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Nonclassical potential symmetries for the Burgers equation
Nonlinear Analysis 71 (2009) e1826–e1834
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Nonclassical potential symmetries for the Burgers equation M.L. Gandarias ∗ , M.S. Bruzon Departamento de Matematicas, Universidad de Cadiz, PO.BOX 40, 11510 Puerto Real, Cadiz, Spain
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Keywords: Potential symmetries Nonclassical potential symmetries Partial differential equations
abstract In this work classes of symmetries for partial differential equations which can be written in a conserved form are found. These nonclassical potential symmetries are realized as nonclassical symmetries of the associated potential system and are neither classical potential symmetries realized as Lie symmetries of a related auxiliary system nor nonclassical symmetries of the considered equation. Some of these symmetries are carried out for the Burgers equation. These symmetries are realized as the nonclassical symmetries of the associated potential system and yield many new nonlocal generators and new explicit solutions of the Burgers equation. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Symmetries admitted by a partial differential equation (PDE) are useful for finding invariant solutions. These solutions are obtained by using group invariants to reduce the number of independent variables. The fundamental basis of the technique is that, when a differential equation is invariant under a Lie group of transformations, a symmetry reduction exists which reduces the equation to a lower dimensional equation. The machinery of Lie group theory provides the systematic method to search for these special group-invariant solutions. For PDE’s with two independent variables, as the Burgers equation
ut +
u2 2
− ux
=0
(1)
x
a single group reduction transforms the PDE into ordinary differential equations (ODE’s), which are generally easier to solve than the original PDE. Most of the required theory and description of the method can be found in for example [1–5]. Local symmetries admitted by a nonlinear PDE are also useful to discover whether or not the equation can be linearized by an invertible mapping and construct an explicit linearization when one exists. A nonlinear scalar PDE is linearizable by an invertible contact (point) transformation if and only if it admits an infinite-parameter Lie group of contact transformations satisfying specific criteria [6,1,7,8]. There have been several generalizations of the classical Lie group method for symmetry reductions. Bluman and Cole [9] developed the nonclassical method to study the symmetry reductions of the heat equation. The basic idea of the method is to require that the N order PDE
∆ x, t , u, u(1) (x, t ), . . . , u(N ) (x, t ) = 0,
(2) (l)
where (x, t ) ∈ R are the independent variables, u ∈ R is the dependent variable and u (x, t ) denote the set of all partial derivatives of l order of u and the invariant surface condition 2
ξ ∗
∂u ∂u +τ − φ = 0, ∂x ∂t
Corresponding author. E-mail address: [email protected] (M.L. Gandarias).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.078
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M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
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which is associated with the vector field v = ξ (x, t , u)
∂ ∂ ∂ + τ (x, t , u) + φ(x, t , u) , ∂x ∂t ∂u
(4)
are both invariant under the transformation with infinitesimal generator (4). Since then, a great number of papers have been devoted to the study of nonclassical symmetries of nonlinear PDE’s in both one and several dimensions. An obvious limitation of group-theoretic methods based in local symmetries, in their utility for particular PDE’s, is that there are partial differential equations (PDE’s) of physical interest possessing few symmetries or none at all [3]. It turns out that PDE’s can admit nonlocal symmetries whose infinitesimal generators depend on integrals of the dependent variables in some specific manner. It also happens that if a nonlinear scalar PDE does not admit an infinite-parameter Lie group of contact transformations is not linearizable by an invertible contact transformation. However most of the interesting linearizations involve non-invertible transformations, such linearizations can be found by embedding given nonlinear PDE’s in auxiliary systems of PDE’s [1]. Krasil’shchik and Vinogradov [10–12] gave criteria which must be satisfied by nonlocal symmetries of a PDE when realized as local symmetries of a system of PDE’s which ‘covers’ the given PDE. Akhatov, Gazizov and Ibragimov [13] gave nontrivial examples of nonlocal symmetries generated by heuristic procedures. In [14] Bluman introduced a method to find a new class of symmetries for a PDE. Suppose a given scalar second order PDE F (x, t , u, ux , ut , uxx , uxt , utt ) = 0,
(5)
where the subscripts denote the partial derivatives of u, can be written as a conservation law D Dt
f ( x , t , u, ux , ut ) −
D Dx
g (x, t , u, ux , ut ) = 0,
for some functions f and g of the indicated arguments. Here
(6) D Dx
and
D Dt
are total derivative operators defined by
∂ ∂ ∂ ∂ + ux + uxx + uxt + ···, ∂x ∂u ∂ ux ∂ ut ∂ ∂ ∂ ∂ = + ut + uxt + utt + ···. Dt ∂t ∂u ∂ ux ∂ ut D
Dx D
=
Through the conservation law (6) one can introduce an auxiliary potential variable v and form an auxiliary potential system (system approach)
vx = f (x, t , u, ux , ut ), vt = g (x, t , u, ux , ut ).
