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Lie symmetry analysis of the Heisenberg equation
Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
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Research paper
Lie symmetry analysis of the Heisenberg equation Zhonglong Zhao∗, Bo Han Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
a r t i c l e
i n f o
Article history: Received 24 November 2015 Revised 4 July 2016 Accepted 4 October 2016 Available online 21 October 2016 Keywords: Heisenberg equation Optimal system Invariant solutions Conservation laws
a b s t r a c t The Lie symmetry analysis is performed on the Heisenberg equation from the statistical physics. Its Lie point symmetries and optimal system of one-dimensional subalgebras are determined. The similarity reductions and invariant solutions are obtained. Using the multipliers, some conservation laws are obtained. We prove that this equation is nonlinearly self-adjoint. The conservation laws associated with symmetries of this equation are constructed by means of Ibragimov’s method. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Lie symmetry analysis method has recently penetrated into most of the areas related to mathematics, such as differential equations [1], Lie algebras [2,3], classical mechanics [4] and rogue wave [5]. As a matter of fact, many practical problems arising from natural phenomenon and engineering technology can be modeled by the nonlinear partial differential equation (NLPDEs). Investigating the explicit solutions of NLPDEs is beneficial for solving these problems. Lie symmetry analysis method can be regarded as one of the most effective methods to derive the explicit solutions. In addition, on the base of symmetries, the integrability of the NLPDEs, such as group classification, optimal system and conservation laws, can be considered successively [6–8]. The conservation law has drawn great attentions of the mathematical physicists. In the past decades, many methods for dealing with the conservation laws are derived, such as the multiplier approach [9], Noether’s approach, the partial Noether’s approach [10], Ibragimov’s method [11]. Noether’s approach and the partial Noether’s approach produce a connection between the symmetries of NLPDEs and the conservation laws. However, they are not applicable to the nonlinear partial differential equations that do not admit a Lagrangian. In order to overcome such difficulties, Ibragimov’s method was proposed [11]. Ibragimov’s method does not require the existence of a Lagrangian and it is based on the concept of an adjoint equation for the nonlinear equations. The resultant conservation laws involve not only the solutions of the original equation, but also the solutions of the adjoint equation. If the equation under consideration is nonlinear self-adjoint [12,13], the conservation vectors obtained are the conservation vectors of the original equation. In this paper, we consider the Heisenberg equation [14] of the form
(ut − uxx )(u + v) + 2u2x = 0, (vt + vxx )(u + v) − 2v2x = 0, ∗
Corresponding author. E-mail address: [email protected] (Z. Zhao).
http://dx.doi.org/10.1016/j.cnsns.2016.10.008 1007-5704/© 2016 Elsevier B.V. All rights reserved.
(1)
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
221
where the subscripts denote the partial derivatives, u(x, t) and v(x, t) are functions with two independent variables x, t and u(x, t ) + v(x, t ) = 0. This equation relates to the famous Heisenberg model St = [S, Sxx ] via the stereographic projection [14]. Heisenberg equation is one of the models used in statistical physics to model ferromagnetism and other phenomena. The present paper aims at obtaining the optimal system, similarity reductions, invariant solutions and conservation laws of the Heisenberg equation. The organization of the paper is as follows. In Section 2, we obtain the Lie point symmetries of the Heisenberg equation using Lie group analysis. Section 3 devotes to constructing an optimal system of one-dimensional subalgebra. In Section 4, we consider the similar reductions and group-invariant solutions of this equation. In Section 5, three local conservations laws of Eq. (1) are obtained via the multipliers. In Section 6, Eq. (1) is proved nonlinearly selfadjoint. The conservation laws of Eq. (1) are established in Section 7 using Ibragimov’s method. The last section contains a summary and discussion. 2. Lie point symmetries In this section, we perform Lie symmetry analysis on the Heisenberg equation. Consider a one-parameter Lie group of transformations
t → t + ε ξ 1 (x, t, u, v), x → x + ε ξ 2 (x, t, u, v), u → u + ε η1 (x, t, u, v),
v → v + ε η2 (x, t, u, v), with a small parameter ε 1. The corresponding generator of the Lie algebra is of the form
X = ξ 1 (t, x, u, v)
∂ ∂ ∂ ∂ + ξ 2 (t, x, u, v ) + η1 (t, x, u, v ) + η2 (t, x, u, v) . ∂t ∂x ∂u ∂v
Thus the second prolongation pr(2) X is
pr (2 ) X = X + ηt1
∂ ∂ ∂ ∂ ∂ ∂ 1 2 + ηt2 + ηx1 + ηx2 + ηxx + ηxx , ∂ ut ∂ vt ∂ ux ∂ vx ∂ uxx ∂ vxx
where
ηt1 = Dt (η1 ) − ut Dt (ξ 1 ) − ux Dt (ξ 2 ), ηt2 = Dt (η2 ) − vt Dt (ξ 1 ) − vx Dt (ξ 2 ), ηx1 = Dx (η1 ) − ut Dx (ξ 1 ) − ux Dx (ξ 2 ), ηx2 = Dx (η2 ) − vt Dx (ξ 1 ) − vx Dx (ξ 2 ), 1 ηxx = Dx (ηx1 ) − uxt Dx (ξ 1 ) − uxx Dx (ξ 2 ), 2 ηxx = Dx (ηx2 ) − vxt Dx (ξ 1 ) − vxx Dx (ξ 2 ), and the operators Dx and Dt are the total derivatives with respect to t and x. The determining equations of Eq. (1) arises from the following invariance condition
pr (2) X (1 )|1 =0 = 0, pr (2 ) X (2 )|2 =0 = 0, where 1 = (ut − uxx )(u + v ) + 2u2x = 0, the partial differential equations
2 = (vt + vxx )(u + v) − 2v2x = 0. Then we obtain the overdetermined system of
ξu1 = 0, ξv1 = 0, ξx1 = 0, ξtt1 = 0, 1 2 2 ηvvv = 0,
ξt2 = 0, ξu2 = 0, ξv2 = 0, ξx2 − ξt1 = 0, ηt2 = 0, ηu2 = 0, ηx2 = 0, 1 2
2 η1 = − (u + v)2 ηvv + (u + v )ηv2 − η1 .
Solving the system, one can get
ξ 1 = c1 t + c5 , c1 x + c6 , 2 c3 2 = − u − c4 + c2 u, 2 c3 2 = v + c2 v + c4 , 2
ξ2 = η1 η2
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where c1 , c2 , c3 , c4 , c5 and c6 are arbitrary constants. Hence the infinitesimal symmetries of Eq. (1) form the six dimensional Lie algebra L6 spanned by the following independent operators:
∂ 1 ∂ + x , ∂t 2 ∂ x ∂ ∂ =u +v , ∂u ∂v 1 ∂ 1 ∂ = − u2 + v2 , 2 ∂u 2 ∂v ∂ ∂ =− + , ∂ u ∂v ∂ = , ∂t ∂ = . ∂x
X1 = t X2 X3 X4 X5 X6
(2)
By solving the following system of ordinary differential equations with initial condition
dt ∗ dε d x∗ dε d u∗ dε d v∗ dε
=
ξ 1 (t ∗ , x∗ , u∗ , v∗ ), t ∗ |ε=0 = t,
=
ξ 2 (t ∗ , x∗ , u∗ , v∗ ), x∗ |ε=0 = x,
=
η1 (t ∗ , x∗ , u∗ , v∗ ), u∗ |ε=0 = u,
=
η2 (t ∗ , x∗ , u∗ , v∗ ), v∗ |ε=0 = v,
we obtain the group transformation which is generated by the infinitesimal generator Xi (i = 1, · · · , 6), respectively
G1 : (t, x, u, v ) → eε t, e 2 ε x, u, v , 1
G2 : u(t, x, u, v ) → (t, x, eε u, eε v ), 2u 2v G3 : (t, x, u, v ) → t, x, , , ε u + 2 −εv + 2 G4 : (t, x, u, v ) → (t, x, u − ε , v + ε ), G5 : (t, x, u, v ) → (t + ε , x, u, v), G6 : (t, x, u, v ) → (t, x + ε , u, v).
(3)
Thus the theorem holds: Theorem 1. If {u(x, t),
v(x, t)} is a solution of Heisenberg equation, so are the functions
G1 (ε ) · u(x, t ) = u e− 2 ε x, e−ε t , 1
G1 (ε ) · v(x, t ) = v e− 2 ε x, e−ε t , 1
G2 (ε ) · u(x, t ) = eε u(x, t ), G2 (ε ) · v(x, t ) = eε v(x, t ), 2u(x, t ) 2v(x, t ) G3 (ε ) · u(x, t ) = , G3 (ε ) · v(x, t ) = , ε u(x, t ) + 2 −εv(x, t ) + 2 G4 (ε ) · u(x, t ) = u(x, t ) − ε , G4 (ε ) · v(x, t ) = v(x, t ) + ε , G5 (ε ) · u(x, t ) = u(x, t − ε ),
G5 (ε ) · v(x, t ) = v(x, t − ε ),
G6 (ε ) · u(x, t ) = u(x − ε , t ),
G6 (ε ) · v(x, t ) = v(x − ε , t ).
(4)
3. Optimal system of one-dimensional subalgebras To classify all the group-invariant solutions, some effective method to classify the subalgebra of Lie algebra generated by Lie point symmetries were proposed by Ovsiannikov [15] and Olver [16] from different angles. In 2010, Ibragimov et al. presented a concise method to get the optimal system [17], which only relies on the commutator table. Then Abdulwahhab constructed an optimal system of the 2-dimensional Burgers equations [18]. We investigated the optimal system of Broer–Kaup system with the aid of the method given in [19]. In this section, we shall construct an optimal system of onedimensional subalgebra of the Lie algebra L6 for Eq. (1).
