ðxc4 þ c2 Þ þ c 5 e b1 x þ 2 : þ c7 sin 2 2 b1
2 c x c c5 x 4 2 þ c6 x þ c7 b2 x3 þ : 24 6 2
4a22
a21
.
In this case we distinguish the following cases. (a) a1 a2 > 0
/ðt; xÞ ¼
A4 t
c1 e
þ
c2 eb2 t A4 b2
!
A4 t
sinðAxÞ þ c3 e
þ
c4 eb2 t A4 b2
! cosðAxÞ
ð7Þ
and wðt; xÞ ¼ f ðxÞeb2 t where f ðxÞ is solution of
f 000 b1 f ¼ c2 sinðAxÞ þ c4 cosðAxÞ; rffiffiffiffiffiffiffiffi ffi where A ¼ 2aa12 . (b) a1 a2 < 0
ð8Þ
R. Tracinà et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 3036–3043
/ðt; xÞ ¼
4
c1 eA t þ
c2 eb2 t A4 b2
!
c4 eb2 t
4
eAx þ c3 eA t þ
3039
!
A4 b2
eAx
ð9Þ
and wðt; xÞ ¼ f ðxÞeb2 t where f ðxÞ is solution of
f 000 b1 f ¼ c2 eAx þ c4 eAx : 4a22
5. a1 a2 – 0; b2 ¼
a21
ð10Þ
.
In this case we distinguish the following cases: (a) a1 a2 > 0
/ðt; xÞ ¼ ðc1 þ c2 tÞeb2 t sinðAxÞ þ ðc3 þ c4 tÞeb2 t cosðAxÞ and wðt; xÞ ¼ f ðxÞeb2 t where f ðxÞ is a solution of Eq. (8). (b) a1 a2 < 0
/ðt; xÞ ¼ ðc1 þ c2 tÞeb2 t eAx þ ðc3 þ c4 tÞeb2 t eAx and wðt; xÞ ¼ f ðxÞeb2 t where f ðxÞ is a solution of Eq. (10). 3. Lie symmetries In order to derive conservation laws for system (1) we perform the corresponding Lie symmetry analysis. The method for finding Lie point symmetries is well known, see for example [3,7,17]. The invariance of system (1) under a Lie group transformations with infinitesimal generator of the form
X ¼ n1
@ @ @ @ þ n2 þ g1 þ g2 @t @x @u @v
ð11Þ
yields a system of equations for the coordinates n1 ðt; x; u; v Þ; n2 ðt; x; u; v Þ; g1 ðt; x; u; v Þ and g2 ðt; x; u; v Þ. By solving this system we get that
n1 ¼ 4p1 t þ p3 ;
n2 ¼ p1 x þ p2 ;
g1 ¼ p4 u þ gðt; xÞ; g2 ¼ ðp1 þ p4 Þv ;
where the constants p1 ; p2 ; p3 ; p4 and the function gðt; xÞ must satisfy the following conditions
b1 p1 ¼ 0; b2 p1 ¼ 0;
a1 ð4p1 þ p4 Þ ¼ 0; a2 ð6p1 þ p4 Þ ¼ 0; g t g xxxxx ¼ 0; b1 g þ g xxx ¼ 0: We observe that, for each value of the constants a1 ; a2 ; b1 and b2 , system (1) admits the symmetry operator
X ¼ p3
@ @ þ p2 : @t @x
ð12Þ
Taking into account that ja1 j þ ja2 j – 0, we obtain the following expressions for the constants p1 ; p4 and for the function g depending on the values of the constants a1 ; a2 ; b1 and b2 . 1. If b1 – 0 we have
p1 ¼ p4 ¼ 0; 1=3
b1 ðxb1 tÞ
gðt; xÞ ¼ p5 e
þe
1=3 b 1 ðxb tÞ 1 2
! pffiffiffi pffiffiffi 3 b11=3 3 b1=3 1 p6 cos ðx b1 tÞ þ p7 sin ðx b1 tÞ ; 2 2
where p5 ; p6 and p7 are arbitrary constants. 2. If b1 ¼ 0 we obtain
gðt; xÞ ¼ p5 x2 þ p6 x þ p7 ; where p5 ; p6 and p7 are arbitrary constants, while for the constants p1 and p4 we must distinguish the following subcases
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(a) b2 – 0 or a1 a2 – 0 we have
p1 ¼ p4 ¼ 0: (b) b2 ¼ 0 and a1 ¼ 0 we have
p4 ¼ 6p1 : (c) b2 ¼ 0 and a2 ¼ 0 we have
p4 ¼ 4p1 : 4. Conservation laws In order to get conservation laws we use the following statement proved in [10,9]. Theorem 1. Any symmetry (Lie point, Lie–Bäcklund, nonlocal symmetry)
X ¼ ni ðx; u; uð1Þ ; . . .Þ
@ @ þ ga ðx; u; uð1Þ ; . . .Þ a @xi @u
of a system of m equations
F a ðx; u; uð1Þ ; . . . ; uðsÞ Þ ¼ 0;