(7)
For many physical equations one can eliminate u from the potential system (7) and form an auxiliary integrated or potential equation (integrated equation approach) G(x, t , v, vx , vt , vxx , vxt , vtt ) = 0,
(8)
for some function G of the indicated arguments. Any Lie group of point transformations w = ξ (x, t , u, v)
∂ ∂ ∂ ∂ + τ (x, t , u, v) + φ(x, t , u, v) + ψ(x, t , u, v) , ∂x ∂t ∂u ∂v
(9)
admitted by (7) yields a nonlocal symmetry potential symmetry of the given PDE (6) if and only if the following condition is satisfied
ξv2 + τv2 + φv2 6= 0.
(10)
We point out that if we consider a Lie group of point transformations u = ξ (x, t , v)
∂ ∂ ∂ + τ (x, t , v) + ψ(x, t , v) ∂x ∂t ∂v
(11)
admitted by (8) the condition
ξv2 + τv2 6= 0
(12)
is a sufficient but not necessary condition in order to yield nonlocal symmetries of (6). The nature of potential symmetries allows one to extend the uses of point symmetries to such nonlocal symmetries. In particular: 1. Invariant solutions of (7), yield solutions of (5) which are not invariant solutions for any local symmetry admitted by (5). 2. If (5) admits a potential symmetry leading to the linearization of (7), then (5) is linearized by a noninvertible mapping.
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Bluman [1] gave theorems which give necessary and sufficient conditions under which nonlinear partial differential equations (scalar or systems) can be transformed to linear PDE’s by invertible mappings. In particular such an invertible mapping does not exist if: 1. a nonlinear scalar PDE does not admit an infinite-parameter Lie group of contact transformations; 2. a nonlinear system of PDE’s does not admit an infinite-parameter Lie group of point transformations. Suppose (5) cannot be linearized by an invertible mapping but an associated system (7), admits an infinite-parameter Lie group of point transformations which leads to its linearization by an invertible mapping; then (5) is linearized by a noninvertible mapping. In [15] two algorithms were proposed which extend the nonclassical method to a potential system (7) or a potential equation (8):
• Algorithm I Nonclassical potential system approach: The nonclassical method is applied to the associated potential system (7). Any Lie group of point transformations v = ξ (x, t , u, v)
∂ ∂ ∂ ∂ + τ (x, t , u, v) + φ(x, t , u, v) + ψ(x, t , u, v) , ∂x ∂t ∂u ∂v
(13)
admitted by (7) yields a nonlocal symmetry potential symmetry of the given PDE (6) if the following condition is satisfied
ξv2 + τv2 + φv2 6= 0.
(14)
• Algorithm II Nonclassical potential equation approach: The nonclassical method is applied to the associated potential Eq. (8). Any Lie group of point transformations X = ξ (x, t , v)
∂ ∂ ∂ + τ (x, t , v) + ψ(x, t , v) ∂x ∂t ∂v
(15)
admitted by (8) yields a nonlocal symmetry potential symmetry of the given PDE (6) if the following condition is satisfied
ξv2 + τv2 6= 0.