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Table 1 Table of Lie brackets. [Xi , Xj ]
X1
X2
X3
X4
X5
X6
X1 X2 X3 X4 X5 X6
0 0 0 0 X5 1 X 2 6
0 0 −X3 X4 0 0
0 X3 0 X2 0 0
0 −X4 −X2 0 0 0
−X5 0 0 0 0 0
− 12 X6 0 0 0 0 0
An arbitrary operator X ∈ L6 is written as
X = l 1 X1 + l 2 X2 + l 3 X3 + l 4 X4 + l 5 X5 + l 6 X6 ,
(5)
The Lie brackets table of the operators 2 of Eq. (1) is given in Table 1. To find the linear transformations of the vector l = l 1 , l 2 , l 3 , l 4 , l 5 , l 6 , we use the following generators
Ei = ciλj l j
∂ , i = 1, 2, 3, 4, 5, 6, ∂ lλ
(6)
where ciλj is defined by the formula [Xi , X j ] = ciλj Xλ . According to Eq. (6) and Table 1, E1 , E2 , E3 , E4 , E5 and E6 can be written as
1 ∂ ∂ − l6 , ∂ l5 2 ∂ l6 ∂ ∂ E2 = l 3 3 − l 4 4 , ∂l ∂l ∂ ∂ E3 = −l 4 2 − l 2 3 , ∂l ∂l ∂ ∂ E4 = l 3 2 + l 2 4 , ∂l ∂l ∂ E5 = l 1 5 , ∂l 1 ∂ E6 = l 1 6 . 2 ∂l E1 = −l 5
(7)
For the generators E1 , E2 , E3 , E4 , E5 and E6 , the Lie equations with parameters a1 a2 , a3 , a4 , a5 and a6 with the initial condition l˜ = l, i = 1 · · · 6 are written as ai =0
dl˜1 = 0, d a1
dl˜2 = 0, d a1
dl˜3 = 0, d a1
dl˜4 = 0, d a1
dl˜1 = 0, d a2
dl˜2 = 0, d a2
dl˜3 = l˜3 , d a2
dl˜4 = −l˜4 , d a2
dl˜1 = 0, d a3
dl˜2 = −l˜4 , d a3
dl˜1 = 0, d a4
dl˜2 = l˜3 , d a4
dl˜3 = 0, d a4
dl˜4 = l˜2 , d a4
dl˜1 = 0, d a5
dl˜2 = 0, d a5
dl˜3 = 0, d a5
dl˜1 = 0, d a6
dl˜2 = 0, d a6
dl˜3 = 0, d a6
dl˜3 = −l˜2 , d a3
dl˜5 = −l˜5 , d a1
1 dl˜6 = − l˜6 , d a1 2
(8)
dl˜6 = 0, d a2
(9)
dl˜5 = 0, d a2
dl˜4 = 0, d a3
dl˜5 = 0, d a3
dl˜6 = 0, d a3
(10)
dl˜5 = 0, d a4
dl˜6 = 0, d a4
(11)
dl˜4 = 0, d a5
dl˜5 = l˜1 , d a5
dl˜6 = 0, d a5
(12)
dl˜4 = 0, d a6
dl˜5 = 0, d a6
dl˜6 1 = l˜1 . d a6 2
(13)
The solutions of these equations give the transformations
T1 :
l˜1 = l 1 ,
l˜2 = l 2 ,
l˜3 = l 3 ,
T2 :
l˜1 = l 1 ,
l˜2 = l 2 ,
l˜3 = ea2 l 3 ,
l˜4 = l 4 ,
l˜5 = e−a1 l 5 ,
l˜4 = e−a2 l 4 ,
l˜5 = l 5 ,
1
l˜6 = e− 2 a1 l 6 ,
(14)
l˜6 = l 6 ,
(15)
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1 l˜3 = a23 l 4 − a3 l 2 + l 3 , 2
T3 :
l˜1 = l 1 ,
l˜2 = −a3 l 4 + l 2 ,
T4 :
l˜1 = l 1 ,
l˜2 = a4 l 3 + l 2 ,
T5 :
l˜1 = l 1 ,
l˜2 = l 2 ,
l˜3 = l 3 ,
l˜4 = l 4 ,
l˜5 = a5 l 1 + l 5 ,
T6 :
l˜1 = l 1 ,
l˜2 = l 2 ,
l˜3 = l 3 ,
l˜4 = l 4 ,
l˜5 = l 5 ,
l˜3 = l 3 ,
l˜4 = l 4 ,
1 l˜4 = a24 l 3 + a4 l 2 + l 4 , 2
l˜5 = l 5 , l˜5 = l 5 ,
l˜6 = l 6 , l˜6 = l 6 ,
l˜6 = l 6 ,
1 l˜6 = a6 l 1 + l 6 . 2
(16) (17) (18) (19)
The construction of the optimal system requires a simplification of the vector
l = l1, l2, l3, l4, l5, l6 ,
(20)
by means of the transformations T1 − T6 . We focus on finding a simplest representative of each class of similar vectors (20). The construction will be divided into two cases. Case 1. l1 = 0 5 By taking a5 = − ll 1 in the transformations T5 , we can make l˜5 = 0. The vector (20) is hence reduced to the form
l = l 1 , l 2 , l 3 , l 4 , 0, l 6 .
(21)
6
We take a6 = − 2ll1 in T6 and reduce the vector (21) to the form
l = l 1 , l 2 , l 3 , l 4 , 0, 0 .
(22)
l3
1.1 = 0 2 By taking a4 = − ll 3 in the transformations T4 , we can make l˜2 = 0. The vector (22) is hence reduced to the form
l = l 1 , 0, l 3 , l 4 , 0, 0 .
(23) l1
Without loss of generality, we assume = 1 and make following representatives for the optimal system
X1 ± X3 ,
l3
=
±1, l 4
= ±1 by the transformation T2 . Thus we can obtain the
X1 ± X3 ± X4 .
(24)
l3
1.2 = 0 We consider the vector (22) of the form
l = l 1 , l 2 , 0, l 4 , 0, 0 . 1.2.1 l4 = 0 We take a3 =
1
l2 l4
(25)
in the transformations T3 and reduce the vector (25) to the form
l = l , 0, 0, l 4 , 0, 0 .
(26)
Taking all the possible combinations, we obtain the following representatives
X1 ± X4 .
(27)
l4
1.2.2 = 0 The vector (25) is reduced to the form
l = l 1 , l 2 , 0, 0, 0, 0 .
(28)
Taking all the possible combinations, we obtain the following representatives
X1 ,
X1 ± X2 .
(29)
l1
Case 2. = 0 We consider the vector (20) of the form
l = 0, l 2 , l 3 , l 4 , l 5 , l 6 .
(30)
l3
2.1 = 0 We consider the vector (20) of the form
l = 0, l 2 , 0, l 4 , l 5 , l 6 . 2.1.1
l4
= 0
(31)
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
By taking a3 =
l2 l4
225
in the transformations T3 , we can make l˜2 = 0. The vector (30) is hence reduced to the form
l = 0, 0, 0, l 4 , l 5 , l 6 .
(32)
Taking all the possible combinations, we obtain the following representatives
X4 ,
X4 ± X5 ,
X4 ± X6 ,
X4 ± X5 ± X6 .
(33)
l4
2.1.2 = 0 2.1.2.1 l 2 = 0 Taking all the possible combinations, we obtain the following representatives
X5 ,
X6 ,
X5 ± X6 .
(34)
l2
2.1.2.2 = 0 The vector (31) is reduced to the form
l = 0, l 2 , 0, 0, l 5 , l 6 ,
(35)
that cannot be simplified. Taking all the possible combinations, we obtain the following representatives
X2 ,
X2 ± X5 ,
X2 ± X6 ,
X2 ± X5 ± X6 .
(36)
l3
2.2 = 0 We consider the vector (30) of the form
l = 0, l 2 , l 3 , l 4 , l 5 , l 6 , 2 By taking a4 = − ll 3 in the transformations T4 , we can make l˜2 = 0. The vector (30) is reduced to the form
l = 0, 0, l 3 , l 4 , l 5 , l 6 ,
3
l = 0
l 4 = 0, l 4 = 0,
X3 ± X4 , X3 ± X4 ± X5 , X3 ± X4 ± X6 , X3 , X3 ± X5 , X3 ± X6 X3 ± X5 ± X6 .
X3 ± X4 ± X5 ± X6 ,
(37)
Finally, by collecting the operators (24,27,29,33,34,36 and 37), we arrive at the following theorem: Theorem 2. The following operators provide an optimal system of one-dimensional subalgebras of the Lie algebra spanned by X1 , X2 , X3 , X4 , X5 , X6 of Eq. (1):
X1 ± X3 ,
X1 ± X2 ,
X3 ± X6 ,
X2 ,
X1 ,
X1 ± X4 ,
X1 ± X3 ± X4 ,
X3 ,
X2 ± X5 ,
X2 ± X6 ,
X2 ± X5 ± X6 ,
X3 ± X4 ,
X3 ± X4 ± X5 ,
X4 ± X6 ,
X4 ± X5 ± X6 ,
X3 ± X4 ± X6 ,
X3 ± X4 ± X5 ± X6 ,
X5 ,
X6 .
X5 ± X6 ,
X4 ,
X4 ± X5 ,
X3 ± X5 ± X6 ,
X3 ± X5
(38)
4. Reductions and the invariant solutions of Eq. (1) Given the optimal system of the one-dimensional subalgebras, we can investigate the symmetry reductions of Eq. (1) by integrating the characteristic equations [20]. The characteristic equation related to determine the invariant surface condition. The characteristic equation is actually the associated Lagrange’s system for the calculation of the characteristics. Lagrange uses the characteristics to reduce the order of ordinary differential equations. The solutions of Eq. (1) can be obtained by solving the reduced equations. 4.1. Solutions through X5 + X6 For the generator X5 + X6 , the characteristic equation is written as
dt dx du dv = = = . 1 1 0 0
(39)
The solution of the characteristic equations yields three invariants of X5 + X6
ξ = −x + t, θ 1 = u, θ 2 = v.
(40)
Thus, the solution of Eq. (1) is given by the invariant form
u = f (−x + t ),
v = g(−x + t ).
(41)
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Fig. 1. Plots of the u(x, t) given by Eq. (43) for parameters c1 = 0.2, c2 = 0.3, c3 = −0.3, c4 = 1. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
Fig. 2. Plots of the v(x, t) given by Eq. (44) for parameters c1 = 0.2, c2 = 0.3, c3 = −0.3, c4 = 1. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
Substitution of Eq. (41) into the system (1) leads to f(ξ ) and g(ξ ) satisfying the reduced ordinary differential equations (ODEs)
f f + f g − f f − f g + 2 f = 0, 2
f g + gg + g f + g g − 2g = 0. 2
(42)
Solving this reduced equation, we obtain the closed-form solution of Eq. (1) as follows
u(x, t ) =
1 −x+t+c3 2 c1 1 −x + c4 , +t+c3
−2 c2 sinh
c1 cosh
2
(43)
c1
2 c3 c3 − c1 c4 − 2c2 cosh 12 −x+ct+ (2 c1 c2 − c1 c4 ) sinh 12 −x+ct+ 1 1 1 −x+t+c3 1 −x+t+c3 v(x, t ) = , c1 cosh
2
c1
+ sinh
2
c1
c1
(44)
where ci , i = 1, 2, 3, 4 are arbitrary constants. Fig. 1 displays the kink type of solution structure of u(x, t) determined by (43). Fig. 2 illustrates that the Heisenberg equation has the hyperbolic solution v(x, t) (44). 4.2. Solutions through X2 + X6 The symmetry X2 + X6 gives rise to the group-invariant solution of the form
u = f (t )ex ,
v = g(t )ex .
(45)
Substituting Eq. (45) into Eq. (1) results in the ODEs where f and g satisfy
f f + f g + f 2 − f g = 0, g f + g g + f g − g2 = 0.
(46)
Solving this reduced equation, we obtain the closed-form solution of Eq. (1) as follows
u(x, t ) = c2 ec1 t+x ,
v(x, t ) = −
c2 (1 + c1 )ec1 t+x , c1 − 1
(47) (48)
where ci , i = 1, 2, are arbitrary constants. Figs. 3 and 4 display that the Heisenberg equation has the exponential solutions u(x, t) (47) and v(x, t) (48).
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
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Fig. 3. Plots of the u(x, t) given by Eq. (47) for parameters c1 = 0.2, c2 = 0.3. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
Fig. 4. Plots of the v(x, t) given by Eq. (48) for parameters c1 = 0.2, c2 = 0.3. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
Fig. 5. Plots of the u(x, t) given by Eq. (51) for parameters c1 = 0.2, c2 = 0.3. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
4.3. Solutions through X3 + X6 The symmetry X3 + X6 gives rise to the group-invariant solution of the form
u(x, t ) =
2 , −x + 2g(t )
v(x, t ) =
2 , x + 2 f (t )
(49)
Substituting Eq. (49) into Eq. (1) results in the ODEs where f and g satisfy
−16 f f − 16 f g + 8 = 0, −16g f − 16g g − 8 = 0.
(50)
Solving this reduced equation, we obtain the closed-form solution of Eq. (1) as follows
u(x, t ) =
−2c1 , −c1 x + 2c12 t + 2 c1 c2 − 1
(51)
v(x, t ) =
2 , −x + 2 c1 t + 2 c2
(52)
where ci , i = 1, 2, are arbitrary constants. Figs. 5 and 6 present that the Heisenberg equation has inversely proportional solutions u(x, t) (51) and v(x, t) (52).
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Fig. 6. Plots of the v(x, t) given by Eq. (52) for parameters c1 = 0.2, c2 = 0.3. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis. Table 2 Reductions of the Heisenberg equation. Case
Similarity variable
Reduced equations − f g − f f − 2 + 8ξ f − 8ξ 2 f − g − f + 4ξ 2 f g +4ξ 2 f f + 6ξ f g + 6ξ f f = 0, 2 −g g − g f + 2 + 8ξ g + 8ξ 2 g − g − f − 4ξ 2 g g −4ξ 2 g f − 6ξ g g − 6ξ g f = 0. 2
(1) X1 + X3
ξ = xt2 , u(x, t ) = v(x, t ) =
(2) X1 + X2
ξ = xt2 , u(x, t ) = f (ξ )x2 , v(x, t ) = g(ξ )x2 .
f f + f g − 4ξ 2 f f − 4ξ 2 f g − 14ξ f f + 2ξ f g 2 +6 f 2 − 2 f g + 8ξ 2 f = 0, g f + g g + 4ξ 2 g f + 4ξ 2 g g − 2ξ g f + 14ξ g g 2 +2 f g − 6g2 − 8ξ 2 g = 0.