a ¼ 1; . . . m
ð13Þ
with n independent variables x ¼ ðx1 ; . . . ; xn Þ , m dependent variables u ¼ ðu1 ; . . . ; um Þ and where uðsÞ denotes the set of the partial derivatives of s-th order of u, is inherited by the adjoint system. Specifically the operator
Y ¼ ni
@ @ @ þ ga a þ ga a @xi @u @v
with appropriately chosen coefficients ga , is admitted by the system of equations consisting of Eqs. (13) and adjoint equations
F a ðx; u; v ; . . . ; uðsÞ ; v ðsÞ Þ
dðv b F b Þ ¼ 0; dua
a ¼ 1; . . . m:
ð14Þ
Furthermore, the combined system (13), (14) has the conservation law Di ðC i Þ ¼ 0, where
"
! ! # " ! # @L @L @L @L @L a þ D þ D þ . . . D D ðW Þ D j j j k k x x x x x @uai @uaij @uaijk @uaij @uaijk " # @L þ Dxj Dxk ðW a Þ ... þ ; @uaijk
C i ¼ ni L þ W a
ð15Þ
with
W a ¼ ga ni uai ;
a ¼ 1; . . . m:
We apply this theorem to look for conservation laws of system (1). Due to the fact that system (1) is nonlinearly self-adjoint, we will provide conserved vectors for system (1) by substituting (5) into the vector (15). By considering the symmetry operator of system (1) in the form (11) and the formal Lagragian (2) for system (1), Eq. (15) become
@L @L þ W2 ; @ut @v t @L @L @L @L @L @L @L D3x Dx þ D2x þ W2 þ Dx ðW 1 Þ Dx þ Dx ðW 2 Þ C 2 ¼ n2 L þ W 1 D2x @uxxx @uxxxx @v x @ v xx @uxxx @uxxxx @ v xx @L @L @L þ D3x ðW 1 Þ ; Dx þD2x ðW 1 Þ @uxxx @uxxxx @uxxxx
C 1 ¼ n1 L þ W 1
ð16Þ
where
W 1 g1 n1 ut n2 ux ; W 2 g2 n1 v t n2 v x : We compute conservation laws for system (1) in the case 4 of the Section 2, i.e. a1 a2 – 0 and b2 – 1=3
xb1
For sake of simplicity, we consider in (7)–(10) c2 ¼ c3 ¼ c4 ¼ 0 and choose f ðxÞ ¼ c5 e become
4a22
a21
.
. Then the functions / and w
R. Tracinà et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 3036–3043
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4 /ðt; xÞ ¼ c1 eA t sinðAxÞ; 1=3
wðt; xÞ ¼ c5 exb1
b2 t
;
where we recall that A ¼
rffiffiffiffiffiffiffiffi ffi 2a2 a1 and the constants c1 and c5 must satisfy the condition
jc1 j þ jc5 j – 0: Considering the symmetry operator (12) with jp2 j þ jp3 j – 0 we obtain
W 1 ¼ p3 ut p2 ux ; W 2 ¼ p3 v t p2 v x : By setting
s1 ¼ c5 p2 b1=3 p3 b2 ; 1
s2 ¼ c1 p2 ;
s3 ¼ c1 p3
from (16), (1) and (2) we obtain the conserved vector
C ðC 1 ; C 2 Þ s1 C 11 þ s2 C 12 þ s3 C 13 ; s1 C 21 þ s2 C 22 þ s3 C 23 ; where 1=3
C 11 ¼ exb1 C 21
b2 t
1=3 xb1 b2 t
¼e
v;
2=3 b1=3 1 ux uxx b1 u
ð17Þ
and, for a1 a2 > 0, 4
C 12 ¼ AeA t cosðAxÞu; C 22 ¼
a2 A4 t 2 e 4a2 u 2a1uxx a21 v 2 sinðAxÞ þ 2a2 ux a1 uxxx a21 vv x cosðAxÞ ; 2 A a1 4
C 13 ¼ A4 eA t sinðAxÞu; C 23 ¼ 2
a22 A4 t e 2 a1 uxxx 2a2 ux þ a21 vv x sinðAxÞ A 2a1 uxx þ a21 v 2 4a2 u cosðAxÞ ; a31
while, for a1 a2 < 0,
C 12 ¼ AeA C 22 ¼ eA
4
4
tþAx
tþAx
u; a1 Avv x a2 v 2 Auxxx þ A2 uxx A3 ux þ A4 u ;
C 13 ¼ A3 C 12 ; C 23 ¼ A3 C 22 : 5. Exact solutions from conservation laws Given a conservation law
Dt ðC 1 Þ þ Dx ðC 2 Þ ¼ 0 it is possible to look for solutions for system (1) that satisfy the conditions
C 1 ¼ f ðxÞ;
C 2 ¼ gðtÞ;