(16)
Algorithm I has been considered in Bluman and Shtelen [16] and Saccomandi [17], Algorithm II has been considered in [18] for a dissipative KdV equation, but neither of these papers exhibited nonclassical potential solutions. The nonclassical symmetries for the Burgers have been considered in [19,20]. The nonclassical potential symmetries for the Burgers equation have been derived in [21] as nonclassical symmetries of the integrated equation (Algorithm II). In [15] it was pointed out that often the nonclassical method when it is applied to the potential system (Algoritm I) yields a set of undetermined determining equations while the nonclassical method is much easier to apply to the potential equation (Algorithm II). However we point out that a great disadvantage of Algoritm II is that condition (16) is a sufficient but not necessary condition in order to see if a generator is a nonclassical potential generator or not. In [22] we proposed a modification to the nonclassical potential system approach, in a way in which is easy to apply and we can give a sufficient and necessary condition in order to see if a generator is a nonclassical potential generator. Note that if the generator considered is not a nonclassical potential generator then no new solutions are found, i.e. all such solutions can be obtained from the nonclassical method applied to the given PDE (6).
• Modified Algorithm I Modified nonclassical potential system approach: The nonclassical method is applied to the associated potential system (7). Any Lie group of point transformations v = ξ (x, t , v)
∂ ∂ ∂ ∂ + τ (x, t , v) + φ(x, t , u, v) + ψ(x, t , v) , ∂x ∂t ∂u ∂v
(17)
admitted by (7) yields a nonlocal symmetry potential symmetry of the given PDE (6) if condition (14) is satisfied. In [22] by using the Modified Algorithm I, some nonclassical potential symmetries for the Burgers’ equation were derived. In [23] the authors constructed a hierarchy of coupled integrable systems. They pointed out that it might be very difficult to find a reduction of the integrable hierarchy to scalar Burgers’ equations. It could be interesting and the goal of a future work to apply modified nonclassical potential system approach to the first two nonlinear systems of that hierarchy. The aim of this work is to obtain new nonclassical potential symmetries and reductions for the Burgers’ equation (1) by applying the modified potential system approach to (1). We note that another procedure could be to apply the nonclassical potential method to the heat equation ft − fxx = 0 and generate the solutions of the Burgers’ equation through the Cole–Hopf transformation u = −(ln f )x . In [23] the authors constructed a hierarchy of coupled integrable systems. They pointed out that it might be very difficult to find a reduction of the integrable hierarchy to scalar Burgers’ equations. It could be interesting and our goal for a future work to apply the modified nonclassical potential system approach to the first two nonlinear systems of that hierarchy.
M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
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2. Nonclassical potential symmetries for the Burgers equation Let (1) be Burgers’ equation, in order to find the potential symmetries we write the equation in the conserved form. From this conserved form the associated auxiliary system
v x = u,
1
v t = ux − u2
(18)
2 with potentials as additionals dependent variables is given. If (u(x), v(x)) satisfies (18) then u(x) solves (1) and v(x) solves 1
vt = vxx − (vx )2 .
(19) 2 To obtain nonclassical potential symmetries for the Burgers’ equation, we apply the nonclassical method to (18) and so we require (18) and the invariant surface condition
ξ
∂v ∂v +τ − ψ = 0, ∂x ∂t
(20)
to be invariant under the infinitesimal generator (13). In the case τ 6= 0, without loss of generality, we may set τ (x, t , u, v) = 1. The nonclassical method applied to (18) gives rise to two determining equations for the infinitesimals ξ (x, t , u, v), φ(x, t , u, v) and ψ(x, t , u, v). If we require that
∂ξ ∂ψ = =0 ∂u ∂u
(modified Algorithm I) then ξ = ξ (x, t , v), ψ = ψ(x, t , v) and
∂ξ 2 ∂ψ ∂ξ ∂ψ u + − u+ , ∂v ∂v ∂x ∂x where ξ = ξ (x, t , v) and ψ = ψ(x, t , v) are related by the following conditions φ=−
(21)
∂ 2ξ ∂ξ + = 0, ∂v 2 ∂v ∂ 2ψ ∂ψ ∂ 2ξ ∂ξ −2 2 − +4 + 4ξ = 0, ∂v ∂v ∂v∂ x ∂v ∂ψ ∂ 2ψ ∂ξ ∂ 2ξ ∂ξ ∂ξ − −2 −2 ψ + 2 + 2ξ + = 0, ∂x ∂v∂ x ∂v ∂x ∂x ∂t ∂ 2ψ ∂ψ ∂ξ + + 2 ψ = 0. 2 ∂x ∂t ∂x 2
Solving these equations we obtain
ξ = ξ (x, t ), ψ = α(x, t )ev/2 + β(x, t ),
(22)
where ξ , α , and β , are related by
∂β ∂ 2ξ ∂ξ ∂ξ + 2 − 2ξ − = 0, ∂x ∂x ∂x ∂t ∂ 2 α ∂α ∂ξ − 2 + +2 α = 0, ∂x ∂t ∂x ∂β ∂ξ ∂ 2β − −2 β + 2 = 0. ∂t ∂x ∂x
(23) (24) (25)
Although we are not able to solve this system in general we can obtain some solutions, setting ξ = ξ (x), α = α(x) and β = β(x), we obtain nontrivial solutions given in Table 1 where a, b, c, d and k are arbitrary constants. 3 3 gauss_b −2, − 13 , 23 , − ax k2 + gauss_a −2, − 13 , 23 , − ax k1 3c 3c F (x) = . 3 − ax3c − 1 From Table 1 we can derive sixteen nonlocal generators vi = ξi ∂x + ∂t + ψi ∂v + φi ∂u with i = 1, . . . , 16 where φi is given by (21), ψi is given by (22), ξi , αi and βi with i = 1, . . . , 16 are given in Table 1.