(3) X1 + X4
ξ = xt2 , u(x, t ) = −2 ln (x ) + f (ξ ), v(x, t ) = 2 ln (x ) + g(ξ ).
(4) X2 + X5 + X6
ξ = −x + t, u(x, t ) = f (ξ )ex , v(x, t ) = g(ξ )ex .
f f + f g − 2 f − 2g − 4ξ 2 f f − 4ξ 2 f g − 6ξ f f 2 −6ξ f g + 8 + 16ξ f + 8ξ 2 f = 0, g f + g g − 2 f − 2g + 4ξ 2 g f + 4ξ 2 g g + 6ξ g f 2 +6ξ g g − 8 + 16ξ g − 8ξ 2 g = 0.
(5) X3 + X4 + X5
ξ = x, √ √ √ u(x, t ) = − 2 tan 22 t + 22 f (ξ ) , √ √ √ v(x, t ) = 2 tan 22 t + 22 g(ξ ) .
(6) X4 + X5 + X6
ξ = −x + t, u(x, t ) = −x + f (ξ ), v(x, t ) = x + g(ξ ).
1 , ln (x )+ f (ξ ) 1 , − ln (x )+g(ξ )
− f f + 3 f g − f f − f g + f 2 − f g + 2 f = 0, 2 −g f + 3g g + g f + g g + f g − g2 − 2g = 0. √ √ √ √ √ 2 − 2 f sin 22 g cos 22 f − 2 f cos 22 f cos 22 g √ √ √ √ √ √ + 2 sin 22 g cos 22 f + 2 f sin 22 f cos 22 g √ √ √ √ √ 2 − 2 sin 22 f cos 22 g − 2 f sin 22 f sin 22 g = 0, √ √ √ √ √ 2 2g sin 22 f cos 22 g + 2g cos 22 g cos 22 f √ √ √ √ √ 2 +2g sin 22 g sin 22 f − 2g sin 22 g cos 22 f √ √ √ √ √ √ − 2 sin 22 g cos 22 f + 2 sin 22 f cos 22 g = 0. 2
f f + f g − f f − f g + 2 + 4 f + 2 f = 0, 2 g f + g g + g f + g g − 2 + 4g − 2g = 0. 2
4.4. Some other of similarity reductions and invariant solutions of Eq. (1) Some other of the similarity reductions for the optimal system of the subalgebras and invariant solutions are listed in Tables 2 and 3, respectively. 5. Construction of conservation laws for Eq. (1) via multipliers α a kth-order system of the partial 1 Consider 1differential equations (PDEs) R [u] with n independent variables x = 2 n 2 m
x ,x ,··· ,x
and m dependent variables u = u , u , · · · , u
Rα [u] = Rα x, u, u(1 ) , · · · , u(k ) = 0,
α = 1, 2, · · · , m,
(53)
where u(1) , u(2) , , u(k) denote the collections of all first, second,...,kth-order derivatives. A local divergence-type conservation law of the PDE system (53) is a divergence expression of the form
Di i [u] = D1 1 [u] + · · · + Dn n [u],
(54)
in terms of the total derivative operators holding for the solutions of (53). There exists a set of multipliers, called local conservation laws multipliers,
α [u] = α x, u, ∂ u, · · · , ∂ l u ,
α = 1, 2, · · · m,
(55)
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
229
Table 3 Corresponding invariant solutions of similarity reductions in Table 2. Case
Invariant solutions u(x, t ) = ln x+1f (ξ ) , v (x, t ) = − ln x1+g(ξ ) , f (ξ ) = h (ξ )dξ + c2 , 8ξ 2 f + f f + 2 − 8ξ f − 4ξ 2 f f − 6ξ f f + f , g(ξ ) = − f − 1 + 4ξ 2 f + 6ξ f 1 1 3 2 2 h = − 16 ξ − 12 ξ ξ h −64 c ξ + 1 − 20ξ 2 h + 64 c + 1 1 4 ξ (2ξ h−1)
Case 1
−48ξ 5 h − 48ξ 3 h + 16c1 + ξ , 2
ξ=
t x2
.
u(x, t ) = 0,
1 x2
3
Case 2
v(x, t ) =
t 2 e− 4 t √ √ −c1 x − c2 π x + c2 x π erf 12
x √ t
1 √ + 2 c 2 te − 4
x2 t
.
u(x, t ) = −2 ln x + f (ξ ), v (x, t ) = 2 ln x + g(ξ ), f (ξ ) = h (ξ )dξ + c2 2 f f + 16ξ f − 2 f − 4ξ 2 f f − 6ξ f f + 8ξ 2 f + 8 g(ξ ) = , 2 − f + 6ξ f + 4ξ f + 2 1 1 ξ − 12ξ 3 + 32c1 ξ 2 h2 + 64c1 ξ − 2 + 40ξ 2 h h = − 32 ξ 4 (1 + hξ )
Case 3
+96ξ 3 h − 48ξ 5 h + 32c1 + 4ξ , 2
ξ=
t x2
.
u(x, t ) = 0, Case 4
v(x, t ) =
√ − 5e
√
5 1 x− x+ 2 2
√
5 3 + 2 2
t
. c1 − c2 √ √ √ u(x, t ) = − 2 tan 22 t + 22 f (x ) , Z 1 √ g(x ) = RootOf ±
Case 5
f (x ) =
√ e 5(−x+t )
√ 2 arctan
√ √ √ v(x, t ) = 2 tan 22 t + 22 g(x ) , dα + x + c3 dx + c4 , 4 2 3 −2 α√ +1+2 √c1 α −2 c√2 α √ √
2 2 2
−2g cos 2
2g 2 sin
2
g + 2 sin
√ √ 2 2
g + 2 cos
g + 2g sin
2
√ √ 2 2
g + 2g cos
2
g
√
, v (x, t ) = x + g(ξ ), u(x, t ) = −x + f (ξ ) Z √ f (ξ ) = RootOf ± 3
2 2
g
.
1 −2c1 α + (2c2 −6c1 )α 2 + (−6c1 +4c2 −2)α −1+2c2 −2c1
dα
+ ξ + c 3 )d ξ + c 4 , 2 − f f − 4 f + f f − 2 − 2 f , g( ξ ) = − f + f ξ = −x + t.
Case 6
such that
Di i [u] ≡ α [u]Rα [u],
(56)
holds for arbitrary u(x). Theorem 3 [21]. For any divergence expression Di i [u], one has
Eu j Di i [u] ≡ 0,
j = 1, 2, · · · m,
(57)
where Eu j = ∂ j − Di ∂ j + · · · + (−1 )s Di1 · · · Dis ∂u
∂ ui
∂ + · · · is the Euler operator with respect to uj . ∂ uij ···i 1
s
Theorem 4 [21]. A set of local multipliers α (x, u, ∂ u, , ∂ l u) yields a divergence expression for PDE system (53) if and only if
Eu j α x, u, ∂ u, · · · , ∂ l u Rα x, u, ∂ u, · · · , ∂ k u
≡ 0,
j = 1, 2, · · · m,
(58)
holds for arbitrary u(x). According to Theorems 3 and 4, for Eq. (1), it follows that all the local conservation laws multipliers are of the form
1 = 1 (x, t, u, v), 2 = 2 (x, t, u, v). Then
(59)
Eu 1 (ut − uxx )(u + v ) + 2u2x + 2 (vt + vxx )(u + v ) − 2v2x Ev 1 (ut − uxx )(u + v ) +
2u2x
+ 2 (vt + vxx )(u + v ) − 2v
2 x
≡ 0, ≡ 0,
(60)
230
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234 Table 4 Conservation laws of the Heisenberg equation. Case
Multipliers
1 2 1 2 1 2
Case 1 Case2 Case3
= =
Conservation Laws
1 v2 2 ( u + v )3 1 u2 2 ( u + v )3
,
Dt
,
= v 3, ( u+v ) = = =
−u ( u + v )3 1 ( u + v )3 1 ( u + v )3
1
uv 2 u+v
+ Dx
1 −v 2 u x + u 2 v x 2 ( u + v )2
Dt − u+v v + Dx − vux +uv2x
,
( u+v )
,
Dt − u+1 v + Dx
,
vx −ux (u+v)2
= 0.
= 0.
= 0.
where the Euler operators Eu and Ev are given by
∂ ∂ ∂ ∂ − Dt − Dx + D2x ··· , ∂u ∂ ut ∂ ux ∂ uxx ∂ ∂ ∂ ∂ Ev = − Dt − Dx + D2x + ··· . ∂v ∂ vt ∂ vx ∂ vxx Eu =
Eq. (60) splits with respect to the second derivatives of u and v with the aid of the Maple package GeM [22] to yield the following determining system
−2 1 u v − 6 1 − 2 1 u u = 0, 2 2 v v + 6 2 + 2 2 v u = 0,
2 vv v + 4 2 v + 2 vv u = 0, −4 1 u − 1 uu v − 1 uu u = 0, −6 1 x − 2 1 xu u − 2 1 xu v = 0, −2 1 x − 2 1 xv v − 2 1 xv u = 0, 2 2 x + 2 2 xu u + 2 2 xu v = 0, 6 2 x + 2 2 xv v + 2 2 xv u = 0, −2 1 uv v − 2 1 uv u − 2 1 u − 6 1 v = 0, − 1 vv v − 1 vv u − 2 1 v − 2 2 u = 0, − 1 xx v − 1 t v − 1 xx u − 1 t u = 0, 2 2 uv v + 2 2 uv u + 2 2 v + 6 2 u = 0, 2 xx v + 2 xx u − 2 t v − 2 t u = 0, 2 1 v + 2 2 u + 2 uu v + 2 uu u = 0, 2 u u − 1 v u − 1 v v + 2 u v − 1 + 2 = 0, 1 + 1 v u − 2 + 1 v v − 2 u v − 2 u u = 0.
(61)
The solutions of the determining system (61) are given by
1 (x, t, u, v) =
1 c 2 2 1
1 v + c2 v + c3 c1 u 2 − c2 u + c3 , 2 (x, t, u, v) = 2 , 3 (u + v ) (u + v )3
(62)
where c1 , c2 and c3 are arbitrary constants. The solution yields the three local conservation laws multipliers
1 = 2 =
(1 )
1 v2 , 2 ( u + v )3 1 u2 , 2 ( u + v )3
1 = (2 ) = 2
v , ( u + v )3 −u , ( u + v )3
1 = (3 ) 2 =
1
( u + v )3
,
1 . ( u + v )3
(63)
By utilizing the direct method given in [10], each multiplier determines a corresponding flux as given in Table 4. 6. Nonlinear self-adjointness In this section, we prove that the Heisenberg Eq. (1) is nonlinear self-adjoint. Consider a system of m PDEs
Rα x, u, · · · , u(k ) = 0,
α = 1, · · · , m.
(64)
The system of the adjoint equations for the kth-order system of PDEs (64) is defined as follows
(Rα )∗ x, u, v, · · · , u(k) , v(k) = 0,
α = 1, · · · , m,
(65)
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
where
(Rα )∗ x, u, v, · · · , u(k) , v(k) =
δ vβ R β , α , β = 1, · · · , m, v = v(x ). δ uα
231
(66)
The Euler–Lagrange operator is defined as
δ
∂
= δ uα
+ ∂ uα
∞
(−1 ) j Di1 · · · Di j
j=1
the formal Lagrangian is
∂ , α = 1, 2, · · · , m, ∂ uαi1 ···i j
(67)
L = vβ Rβ x, u, · · · , u(k ) .