with suitable functions f ðxÞ and gðtÞ. We use this procedure by considering the conserved vectors obtained previously. In particular we consider the conserved vector ðC 11 ; C 21 Þ given by (17). In this case the conservation law becomes
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1=3 1=3 2=3 Dt exb1 b2 t v þ Dx exb1 b2 t b1=3 ¼ 0: 1 ux uxx b1 u Then we look for solutions of system (1) that satisfy the conditions 1=3
exb1
1=3
exb1
b2 t
b2 t
v ¼ f ðxÞ;
ð18Þ
1=3 2=3 b1 ux uxx b1 u ¼ gðtÞ:
ð19Þ
From the condition (18) we obtain
v ðt; xÞ ¼ f ðxÞexb
1=3 þb2 t 1
ð20Þ
;
while, by supposing b1 – 0, from (19) we have xb
uðt; xÞ ¼ e
! !! pffiffiffi pffiffiffi 1=3 3 1=3 3 1=3 exb1 þtb2 g 1 ðtÞ cos x b1 b1 : þ g 2 ðtÞ sin x gðtÞ 2=3 2 2 3b1
1=3 1 2
ð21Þ
Now we determine the unknown functions g 1 ðtÞ; g 2 ðtÞ; gðtÞ and f ðxÞ requiring that u and v given by (21), (20) must satisfy system (1). We observe that the second equation of system (1) is satisfied, while the first reads
1=3 2=3 2 2 6b12=3 a1 4b1=3 b2 g t a2 f 2 þ 2exb1 tb2 g b4=3 1 ffx 2b1 f ffxx fx 1 pffiffiffi 1=3 ! pffiffiffi 1=3 !! 1=3 5xb 4tb2 pffiffiffi pffiffiffi 3b1 3b1 1 4=3 4=3 2 þ g 1 þ 2b1 g 1t þ 3g 2 cos x g 2 þ 2b1 g 2t 3g 1 sin x þ 3b1 e 2 2 2 ¼ 0:
ð22Þ
In order to find new solutions of system (1) we substitute the functions f ; g; g 1 and g 2 that satisfy Eq. (22) into (21) and (20). For example, setting
g 1 ðtÞ ¼ et g 2 ðtÞ ¼ et
pffiffiffi 4=3 pffiffiffi 4=3 ! t 3b1 t 3b1 ; c1 sin 2 2 ! pffiffiffi pffiffiffi t 3b4=3 t 3b14=3 1 ; c1 cos þ c2 sin 2 2
4=3 b 1 2
c2 cos
4=3 b 1 2
ð23Þ
with c1 and c2 arbitrary constants, (22) becomes
1=3 2=3 2 2 3b12=3 a1 4b1=3 a2 f 2 þ exb1 tb2 g b14=3 b2 g t ¼ 0: 1 ffx 2b1 f ffxx fx
ð24Þ
If b4=3 2b2 – 0, a particular solution of this equation is given by 1 xb
f ðxÞ ¼ c4 e
1=3 1 2
4=3
ðb1 b2 Þt
gðtÞ ¼ c3 e
;
2=3 2 2=3 b2 t 3 c4 b1 2a2 þ a1 b1 e þ ; 2 b4=3 2b2 1
ð25Þ
with c3 and c4 arbitrary constants. Taking into account (23) and these forms of f and g, from (21) and (20) we obtain the following solution for system (1) b
u¼e
1=3 1 2
1=3 1=3 ! !# pffiffiffi pffiffiffi 2c3 b4=3 2b2 eb1 r þ 3c24 b12=3 2a2 þ a1 b2=3 e2tb2 xb1 1 1 3 3 1=3 1=3 r þ c2 cos b r þ c1 sin b r ; 2 1 2 1 6b2=3 2b b4=3 "
1
v ¼ c4 etb x 2
1=3 b 1 2
2
1
;
where r ¼ x b1 t. If a1 a2 < 0, another particular solution of Eq. (24) is given by 1=3
f ðxÞ ¼ c4 eðb1
þA2Þx
;
4=3
gðtÞ ¼ c3 eðb1
b2 Þt
;
with c3 and c4 arbitrary constants. Together with (23) from (21) and (20) we obtain the following solution for system (1) b
u¼e
1=3 1 2
" r
! !# pffiffiffi pffiffiffi 1=3 c3 3 1=3 3 1=3 c2 cos b r þ c1 sin b r 2=3 eb1 r ; 2 1 2 1 3b1 A 2
v ¼ c4 eb tþx : 2