(68)
Definition 1 [12,13]. The system (64) is said to be nonlinearly self-adjoint if the adjoint system is satisfied for all the solutions u of system (64) upon a substitution v = ϕ (x, u ) such that ϕ (x, u) = 0. Specifically, the system
(Rα )∗ x, u, ϕ , · · · , u(k) , ϕ(k) = 0, α = 1, · · · , m,
(69)
is identical to the system
λβα Rβ x, u, u, · · · , u(k) , u(k) = 0, β = 1, · · · , m,
(70)
that means
(Rα )∗ v=ϕ (x,u) = λβα Rβ , β = 1, · · · , m,
(71)
β
where λα is a certain function. Theorem 5 [23]. The determining system of the multiplier (x, u) for system (64) is identical to the system for the substitution of nonlinear self-adjointness. If the formal Lagrangian of Eq. (1) is written as
L = ϕ 1 (x, t, u, v) (ut − uxx )(u + v ) + 2u2x + ϕ 2 (x, t, u, v ) (vt + vxx )(u + v ) − 2v2x ,
(72)
according to Theorem 5, one can get
1 c1 v2 + c2 v + c3 ϕ (x, t, u, v) = 1 (x, t, u, v) = 2 , (u + v )3 1 c1 u 2 − c2 u + c3 2 2 ϕ (x, t, u, v) = 2 (x, t, u, v) = . (u + v)3 1
(73)
Therefore, Eq. (1) is nonlinearly self-adjoint with substitution (73). 7. Construction of conservation laws for Eq. (1) via Ibragimov’s method Theorem 6 [12] [Ibragimov’s method]. Let the system of differential Eq. (64) be nonlinearly self-adjoint. Then every Lie point, Lie-Bäcklund, nonlocal symmetry
∂ ∂ + ηα x, u, u(1 ) , · · · , i ∂ uα ∂x
X = ξ i x, u, u(1 ) , · · ·
(74)
admitted by the system of Eq. (64) gives rise to a conservation law, where the components C i of the conserved vector C = 1 C , · · · , C n are determined by
∂L ∂L ∂L − D + D D − · · · j j k ∂ uαi ∂ uαi j ∂ uαi jk
∂L ∂L ∂L α +D j (W α ) − D + · · · + D D W − · · · , ) j k( k ∂ uαi j ∂ uαi jk ∂ uαi jk
Ci = W α
(75)
with W α = ηα − ξ j uαj . The formal Lagrangian L should be written in the symmetric form with respect to all mixed derivatives uα , uα ,···. ij i jk The Lagrangian L is
L = 1 (ut − uxx )(u + v ) + 2u2x + 2 (vt + vxx )(u + v ) − 2v2x .
(76)
232
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
∂ , from Theorem 1 we have W 1 = η1 − ξ 1 u − ξ 2 u , W 2 = η2 − ξ 1 v − ξ 2 v , For the generator X = ξ 1 ∂∂t + ξ 2 ∂∂x + η1 ∂∂u + η2 ∂v x x t t hence we can obtain the following components of the conserved vector
t
C =W
1
Cx = W 1
∂L ∂ ut
+W
2
∂L ∂L − Dx ∂ ux ∂ uxx
∂L , ∂ vt
+ W2
(77)
∂L ∂L − Dx ∂ vx ∂ vxx
+ Dx W 1
∂L ∂L + Dx W 2 . ∂ uxx ∂ vxx
(78)
Taking the formal Lagrangian L given by (76) into (77,78), we can simplify C t , C x as follows
C t = W 1 1 (u + v ) + W 2 2 (u + v ),
(79)
C x = W 1 (4ux + Dx ( 1 (u + v ) ) ) + W 2 (−4vx − Dx ( 2 (u + v ) ) )
+Dx W 1 1 (−u − v ) + Dx W 2 2 (u + v ).
(80)
The Lie point symmetry generator X1 = t ∂∂t + 12 x ∂∂x has the Lie characteristic functions W 1 = −t ut − 12 xux and W 2 = −t vt − 12 xvx . According to the formulas (79,80), the symmetry generator X1 can give rise to the following components of the conserved vector
C1t
C1x
=
−t ut − 12 xux
1
c 2 1
v2 + c2 v + c3
( u + v )2
+
−t vt − 12 xvx
1
c u2 2 1 2
− c2 u + c3
(u + v )
,
v2 + c2 v + c3 ( u x + vx ) 4ux + −2 (u + v )2 (u + v )3 1 2 c u − c2 u + c3 ( u x + vx ) 1 c1 u u x − c2 u x 2 1 + −t vt − xvx −4vx − −2 2 (u + v)2 (u + v )3 −t utx − 12 ux − 12 xuxx 12 c1 v2 + c2 v + c3 −t vtx − 12 vx − 12 xvxx 12 c1 u2 − c2 u + c3 − + . (u + v )2 ( u + v )2
1 = −t ut − xux 2
1
c 2 1
c1 vvx + c2 vx
∂ , we have W 1 = u and W 2 = v. Thus formulas (79,80) yield the following components For the generator X2 = u ∂∂u + v ∂v of the conserved vector
C2t
=
u
1
c 2 1
v2 + c2 v + c3 v 12 c1 u2 − c2 u + c3 + , (u + v)2 ( u + v )2
C2x
v2 + c2 v + c3 ( u x + vx ) = u 4ux + −2 (u + v)2 (u + v)3 1 2 c u − c2 u + c3 ( u x + vx ) vx 12 c1 u2 − c2 u + c3 ux 12 c1 v2 + c2 v + c3 c1 uux −c2 ux 2 1 +v −4vx − +2 − + . (u + v )2 ( u + v )3 (u + v )2 (u + v)2 c1 vvx + c2 vx
1
c 2 1
∂ , we have W = − 1 u2 and W 2 = For the generator X3 = − 21 u2 ∂∂u + 12 v2 ∂v 1 2 components of the conserved vector
1 2 2v .
Thus formulas (79,80) yield the following
2 1 2 v + c2 v + c3 1 v 2 c1 u − c2 u + c3 + , 2 ( u + v )2 (u + v )2 1 2 c v + c v + c u + v ( ) x x 1 2 3 1 c v v + c v x 1 2 x 2 C3x = − u2 4ux + −2 2 (u + v)2 (u + v)3 1 2 c u − c2 u + c3 ( u x + vx ) uux 12 c1 v2 + c2 v + c3 vvx 12 c1 u2 − c2 u + c3 c1 u u x − c2 u x 1 2 2 1 + v −4vx − + 2 + + . 2 ( u + v )2 (u + v)3 (u + v )2 (u + v)2 C3t
1u =− 2
2 1 c 2 2 1
∂ , we have W 1 = −1 and W 2 = 1. Thus formulas (79,80) yield the following components For the generator X4 = − ∂∂u + ∂v of the conserved vector 1
C4t = − 2
c1 v2 + c2 v + c3
(u + v )
2
+
1 c u2 2 1
− c2 u + c3
(u + v)2
,
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
C4x = −4ux −
c1 vvx + c2 vx
(u + v )2
1 +2
c 2 1
233
1 2 v2 + c2 v + c3 ( u x + vx ) c u − c2 u + c3 ( u x + vx ) c1 u u x − c2 u x 2 1 − 4 v − + 2 . x (u + v )3 (u + v)2 (u + v )3
For the generator X5 = ∂∂t , we have W 1 = −ut and W 2 = −vt . Thus formulas (79,80) yield the following components of the conserved vector
v2 + c2 v + c3 vt 12 c1 u2 − c2 u + c3 =− − , ( u + v )2 (u + v)2 1 2 c v + c2 v + c3 ( u x + vx ) c1 vvx + c2 vx 2 1 x C5 = −ut 4ux + −2 (u + v)2 (u + v)3 1 2 c u − c2 u + c3 ( u x + vx ) c1 u u x − c2 u x 2 1 −vt −4vx − +2 ( u + v )2 (u + v)3 1 2 1 2 utx 2 c1 v + c2 v + c3 vtx 2 c1 u − c2 u + c3 + − . (u + v)2 ( u + v )2 C5t
ut
1
c 2 1
For the generator X6 = ∂∂x , we have W 1 = −ux and W 2 = −vx . Thus formulas (79,80) yield the following components of the conserved vector
v2 + c2 v + c3 vx 12 c1 u2 − c2 u + c3 =− − , (u + v )2 (u + v)2 1 2 c v + c2 v + c3 ( u x + vx ) c1 vvx + c2 vx 2 1 x C6 = −ux 4ux + −2 ( u + v )2 ( u + v )3 1 2 c u − c u + c u + v uxx 12 c1 v2 + c2 v + c3 vxx 12 c1 u2 − c2 u + c3 ( ) x x 1 2 3 c1 u u x − c2 u x 2 −vx −4vx − +2 + − . (u + v )2 (u + v )3 (u + v)2 (u + v)2 C6t
ux
1
c 2 1
8. Conclusions Lie symmetry analysis is performed on the Heisenberg equation. We obtain the Lie point symmetries of this equation and discuss the optimal system of one-dimensional subalgebra of the Lie algebra L6 . Some similar reductions and groupinvariant solutions to the equation are considered based on the optimal system. The conservation laws are obtained for Heisenberg equation via the multipliers. Meanwhile, we demonstrate that Heisenberg equation is nonlinearly self-adjoint. Based on Ibragimov’s method, some nontrivial conservation laws for this equation are obtained. Further research may focus on getting some solutions by means of the obtained conservation laws. Acknowledgement The authors are grateful to the reviewers for the suggested revisions and comments. This work is supported by the National Natural Science Foundation of China under Grant No. 41474102. References [1] Bluman GW, Cheviakov AF, Anco SC. Applications of symmetry methods to partial differential equations. Springer; 2010. [2] Ibragimov NH, Gainetdinova AA. Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional vessiot-guldberg-lie algebras. Appl Math Lett 2016;52:126–31. [3] Manno G, Oliveri F, Saccomandi G, Vitolo R. Ordinary differential equations described by their lie symmetry algebra. J Geom Phys 2014;85:2–15. [4] Ibragimov NH, Kovalev VF, Meleshko SV. Group analysis of kinetic equations in a non-linear thermal transport problem. Int J Non-linear Mech 2015;71:1–7. [5] Wang GW. Symmetry analysis and rogue wave solutions for the (2+1)-dimensional nonlinear schrödinger equation with variable coefficients. Appl Math Lett 2016;56:56–64. [6] Sheftel MB. Lie groups and differential equations: symmetries, conservation laws, and exact solutions of mathematical models in physics. Phys Part Nucl 1997;28:241–66. [7] Nadjafikhah M, Ahangari F. Symmetry analysis and conservation laws of the geodesic equations for the reissner-nordström de sitter black hole with a global monopole. Commun Nonlinear Sci Numer Simul 2012;17:2350–61. [8] Nadjafikhah M, Shirvani-Sh V. Lie symmetries and conservation laws of the hirota-ramani equation. Commun Nonlinear Sci Numer Simul 2012;17:4064–73. [9] Naz R. Conservation laws for some compacton equations using the multiplier approach. Appl Math Lett 2012;25:257–61. [10] Naz R, Mahomed FM, Mason DP. Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics. Appl Math Comput 2008;205:212–30. [11] Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333:311–28.
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Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234 Ibragimov N.H.. Nonlinear self-adjointness in constructing conservation laws, 1–99, arXiv:1109.1728v1. Ibragimov NH. Nonlinear self-adjointness and conservation laws. J Phys A 2011;44 432002(18). Svinolupov SI, Sokolov VV. Deformations of triple-jordan systems and integrable equations. Theor Math Phys 1996;108:1160–3. Ovsiannikov LV. Group analysis of differential equations. Academic Press; 1982. Olver PJ. Application of lie groups to differential equations. Springer; 1993. Grigoriev YN, Ibragimov NH, Kovalev VF, Meleshko SV. Symmmetry of integro-differential equations: with applications in mechanics and plasma physica. Springer; 2010. Abdulwahhab MA. Optimal system and exact solutions for the generalized system of 2-dimensional burgers equations with infinite reynolds number. Commun Nonlinear Sci Numer Simul 2015;20:98–112. Zhao ZL, Han B. On optimal system, exact solutions and conservation laws of the broer-kaup system. Eur Phys J Plus 2015;130 223(115). Bluman GW, Anco SC. Symmetry and integration methods for differential equations. Springer; 2002. Ganghoffer JF, Mladenov I. Similarity and symmetry methods. Springer; 2014. Cheviakov AF. Gem software package for computation of symmetries and conservation laws of differential equations. Comp Phys Comm 2007;176:48–61. Zhang ZY. On the existence of conservation law multiplier for partial differential equations. Commun Nonlinear Sci Numer Simul 2015;20:338–51.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research paper
Lie symmetry analysis of the Heisenberg equation Zhonglong Zhao∗, Bo Han Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
a r t i c l e
i n f o
Article history: Received 24 November 2015 Revised 4 July 2016 Accepted 4 October 2016 Available online 21 October 2016 Keywords: Heisenberg equation Optimal system Invariant solutions Conservation laws
a b s t r a c t The Lie symmetry analysis is performed on the Heisenberg equation from the statistical physics. Its Lie point symmetries and optimal system of one-dimensional subalgebras are determined. The similarity reductions and invariant solutions are obtained. Using the multipliers, some conservation laws are obtained. We prove that this equation is nonlinearly self-adjoint. The conservation laws associated with symmetries of this equation are constructed by means of Ibragimov’s method. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Lie symmetry analysis method has recently penetrated into most of the areas related to mathematics, such as differential equations [1], Lie algebras [2,3], classical mechanics [4] and rogue wave [5]. As a matter of fact, many practical problems arising from natural phenomenon and engineering technology can be modeled by the nonlinear partial differential equation (NLPDEs). Investigating the explicit solutions of NLPDEs is beneficial for solving these problems. Lie symmetry analysis method can be regarded as one of the most effective methods to derive the explicit solutions. In addition, on the base of symmetries, the integrability of the NLPDEs, such as group classification, optimal system and conservation laws, can be considered successively [6–8]. The conservation law has drawn great attentions of the mathematical physicists. In the past decades, many methods for dealing with the conservation laws are derived, such as the multiplier approach [9], Noether’s approach, the partial Noether’s approach [10], Ibragimov’s method [11]. Noether’s approach and the partial Noether’s approach produce a connection between the symmetries of NLPDEs and the conservation laws. However, they are not applicable to the nonlinear partial differential equations that do not admit a Lagrangian. In order to overcome such difficulties, Ibragimov’s method was proposed [11]. Ibragimov’s method does not require the existence of a Lagrangian and it is based on the concept of an adjoint equation for the nonlinear equations. The resultant conservation laws involve not only the solutions of the original equation, but also the solutions of the adjoint equation. If the equation under consideration is nonlinear self-adjoint [12,13], the conservation vectors obtained are the conservation vectors of the original equation. In this paper, we consider the Heisenberg equation [14] of the form
(ut − uxx )(u + v) + 2u2x = 0, (vt + vxx )(u + v) − 2v2x = 0, ∗
Corresponding author. E-mail address: [email protected] (Z. Zhao).
http://dx.doi.org/10.1016/j.cnsns.2016.10.008 1007-5704/© 2016 Elsevier B.V. All rights reserved.
(1)
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where the subscripts denote the partial derivatives, u(x, t) and v(x, t) are functions with two independent variables x, t and u(x, t ) + v(x, t ) = 0. This equation relates to the famous Heisenberg model St = [S, Sxx ] via the stereographic projection [14]. Heisenberg equation is one of the models used in statistical physics to model ferromagnetism and other phenomena. The present paper aims at obtaining the optimal system, similarity reductions, invariant solutions and conservation laws of the Heisenberg equation. The organization of the paper is as follows. In Section 2, we obtain the Lie point symmetries of the Heisenberg equation using Lie group analysis. Section 3 devotes to constructing an optimal system of one-dimensional subalgebra. In Section 4, we consider the similar reductions and group-invariant solutions of this equation. In Section 5, three local conservations laws of Eq. (1) are obtained via the multipliers. In Section 6, Eq. (1) is proved nonlinearly selfadjoint. The conservation laws of Eq. (1) are established in Section 7 using Ibragimov’s method. The last section contains a summary and discussion. 2. Lie point symmetries In this section, we perform Lie symmetry analysis on the Heisenberg equation. Consider a one-parameter Lie group of transformations
t → t + ε ξ 1 (x, t, u, v), x → x + ε ξ 2 (x, t, u, v), u → u + ε η1 (x, t, u, v),
v → v + ε η2 (x, t, u, v), with a small parameter ε 1. The corresponding generator of the Lie algebra is of the form
X = ξ 1 (t, x, u, v)
∂ ∂ ∂ ∂ + ξ 2 (t, x, u, v ) + η1 (t, x, u, v ) + η2 (t, x, u, v) . ∂t ∂x ∂u ∂v
Thus the second prolongation pr(2) X is
pr (2 ) X = X + ηt1
∂ ∂ ∂ ∂ ∂ ∂ 1 2 + ηt2 + ηx1 + ηx2 + ηxx + ηxx , ∂ ut ∂ vt ∂ ux ∂ vx ∂ uxx ∂ vxx
where
ηt1 = Dt (η1 ) − ut Dt (ξ 1 ) − ux Dt (ξ 2 ), ηt2 = Dt (η2 ) − vt Dt (ξ 1 ) − vx Dt (ξ 2 ), ηx1 = Dx (η1 ) − ut Dx (ξ 1 ) − ux Dx (ξ 2 ), ηx2 = Dx (η2 ) − vt Dx (ξ 1 ) − vx Dx (ξ 2 ), 1 ηxx = Dx (ηx1 ) − uxt Dx (ξ 1 ) − uxx Dx (ξ 2 ), 2 ηxx = Dx (ηx2 ) − vxt Dx (ξ 1 ) − vxx Dx (ξ 2 ), and the operators Dx and Dt are the total derivatives with respect to t and x. The determining equations of Eq. (1) arises from the following invariance condition
pr (2) X (1 )|1 =0 = 0, pr (2 ) X (2 )|2 =0 = 0, where 1 = (ut − uxx )(u + v ) + 2u2x = 0, the partial differential equations
2 = (vt + vxx )(u + v) − 2v2x = 0. Then we obtain the overdetermined system of
ξu1 = 0, ξv1 = 0, ξx1 = 0, ξtt1 = 0, 1 2 2 ηvvv = 0,
ξt2 = 0, ξu2 = 0, ξv2 = 0, ξx2 − ξt1 = 0, ηt2 = 0, ηu2 = 0, ηx2 = 0, 1 2
2 η1 = − (u + v)2 ηvv + (u + v )ηv2 − η1 .
Solving the system, one can get
ξ 1 = c1 t + c5 , c1 x + c6 , 2 c3 2 = − u − c4 + c2 u, 2 c3 2 = v + c2 v + c4 , 2
ξ2 = η1 η2
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where c1 , c2 , c3 , c4 , c5 and c6 are arbitrary constants. Hence the infinitesimal symmetries of Eq. (1) form the six dimensional Lie algebra L6 spanned by the following independent operators:
∂ 1 ∂ + x , ∂t 2 ∂ x ∂ ∂ =u +v , ∂u ∂v 1 ∂ 1 ∂ = − u2 + v2 , 2 ∂u 2 ∂v ∂ ∂ =− + , ∂ u ∂v ∂ = , ∂t ∂ = . ∂x
X1 = t X2 X3 X4 X5 X6
(2)
By solving the following system of ordinary differential equations with initial condition
dt ∗ dε d x∗ dε d u∗ dε d v∗ dε
=
ξ 1 (t ∗ , x∗ , u∗ , v∗ ), t ∗ |ε=0 = t,
=
ξ 2 (t ∗ , x∗ , u∗ , v∗ ), x∗ |ε=0 = x,
=
η1 (t ∗ , x∗ , u∗ , v∗ ), u∗ |ε=0 = u,
=
η2 (t ∗ , x∗ , u∗ , v∗ ), v∗ |ε=0 = v,
we obtain the group transformation which is generated by the infinitesimal generator Xi (i = 1, · · · , 6), respectively
G1 : (t, x, u, v ) → eε t, e 2 ε x, u, v , 1
G2 : u(t, x, u, v ) → (t, x, eε u, eε v ), 2u 2v G3 : (t, x, u, v ) → t, x, , , ε u + 2 −εv + 2 G4 : (t, x, u, v ) → (t, x, u − ε , v + ε ), G5 : (t, x, u, v ) → (t + ε , x, u, v), G6 : (t, x, u, v ) → (t, x + ε , u, v).
(3)
Thus the theorem holds: Theorem 1. If {u(x, t),
v(x, t)} is a solution of Heisenberg equation, so are the functions
G1 (ε ) · u(x, t ) = u e− 2 ε x, e−ε t , 1
G1 (ε ) · v(x, t ) = v e− 2 ε x, e−ε t , 1
G2 (ε ) · u(x, t ) = eε u(x, t ), G2 (ε ) · v(x, t ) = eε v(x, t ), 2u(x, t ) 2v(x, t ) G3 (ε ) · u(x, t ) = , G3 (ε ) · v(x, t ) = , ε u(x, t ) + 2 −εv(x, t ) + 2 G4 (ε ) · u(x, t ) = u(x, t ) − ε , G4 (ε ) · v(x, t ) = v(x, t ) + ε , G5 (ε ) · u(x, t ) = u(x, t − ε ),
G5 (ε ) · v(x, t ) = v(x, t − ε ),
G6 (ε ) · u(x, t ) = u(x − ε , t ),
G6 (ε ) · v(x, t ) = v(x − ε , t ).
(4)
3. Optimal system of one-dimensional subalgebras To classify all the group-invariant solutions, some effective method to classify the subalgebra of Lie algebra generated by Lie point symmetries were proposed by Ovsiannikov [15] and Olver [16] from different angles. In 2010, Ibragimov et al. presented a concise method to get the optimal system [17], which only relies on the commutator table. Then Abdulwahhab constructed an optimal system of the 2-dimensional Burgers equations [18]. We investigated the optimal system of Broer–Kaup system with the aid of the method given in [19]. In this section, we shall construct an optimal system of onedimensional subalgebra of the Lie algebra L6 for Eq. (1).