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6. Conclusions We verified that the nonlinear system of dispersive evolution equation (1) is nonlinearly self-adjoint. By using the conservation theorem proposed by Ibragimov, this property was used to obtain conservation laws from the classical symmetries admitted by this system even if the evolution equations (1) do not possess an usual Lagrangian. The knowledge of conservation laws was used to obtain exact solutions of system (1). Acknowledgments M.S.B. and M.L.G. acknowledge the financial support from Junta de Andalucia group FQM-201. M.T. and R.T. thank the support from University of Catania through PRA and from MIUR through PRIN ‘‘Kinetic and macroscopic models for particle transport in gases and semiconductors: analytical and computational aspects’’. M.T. also would like to thank GNFM (Gruppo Nazionale per Fisica-Matematica) for its support. We would like to thank the anonymous referee for his useful comments and interesting suggestions. References [1] Anco SC, Bluman G. Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications. Eur J Appl Math 2002;13:545–66. [2] Anco SC, Bluman G. Direct construction method for conservation laws for partial differential equations. Part II: General treatment. Eur J Appl Math 2002;41:567–85. [3] Bluman GW, Kumei S. Symmetries and differential equations. Applied mathematical sciences. Berlin: Springer; 1989. [4] Galaktionov VA. Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc R Soc Edinburgh Sec A Math 1995;125:225–46. [5] Galaktionov VA, Svirshchevskii SR. Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. Chapman & Hall/CRC; 2007 (Applied mathematics & nonlinear science, Taylor & Francis). [6] Gandarias ML. Weak self-adjoint differential equations. J Phys A Math Theor 2011;44(26):262001. [7] Ibragimov NH. Elementary Lie group analysis and ordinary differential equations. New York: Wiley; 1999. [8] Ibragimov NH. Integrating factors, adjoint equations and Lagrangians. J Math Anal Appl 2006;318(2):742–57. [9] Ibragimov NH. The answer to the question put to me by L.V. Ovsiannikov 33 years ago. Arch ALGA 2006;3:53–80. [10] Ibragimov NH. Quasi-self-adjoint differential equations. Arch ALGA 2007;4:55–60. [11] Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333(1):311–28. [12] Ibragimov NH, Torrisi M, Tracinà R. Self-adjointness and conservation laws of a generalized Burgers equation. J Phys A Math Theor 2011;44(14):145201. [13] Ibragimov NH. Nonlinear self-adjointness and conservation laws. J Phys A Math Theor 2011;44(43):432002. [14] Ibragimov NH. Nonlinear self-adjointness in constructing conservation laws. Arch ALGA 2011;7(8):1–99. [15] Ibragimov NH. Method of conservation laws for constructing solutions to systems of PDEs. Discontinuity, Nonlinearity Complexity 2012;1(4):353–65. [16] Ma WX, Liu Y. Invariant subspaces and exact solutions of a class of dispersive evolution equations. Commun Nonlinear Sci Numer Simul 2012;17(10):3795–801. [17] Olver PJ. Applications of Lie groups to differential equations. Berlin: Springer; 1986. [18] Titov SS. In: Ivanova TP, editor. Aerodynamics. Saratov: Saratov University; 1988. p. 104–9. [19] Tracinà R, Torrisi M. Quasi self-adjoint reaction diffusion systems, in: ICNAAM 2011 International Conference on Numerical and Applied Mathematics, AIP-Conference Proceedings, vol. 1389; 2011. p. 1386–9.