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Table 1 Table of Lie brackets. [Xi , Xj ]
X1
X2
X3
X4
X5
X6
X1 X2 X3 X4 X5 X6
0 0 0 0 X5 1 X 2 6
0 0 −X3 X4 0 0
0 X3 0 X2 0 0
0 −X4 −X2 0 0 0
−X5 0 0 0 0 0
− 12 X6 0 0 0 0 0
An arbitrary operator X ∈ L6 is written as
X = l 1 X1 + l 2 X2 + l 3 X3 + l 4 X4 + l 5 X5 + l 6 X6 ,
(5)
The Lie brackets table of the operators 2 of Eq. (1) is given in Table 1. To find the linear transformations of the vector l = l 1 , l 2 , l 3 , l 4 , l 5 , l 6 , we use the following generators
Ei = ciλj l j
∂ , i = 1, 2, 3, 4, 5, 6, ∂ lλ
(6)
where ciλj is defined by the formula [Xi , X j ] = ciλj Xλ . According to Eq. (6) and Table 1, E1 , E2 , E3 , E4 , E5 and E6 can be written as
1 ∂ ∂ − l6 , ∂ l5 2 ∂ l6 ∂ ∂ E2 = l 3 3 − l 4 4 , ∂l ∂l ∂ ∂ E3 = −l 4 2 − l 2 3 , ∂l ∂l ∂ ∂ E4 = l 3 2 + l 2 4 , ∂l ∂l ∂ E5 = l 1 5 , ∂l 1 ∂ E6 = l 1 6 . 2 ∂l E1 = −l 5
(7)
For the generators E1 , E2 , E3 , E4 , E5 and E6 , the Lie equations with parameters a1 a2 , a3 , a4 , a5 and a6 with the initial condition l˜ = l, i = 1 · · · 6 are written as ai =0
dl˜1 = 0, d a1
dl˜2 = 0, d a1
dl˜3 = 0, d a1
dl˜4 = 0, d a1
dl˜1 = 0, d a2
dl˜2 = 0, d a2
dl˜3 = l˜3 , d a2
dl˜4 = −l˜4 , d a2
dl˜1 = 0, d a3
dl˜2 = −l˜4 , d a3
dl˜1 = 0, d a4
dl˜2 = l˜3 , d a4
dl˜3 = 0, d a4
dl˜4 = l˜2 , d a4
dl˜1 = 0, d a5
dl˜2 = 0, d a5
dl˜3 = 0, d a5
dl˜1 = 0, d a6
dl˜2 = 0, d a6
dl˜3 = 0, d a6
dl˜3 = −l˜2 , d a3
dl˜5 = −l˜5 , d a1
1 dl˜6 = − l˜6 , d a1 2
(8)
dl˜6 = 0, d a2
(9)
dl˜5 = 0, d a2
dl˜4 = 0, d a3
dl˜5 = 0, d a3
dl˜6 = 0, d a3
(10)
dl˜5 = 0, d a4
dl˜6 = 0, d a4
(11)
dl˜4 = 0, d a5
dl˜5 = l˜1 , d a5
dl˜6 = 0, d a5
(12)
dl˜4 = 0, d a6
dl˜5 = 0, d a6
dl˜6 1 = l˜1 . d a6 2
(13)
The solutions of these equations give the transformations
T1 :
l˜1 = l 1 ,
l˜2 = l 2 ,
l˜3 = l 3 ,
T2 :
l˜1 = l 1 ,
l˜2 = l 2 ,
l˜3 = ea2 l 3 ,
l˜4 = l 4 ,
l˜5 = e−a1 l 5 ,
l˜4 = e−a2 l 4 ,
l˜5 = l 5 ,
1
l˜6 = e− 2 a1 l 6 ,
(14)
l˜6 = l 6 ,
(15)
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1 l˜3 = a23 l 4 − a3 l 2 + l 3 , 2
T3 :
l˜1 = l 1 ,
l˜2 = −a3 l 4 + l 2 ,
T4 :
l˜1 = l 1 ,
l˜2 = a4 l 3 + l 2 ,
T5 :
l˜1 = l 1 ,
l˜2 = l 2 ,
l˜3 = l 3 ,
l˜4 = l 4 ,
l˜5 = a5 l 1 + l 5 ,
T6 :
l˜1 = l 1 ,
l˜2 = l 2 ,
l˜3 = l 3 ,
l˜4 = l 4 ,
l˜5 = l 5 ,
l˜3 = l 3 ,
l˜4 = l 4 ,
1 l˜4 = a24 l 3 + a4 l 2 + l 4 , 2
l˜5 = l 5 , l˜5 = l 5 ,
l˜6 = l 6 , l˜6 = l 6 ,
l˜6 = l 6 ,
1 l˜6 = a6 l 1 + l 6 . 2
(16) (17) (18) (19)
The construction of the optimal system requires a simplification of the vector
l = l1, l2, l3, l4, l5, l6 ,
(20)
by means of the transformations T1 − T6 . We focus on finding a simplest representative of each class of similar vectors (20). The construction will be divided into two cases. Case 1. l1 = 0 5 By taking a5 = − ll 1 in the transformations T5 , we can make l˜5 = 0. The vector (20) is hence reduced to the form
l = l 1 , l 2 , l 3 , l 4 , 0, l 6 .
(21)
6
We take a6 = − 2ll1 in T6 and reduce the vector (21) to the form
l = l 1 , l 2 , l 3 , l 4 , 0, 0 .
(22)
l3
1.1 = 0 2 By taking a4 = − ll 3 in the transformations T4 , we can make l˜2 = 0. The vector (22) is hence reduced to the form
l = l 1 , 0, l 3 , l 4 , 0, 0 .
(23) l1
Without loss of generality, we assume = 1 and make following representatives for the optimal system
X1 ± X3 ,
l3
=
±1, l 4
= ±1 by the transformation T2 . Thus we can obtain the
X1 ± X3 ± X4 .
(24)
l3
1.2 = 0 We consider the vector (22) of the form
l = l 1 , l 2 , 0, l 4 , 0, 0 . 1.2.1 l4 = 0 We take a3 =
1
l2 l4
(25)
in the transformations T3 and reduce the vector (25) to the form
l = l , 0, 0, l 4 , 0, 0 .
(26)
Taking all the possible combinations, we obtain the following representatives
X1 ± X4 .
(27)
l4
1.2.2 = 0 The vector (25) is reduced to the form
l = l 1 , l 2 , 0, 0, 0, 0 .
(28)
Taking all the possible combinations, we obtain the following representatives
X1 ,
X1 ± X2 .
(29)
l1
Case 2. = 0 We consider the vector (20) of the form
l = 0, l 2 , l 3 , l 4 , l 5 , l 6 .
(30)
l3
2.1 = 0 We consider the vector (20) of the form
l = 0, l 2 , 0, l 4 , l 5 , l 6 . 2.1.1
l4
= 0
(31)
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By taking a3 =
l2 l4
225
in the transformations T3 , we can make l˜2 = 0. The vector (30) is hence reduced to the form
l = 0, 0, 0, l 4 , l 5 , l 6 .
(32)
Taking all the possible combinations, we obtain the following representatives
X4 ,
X4 ± X5 ,
X4 ± X6 ,
X4 ± X5 ± X6 .
(33)
l4
2.1.2 = 0 2.1.2.1 l 2 = 0 Taking all the possible combinations, we obtain the following representatives
X5 ,
X6 ,
X5 ± X6 .
(34)
l2
2.1.2.2 = 0 The vector (31) is reduced to the form
l = 0, l 2 , 0, 0, l 5 , l 6 ,
(35)
that cannot be simplified. Taking all the possible combinations, we obtain the following representatives
X2 ,
X2 ± X5 ,
X2 ± X6 ,
X2 ± X5 ± X6 .
(36)
l3
2.2 = 0 We consider the vector (30) of the form
l = 0, l 2 , l 3 , l 4 , l 5 , l 6 , 2 By taking a4 = − ll 3 in the transformations T4 , we can make l˜2 = 0. The vector (30) is reduced to the form
l = 0, 0, l 3 , l 4 , l 5 , l 6 ,
3
l = 0
l 4 = 0, l 4 = 0,
X3 ± X4 , X3 ± X4 ± X5 , X3 ± X4 ± X6 , X3 , X3 ± X5 , X3 ± X6 X3 ± X5 ± X6 .
X3 ± X4 ± X5 ± X6 ,
(37)
Finally, by collecting the operators (24,27,29,33,34,36 and 37), we arrive at the following theorem: Theorem 2. The following operators provide an optimal system of one-dimensional subalgebras of the Lie algebra spanned by X1 , X2 , X3 , X4 , X5 , X6 of Eq. (1):
X1 ± X3 ,
X1 ± X2 ,
X3 ± X6 ,
X2 ,
X1 ,
X1 ± X4 ,
X1 ± X3 ± X4 ,
X3 ,
X2 ± X5 ,
X2 ± X6 ,
X2 ± X5 ± X6 ,
X3 ± X4 ,
X3 ± X4 ± X5 ,
X4 ± X6 ,
X4 ± X5 ± X6 ,
X3 ± X4 ± X6 ,
X3 ± X4 ± X5 ± X6 ,
X5 ,
X6 .
X5 ± X6 ,
X4 ,
X4 ± X5 ,
X3 ± X5 ± X6 ,
X3 ± X5
(38)
4. Reductions and the invariant solutions of Eq. (1) Given the optimal system of the one-dimensional subalgebras, we can investigate the symmetry reductions of Eq. (1) by integrating the characteristic equations [20]. The characteristic equation related to determine the invariant surface condition. The characteristic equation is actually the associated Lagrange’s system for the calculation of the characteristics. Lagrange uses the characteristics to reduce the order of ordinary differential equations. The solutions of Eq. (1) can be obtained by solving the reduced equations. 4.1. Solutions through X5 + X6 For the generator X5 + X6 , the characteristic equation is written as
dt dx du dv = = = . 1 1 0 0
(39)
The solution of the characteristic equations yields three invariants of X5 + X6
ξ = −x + t, θ 1 = u, θ 2 = v.
(40)
Thus, the solution of Eq. (1) is given by the invariant form
u = f (−x + t ),
v = g(−x + t ).
(41)
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Fig. 1. Plots of the u(x, t) given by Eq. (43) for parameters c1 = 0.2, c2 = 0.3, c3 = −0.3, c4 = 1. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
Fig. 2. Plots of the v(x, t) given by Eq. (44) for parameters c1 = 0.2, c2 = 0.3, c3 = −0.3, c4 = 1. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
Substitution of Eq. (41) into the system (1) leads to f(ξ ) and g(ξ ) satisfying the reduced ordinary differential equations (ODEs)
f f + f g − f f − f g + 2 f = 0, 2
f g + gg + g f + g g − 2g = 0. 2
(42)
Solving this reduced equation, we obtain the closed-form solution of Eq. (1) as follows
u(x, t ) =
1 −x+t+c3 2 c1 1 −x + c4 , +t+c3
−2 c2 sinh
c1 cosh
2
(43)
c1
2 c3 c3 − c1 c4 − 2c2 cosh 12 −x+ct+ (2 c1 c2 − c1 c4 ) sinh 12 −x+ct+ 1 1 1 −x+t+c3 1 −x+t+c3 v(x, t ) = , c1 cosh
2
c1
+ sinh
2
c1
c1
(44)
where ci , i = 1, 2, 3, 4 are arbitrary constants. Fig. 1 displays the kink type of solution structure of u(x, t) determined by (43). Fig. 2 illustrates that the Heisenberg equation has the hyperbolic solution v(x, t) (44). 4.2. Solutions through X2 + X6 The symmetry X2 + X6 gives rise to the group-invariant solution of the form
u = f (t )ex ,
v = g(t )ex .
(45)
Substituting Eq. (45) into Eq. (1) results in the ODEs where f and g satisfy
f f + f g + f 2 − f g = 0, g f + g g + f g − g2 = 0.
(46)
Solving this reduced equation, we obtain the closed-form solution of Eq. (1) as follows
u(x, t ) = c2 ec1 t+x ,
v(x, t ) = −
c2 (1 + c1 )ec1 t+x , c1 − 1
(47) (48)
where ci , i = 1, 2, are arbitrary constants. Figs. 3 and 4 display that the Heisenberg equation has the exponential solutions u(x, t) (47) and v(x, t) (48).
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Fig. 3. Plots of the u(x, t) given by Eq. (47) for parameters c1 = 0.2, c2 = 0.3. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
Fig. 4. Plots of the v(x, t) given by Eq. (48) for parameters c1 = 0.2, c2 = 0.3. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
Fig. 5. Plots of the u(x, t) given by Eq. (51) for parameters c1 = 0.2, c2 = 0.3. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis.
4.3. Solutions through X3 + X6 The symmetry X3 + X6 gives rise to the group-invariant solution of the form
u(x, t ) =
2 , −x + 2g(t )
v(x, t ) =
2 , x + 2 f (t )
(49)
Substituting Eq. (49) into Eq. (1) results in the ODEs where f and g satisfy
−16 f f − 16 f g + 8 = 0, −16g f − 16g g − 8 = 0.
(50)
Solving this reduced equation, we obtain the closed-form solution of Eq. (1) as follows
u(x, t ) =
−2c1 , −c1 x + 2c12 t + 2 c1 c2 − 1
(51)
v(x, t ) =
2 , −x + 2 c1 t + 2 c2
(52)
where ci , i = 1, 2, are arbitrary constants. Figs. 5 and 6 present that the Heisenberg equation has inversely proportional solutions u(x, t) (51) and v(x, t) (52).
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Fig. 6. Plots of the v(x, t) given by Eq. (52) for parameters c1 = 0.2, c2 = 0.3. (a) 3D plot, (b) density plot, (c) along x-axis, (d) along t-axis. Table 2 Reductions of the Heisenberg equation. Case
Similarity variable
Reduced equations − f g − f f − 2 + 8ξ f − 8ξ 2 f − g − f + 4ξ 2 f g +4ξ 2 f f + 6ξ f g + 6ξ f f = 0, 2 −g g − g f + 2 + 8ξ g + 8ξ 2 g − g − f − 4ξ 2 g g −4ξ 2 g f − 6ξ g g − 6ξ g f = 0. 2
(1) X1 + X3
ξ = xt2 , u(x, t ) = v(x, t ) =
(2) X1 + X2
ξ = xt2 , u(x, t ) = f (ξ )x2 , v(x, t ) = g(ξ )x2 .
f f + f g − 4ξ 2 f f − 4ξ 2 f g − 14ξ f f + 2ξ f g 2 +6 f 2 − 2 f g + 8ξ 2 f = 0, g f + g g + 4ξ 2 g f + 4ξ 2 g g − 2ξ g f + 14ξ g g 2 +2 f g − 6g2 − 8ξ 2 g = 0.
(3) X1 + X4
ξ = xt2 , u(x, t ) = −2 ln (x ) + f (ξ ), v(x, t ) = 2 ln (x ) + g(ξ ).
(4) X2 + X5 + X6
ξ = −x + t, u(x, t ) = f (ξ )ex , v(x, t ) = g(ξ )ex .
f f + f g − 2 f − 2g − 4ξ 2 f f − 4ξ 2 f g − 6ξ f f 2 −6ξ f g + 8 + 16ξ f + 8ξ 2 f = 0, g f + g g − 2 f − 2g + 4ξ 2 g f + 4ξ 2 g g + 6ξ g f 2 +6ξ g g − 8 + 16ξ g − 8ξ 2 g = 0.
(5) X3 + X4 + X5
ξ = x, √ √ √ u(x, t ) = − 2 tan 22 t + 22 f (ξ ) , √ √ √ v(x, t ) = 2 tan 22 t + 22 g(ξ ) .
(6) X4 + X5 + X6
ξ = −x + t, u(x, t ) = −x + f (ξ ), v(x, t ) = x + g(ξ ).
1 , ln (x )+ f (ξ ) 1 , − ln (x )+g(ξ )
− f f + 3 f g − f f − f g + f 2 − f g + 2 f = 0, 2 −g f + 3g g + g f + g g + f g − g2 − 2g = 0. √ √ √ √ √ 2 − 2 f sin 22 g cos 22 f − 2 f cos 22 f cos 22 g √ √ √ √ √ √ + 2 sin 22 g cos 22 f + 2 f sin 22 f cos 22 g √ √ √ √ √ 2 − 2 sin 22 f cos 22 g − 2 f sin 22 f sin 22 g = 0, √ √ √ √ √ 2 2g sin 22 f cos 22 g + 2g cos 22 g cos 22 f √ √ √ √ √ 2 +2g sin 22 g sin 22 f − 2g sin 22 g cos 22 f √ √ √ √ √ √ − 2 sin 22 g cos 22 f + 2 sin 22 f cos 22 g = 0. 2
f f + f g − f f − f g + 2 + 4 f + 2 f = 0, 2 g f + g g + g f + g g − 2 + 4g − 2g = 0. 2
4.4. Some other of similarity reductions and invariant solutions of Eq. (1) Some other of the similarity reductions for the optimal system of the subalgebras and invariant solutions are listed in Tables 2 and 3, respectively. 5. Construction of conservation laws for Eq. (1) via multipliers α a kth-order system of the partial 1 Consider 1differential equations (PDEs) R [u] with n independent variables x = 2 n 2 m
x ,x ,··· ,x
and m dependent variables u = u , u , · · · , u
Rα [u] = Rα x, u, u(1 ) , · · · , u(k ) = 0,
α = 1, 2, · · · , m,
(53)
where u(1) , u(2) , , u(k) denote the collections of all first, second,...,kth-order derivatives. A local divergence-type conservation law of the PDE system (53) is a divergence expression of the form
Di i [u] = D1 1 [u] + · · · + Dn n [u],
(54)
in terms of the total derivative operators holding for the solutions of (53). There exists a set of multipliers, called local conservation laws multipliers,
α [u] = α x, u, ∂ u, · · · , ∂ l u ,
α = 1, 2, · · · m,
(55)
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Table 3 Corresponding invariant solutions of similarity reductions in Table 2. Case
Invariant solutions u(x, t ) = ln x+1f (ξ ) , v (x, t ) = − ln x1+g(ξ ) , f (ξ ) = h (ξ )dξ + c2 , 8ξ 2 f + f f + 2 − 8ξ f − 4ξ 2 f f − 6ξ f f + f , g(ξ ) = − f − 1 + 4ξ 2 f + 6ξ f 1 1 3 2 2 h = − 16 ξ − 12 ξ ξ h −64 c ξ + 1 − 20ξ 2 h + 64 c + 1 1 4 ξ (2ξ h−1)
Case 1
−48ξ 5 h − 48ξ 3 h + 16c1 + ξ , 2
ξ=
t x2
.
u(x, t ) = 0,
1 x2
3
Case 2
v(x, t ) =
t 2 e− 4 t √ √ −c1 x − c2 π x + c2 x π erf 12
x √ t
1 √ + 2 c 2 te − 4
x2 t
.
u(x, t ) = −2 ln x + f (ξ ), v (x, t ) = 2 ln x + g(ξ ), f (ξ ) = h (ξ )dξ + c2 2 f f + 16ξ f − 2 f − 4ξ 2 f f − 6ξ f f + 8ξ 2 f + 8 g(ξ ) = , 2 − f + 6ξ f + 4ξ f + 2 1 1 ξ − 12ξ 3 + 32c1 ξ 2 h2 + 64c1 ξ − 2 + 40ξ 2 h h = − 32 ξ 4 (1 + hξ )
Case 3
+96ξ 3 h − 48ξ 5 h + 32c1 + 4ξ , 2
ξ=
t x2
.
u(x, t ) = 0, Case 4
v(x, t ) =
√ − 5e
√
5 1 x− x+ 2 2
√
5 3 + 2 2
t
. c1 − c2 √ √ √ u(x, t ) = − 2 tan 22 t + 22 f (x ) , Z 1 √ g(x ) = RootOf ±
Case 5
f (x ) =
√ e 5(−x+t )
√ 2 arctan
√ √ √ v(x, t ) = 2 tan 22 t + 22 g(x ) , dα + x + c3 dx + c4 , 4 2 3 −2 α√ +1+2 √c1 α −2 c√2 α √ √
2 2 2
−2g cos 2
2g 2 sin
2
g + 2 sin
√ √ 2 2
g + 2 cos
g + 2g sin
2
√ √ 2 2
g + 2g cos
2
g
√
, v (x, t ) = x + g(ξ ), u(x, t ) = −x + f (ξ ) Z √ f (ξ ) = RootOf ± 3
2 2
g
.
1 −2c1 α + (2c2 −6c1 )α 2 + (−6c1 +4c2 −2)α −1+2c2 −2c1
dα
+ ξ + c 3 )d ξ + c 4 , 2 − f f − 4 f + f f − 2 − 2 f , g( ξ ) = − f + f ξ = −x + t.
Case 6
such that
Di i [u] ≡ α [u]Rα [u],
(56)
holds for arbitrary u(x). Theorem 3 [21]. For any divergence expression Di i [u], one has
Eu j Di i [u] ≡ 0,
j = 1, 2, · · · m,
(57)
where Eu j = ∂ j − Di ∂ j + · · · + (−1 )s Di1 · · · Dis ∂u
∂ ui
∂ + · · · is the Euler operator with respect to uj . ∂ uij ···i 1
s
Theorem 4 [21]. A set of local multipliers α (x, u, ∂ u, , ∂ l u) yields a divergence expression for PDE system (53) if and only if
Eu j α x, u, ∂ u, · · · , ∂ l u Rα x, u, ∂ u, · · · , ∂ k u
≡ 0,
j = 1, 2, · · · m,
(58)
holds for arbitrary u(x). According to Theorems 3 and 4, for Eq. (1), it follows that all the local conservation laws multipliers are of the form
1 = 1 (x, t, u, v), 2 = 2 (x, t, u, v). Then
(59)
Eu 1 (ut − uxx )(u + v ) + 2u2x + 2 (vt + vxx )(u + v ) − 2v2x Ev 1 (ut − uxx )(u + v ) +
2u2x
+ 2 (vt + vxx )(u + v ) − 2v
2 x
≡ 0, ≡ 0,
(60)
230
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234 Table 4 Conservation laws of the Heisenberg equation. Case
Multipliers
1 2 1 2 1 2
Case 1 Case2 Case3
= =
Conservation Laws
1 v2 2 ( u + v )3 1 u2 2 ( u + v )3
,
Dt
,
= v 3, ( u+v ) = = =
−u ( u + v )3 1 ( u + v )3 1 ( u + v )3
1
uv 2 u+v
+ Dx
1 −v 2 u x + u 2 v x 2 ( u + v )2
Dt − u+v v + Dx − vux +uv2x
,
( u+v )
,
Dt − u+1 v + Dx
,
vx −ux (u+v)2
= 0.
= 0.
= 0.
where the Euler operators Eu and Ev are given by
∂ ∂ ∂ ∂ − Dt − Dx + D2x ··· , ∂u ∂ ut ∂ ux ∂ uxx ∂ ∂ ∂ ∂ Ev = − Dt − Dx + D2x + ··· . ∂v ∂ vt ∂ vx ∂ vxx Eu =
Eq. (60) splits with respect to the second derivatives of u and v with the aid of the Maple package GeM [22] to yield the following determining system
−2 1 u v − 6 1 − 2 1 u u = 0, 2 2 v v + 6 2 + 2 2 v u = 0,
2 vv v + 4 2 v + 2 vv u = 0, −4 1 u − 1 uu v − 1 uu u = 0, −6 1 x − 2 1 xu u − 2 1 xu v = 0, −2 1 x − 2 1 xv v − 2 1 xv u = 0, 2 2 x + 2 2 xu u + 2 2 xu v = 0, 6 2 x + 2 2 xv v + 2 2 xv u = 0, −2 1 uv v − 2 1 uv u − 2 1 u − 6 1 v = 0, − 1 vv v − 1 vv u − 2 1 v − 2 2 u = 0, − 1 xx v − 1 t v − 1 xx u − 1 t u = 0, 2 2 uv v + 2 2 uv u + 2 2 v + 6 2 u = 0, 2 xx v + 2 xx u − 2 t v − 2 t u = 0, 2 1 v + 2 2 u + 2 uu v + 2 uu u = 0, 2 u u − 1 v u − 1 v v + 2 u v − 1 + 2 = 0, 1 + 1 v u − 2 + 1 v v − 2 u v − 2 u u = 0.
(61)
The solutions of the determining system (61) are given by
1 (x, t, u, v) =
1 c 2 2 1
1 v + c2 v + c3 c1 u 2 − c2 u + c3 , 2 (x, t, u, v) = 2 , 3 (u + v ) (u + v )3
(62)
where c1 , c2 and c3 are arbitrary constants. The solution yields the three local conservation laws multipliers
1 = 2 =
(1 )
1 v2 , 2 ( u + v )3 1 u2 , 2 ( u + v )3
1 = (2 ) = 2
v , ( u + v )3 −u , ( u + v )3
1 = (3 ) 2 =
1
( u + v )3
,
1 . ( u + v )3
(63)
By utilizing the direct method given in [10], each multiplier determines a corresponding flux as given in Table 4. 6. Nonlinear self-adjointness In this section, we prove that the Heisenberg Eq. (1) is nonlinear self-adjoint. Consider a system of m PDEs
Rα x, u, · · · , u(k ) = 0,
α = 1, · · · , m.
(64)
The system of the adjoint equations for the kth-order system of PDEs (64) is defined as follows
(Rα )∗ x, u, v, · · · , u(k) , v(k) = 0,
α = 1, · · · , m,
(65)
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
where
(Rα )∗ x, u, v, · · · , u(k) , v(k) =
δ vβ R β , α , β = 1, · · · , m, v = v(x ). δ uα
231
(66)
The Euler–Lagrange operator is defined as
δ
∂
= δ uα
+ ∂ uα
∞
(−1 ) j Di1 · · · Di j
j=1
the formal Lagrangian is
∂ , α = 1, 2, · · · , m, ∂ uαi1 ···i j
(67)
L = vβ Rβ x, u, · · · , u(k ) .
(68)
Definition 1 [12,13]. The system (64) is said to be nonlinearly self-adjoint if the adjoint system is satisfied for all the solutions u of system (64) upon a substitution v = ϕ (x, u ) such that ϕ (x, u) = 0. Specifically, the system
(Rα )∗ x, u, ϕ , · · · , u(k) , ϕ(k) = 0, α = 1, · · · , m,
(69)
is identical to the system
λβα Rβ x, u, u, · · · , u(k) , u(k) = 0, β = 1, · · · , m,
(70)
that means
(Rα )∗ v=ϕ (x,u) = λβα Rβ , β = 1, · · · , m,
(71)
β
where λα is a certain function. Theorem 5 [23]. The determining system of the multiplier (x, u) for system (64) is identical to the system for the substitution of nonlinear self-adjointness. If the formal Lagrangian of Eq. (1) is written as
L = ϕ 1 (x, t, u, v) (ut − uxx )(u + v ) + 2u2x + ϕ 2 (x, t, u, v ) (vt + vxx )(u + v ) − 2v2x ,
(72)
according to Theorem 5, one can get
1 c1 v2 + c2 v + c3 ϕ (x, t, u, v) = 1 (x, t, u, v) = 2 , (u + v )3 1 c1 u 2 − c2 u + c3 2 2 ϕ (x, t, u, v) = 2 (x, t, u, v) = . (u + v)3 1
(73)
Therefore, Eq. (1) is nonlinearly self-adjoint with substitution (73). 7. Construction of conservation laws for Eq. (1) via Ibragimov’s method Theorem 6 [12] [Ibragimov’s method]. Let the system of differential Eq. (64) be nonlinearly self-adjoint. Then every Lie point, Lie-Bäcklund, nonlocal symmetry
∂ ∂ + ηα x, u, u(1 ) , · · · , i ∂ uα ∂x
X = ξ i x, u, u(1 ) , · · ·
(74)
admitted by the system of Eq. (64) gives rise to a conservation law, where the components C i of the conserved vector C = 1 C , · · · , C n are determined by
∂L ∂L ∂L − D + D D − · · · j j k ∂ uαi ∂ uαi j ∂ uαi jk
∂L ∂L ∂L α +D j (W α ) − D + · · · + D D W − · · · , ) j k( k ∂ uαi j ∂ uαi jk ∂ uαi jk
Ci = W α
(75)
with W α = ηα − ξ j uαj . The formal Lagrangian L should be written in the symmetric form with respect to all mixed derivatives uα , uα ,···. ij i jk The Lagrangian L is
L = 1 (ut − uxx )(u + v ) + 2u2x + 2 (vt + vxx )(u + v ) − 2v2x .
(76)
232
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
∂ , from Theorem 1 we have W 1 = η1 − ξ 1 u − ξ 2 u , W 2 = η2 − ξ 1 v − ξ 2 v , For the generator X = ξ 1 ∂∂t + ξ 2 ∂∂x + η1 ∂∂u + η2 ∂v x x t t hence we can obtain the following components of the conserved vector
t
C =W
1
Cx = W 1
∂L ∂ ut
+W
2
∂L ∂L − Dx ∂ ux ∂ uxx
∂L , ∂ vt
+ W2
(77)
∂L ∂L − Dx ∂ vx ∂ vxx
+ Dx W 1
∂L ∂L + Dx W 2 . ∂ uxx ∂ vxx
(78)
Taking the formal Lagrangian L given by (76) into (77,78), we can simplify C t , C x as follows
C t = W 1 1 (u + v ) + W 2 2 (u + v ),
(79)
C x = W 1 (4ux + Dx ( 1 (u + v ) ) ) + W 2 (−4vx − Dx ( 2 (u + v ) ) )
+Dx W 1 1 (−u − v ) + Dx W 2 2 (u + v ).
(80)
The Lie point symmetry generator X1 = t ∂∂t + 12 x ∂∂x has the Lie characteristic functions W 1 = −t ut − 12 xux and W 2 = −t vt − 12 xvx . According to the formulas (79,80), the symmetry generator X1 can give rise to the following components of the conserved vector
C1t
C1x
=
−t ut − 12 xux
1
c 2 1
v2 + c2 v + c3
( u + v )2
+
−t vt − 12 xvx
1
c u2 2 1 2
− c2 u + c3
(u + v )
,
v2 + c2 v + c3 ( u x + vx ) 4ux + −2 (u + v )2 (u + v )3 1 2 c u − c2 u + c3 ( u x + vx ) 1 c1 u u x − c2 u x 2 1 + −t vt − xvx −4vx − −2 2 (u + v)2 (u + v )3 −t utx − 12 ux − 12 xuxx 12 c1 v2 + c2 v + c3 −t vtx − 12 vx − 12 xvxx 12 c1 u2 − c2 u + c3 − + . (u + v )2 ( u + v )2
1 = −t ut − xux 2
1
c 2 1
c1 vvx + c2 vx
∂ , we have W 1 = u and W 2 = v. Thus formulas (79,80) yield the following components For the generator X2 = u ∂∂u + v ∂v of the conserved vector
C2t
=
u
1
c 2 1
v2 + c2 v + c3 v 12 c1 u2 − c2 u + c3 + , (u + v)2 ( u + v )2
C2x
v2 + c2 v + c3 ( u x + vx ) = u 4ux + −2 (u + v)2 (u + v)3 1 2 c u − c2 u + c3 ( u x + vx ) vx 12 c1 u2 − c2 u + c3 ux 12 c1 v2 + c2 v + c3 c1 uux −c2 ux 2 1 +v −4vx − +2 − + . (u + v )2 ( u + v )3 (u + v )2 (u + v)2 c1 vvx + c2 vx
1
c 2 1
∂ , we have W = − 1 u2 and W 2 = For the generator X3 = − 21 u2 ∂∂u + 12 v2 ∂v 1 2 components of the conserved vector
1 2 2v .
Thus formulas (79,80) yield the following
2 1 2 v + c2 v + c3 1 v 2 c1 u − c2 u + c3 + , 2 ( u + v )2 (u + v )2 1 2 c v + c v + c u + v ( ) x x 1 2 3 1 c v v + c v x 1 2 x 2 C3x = − u2 4ux + −2 2 (u + v)2 (u + v)3 1 2 c u − c2 u + c3 ( u x + vx ) uux 12 c1 v2 + c2 v + c3 vvx 12 c1 u2 − c2 u + c3 c1 u u x − c2 u x 1 2 2 1 + v −4vx − + 2 + + . 2 ( u + v )2 (u + v)3 (u + v )2 (u + v)2 C3t
1u =− 2
2 1 c 2 2 1
∂ , we have W 1 = −1 and W 2 = 1. Thus formulas (79,80) yield the following components For the generator X4 = − ∂∂u + ∂v of the conserved vector 1
C4t = − 2
c1 v2 + c2 v + c3
(u + v )
2
+
1 c u2 2 1
− c2 u + c3
(u + v)2
,
Z. Zhao, B. Han / Commun Nonlinear Sci Numer Simulat 45 (2017) 220–234
C4x = −4ux −
c1 vvx + c2 vx
(u + v )2
1 +2
c 2 1
233
1 2 v2 + c2 v + c3 ( u x + vx ) c u − c2 u + c3 ( u x + vx ) c1 u u x − c2 u x 2 1 − 4 v − + 2 . x (u + v )3 (u + v)2 (u + v )3
For the generator X5 = ∂∂t , we have W 1 = −ut and W 2 = −vt . Thus formulas (79,80) yield the following components of the conserved vector
v2 + c2 v + c3 vt 12 c1 u2 − c2 u + c3 =− − , ( u + v )2 (u + v)2 1 2 c v + c2 v + c3 ( u x + vx ) c1 vvx + c2 vx 2 1 x C5 = −ut 4ux + −2 (u + v)2 (u + v)3 1 2 c u − c2 u + c3 ( u x + vx ) c1 u u x − c2 u x 2 1 −vt −4vx − +2 ( u + v )2 (u + v)3 1 2 1 2 utx 2 c1 v + c2 v + c3 vtx 2 c1 u − c2 u + c3 + − . (u + v)2 ( u + v )2 C5t
ut
1
c 2 1
For the generator X6 = ∂∂x , we have W 1 = −ux and W 2 = −vx . Thus formulas (79,80) yield the following components of the conserved vector
v2 + c2 v + c3 vx 12 c1 u2 − c2 u + c3 =− − , (u + v )2 (u + v)2 1 2 c v + c2 v + c3 ( u x + vx ) c1 vvx + c2 vx 2 1 x C6 = −ux 4ux + −2 ( u + v )2 ( u + v )3 1 2 c u − c u + c u + v uxx 12 c1 v2 + c2 v + c3 vxx 12 c1 u2 − c2 u + c3 ( ) x x 1 2 3 c1 u u x − c2 u x 2 −vx −4vx − +2 + − . (u + v )2 (u + v )3 (u + v)2 (u + v)2 C6t
ux
1
c 2 1
8. Conclusions Lie symmetry analysis is performed on the Heisenberg equation. We obtain the Lie point symmetries of this equation and discuss the optimal system of one-dimensional subalgebra of the Lie algebra L6 . Some similar reductions and groupinvariant solutions to the equation are considered based on the optimal system. The conservation laws are obtained for Heisenberg equation via the multipliers. Meanwhile, we demonstrate that Heisenberg equation is nonlinearly self-adjoint. Based on Ibragimov’s method, some nontrivial conservation laws for this equation are obtained. Further research may focus on getting some solutions by means of the obtained conservation laws. Acknowledgement The authors are grateful to the reviewers for the suggested revisions and comments. This work is supported by the National Natural Science Foundation of China under Grant No. 41474102. References [1] Bluman GW, Cheviakov AF, Anco SC. Applications of symmetry methods to partial differential equations. Springer; 2010. [2] Ibragimov NH, Gainetdinova AA. Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional vessiot-guldberg-lie algebras. Appl Math Lett 2016;52:126–31. [3] Manno G, Oliveri F, Saccomandi G, Vitolo R. Ordinary differential equations described by their lie symmetry algebra. J Geom Phys 2014;85:2–15. [4] Ibragimov NH, Kovalev VF, Meleshko SV. Group analysis of kinetic equations in a non-linear thermal transport problem. Int J Non-linear Mech 2015;71:1–7. [5] Wang GW. Symmetry analysis and rogue wave solutions for the (2+1)-dimensional nonlinear schrödinger equation with variable coefficients. Appl Math Lett 2016;56:56–64. [6] Sheftel MB. Lie groups and differential equations: symmetries, conservation laws, and exact solutions of mathematical models in physics. Phys Part Nucl 1997;28:241–66. [7] Nadjafikhah M, Ahangari F. Symmetry analysis and conservation laws of the geodesic equations for the reissner-nordström de sitter black hole with a global monopole. Commun Nonlinear Sci Numer Simul 2012;17:2350–61. [8] Nadjafikhah M, Shirvani-Sh V. Lie symmetries and conservation laws of the hirota-ramani equation. Commun Nonlinear Sci Numer Simul 2012;17:4064–73. [9] Naz R. Conservation laws for some compacton equations using the multiplier approach. Appl Math Lett 2012;25:257–61. [10] Naz R, Mahomed FM, Mason DP. Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics. Appl Math Comput 2008;205:212–30. [11] Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333:311–28.
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