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M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
Table 1
ξi
βi
αi
1 2 3 4 5
a tan(ax + b) + c −a tanh(ax + b) + c −a cot(ax + b) + c −a coth(ax + b) + c 3a tan(ax + b)
2ac tan(ax + b), −2ac tanh(ax + b) −2ac cot(ax + b) −2ac coth(ax + b) 2a2 3 tan2 (ax + b) + 1
6
−3a tanh(ax + b) −3a cot(ax + b) −3a coth(ax + b) − x+1 b + c − x+3 b
2a2 3 tanh2 (ax + b) − 1
(adx + k) tan(ax + b) + d −(adx + k) tanh(ax + b) + d −(adx + k) cot(ax + b) + d −(adx + k) coth(ax + b) + d k 3 tan2 (ax + b) + 1 k 3 tanh2 (ax + b) − 1 k 3 cot2 (ax + b) − 1 k 3 coth2 (ax + b) − 1 k 4 k3 (x + b)2 − 3(x+ b) k k3 (x + b)3 − (x+4b)2
i
7 8 9 10
2a2 3 cot2 (ax + b) − 1
2a2 3 coth2 (ax + b) − 1
− x2c , +b 6 , 2 (x+b)
− a23x(3ax+)3ac a tan (a(x + 2bt + c )) −a tanh (a(x + 2bt + c )) −a cot (a(x + 2bt + c )) −a coth (a(x + 2bt + c )) 1 − x+2bt +c 2
11 12 13 14 15 16
6a2 x a2 x3 +3ac
,
F (x )
keb(x+bt ) a tan (a(x + 2bt + c )) keb(x+bt ) a tanh (a(x + 2bt + c )) keb(x+bt ) a cot (a(x + 2bt + c )) keb(x+bt ) a coth (a(x + 2bt + c )) 1 keb(x+bt ) x+2bt +c
2ab tan (a(x + 2bt + c )) − 2b2 , −2ab tanh (a(x + 2bt + c )) − 2b2 , 2ab cot (a(x + 2bt + c )) − 2b2 , 2ab coth (a(x + 2bt + c )) − 2b2 , 1 eb(x+bt ) x+2bt +c
Generators v1 , v10 , with c = 0, were derived in [21] by considering algorithm II and in [22] by considering modified Algorithm I. Generators v5 , v6 , and v8 , were derived in [24]. Case 1: Setting in v1 c = 0, solving the invariant surface condition we obtain the nonclassical symmetry reduction z=
log sin(ax + b) a2
− t,
v = k3 − 2 log −2d log sin(ax + b) − a2 dx2 − 2akx + 2a2 h .
(27)
Substitution of (27) into (19) leads to the ODE d2 h dz 2
− a2
dh dz
+ a2 d = 0
(28)
which has solution h(z ) = k1 ea
2z
+
a2 dz + d a2
+ k2 ,
(29)
from which we obtain that an exact solution for (19) is
2 2 2 v = −2 log 2a2 k1 sin(ax + b) − a2 dea t x2 − 2akea t x + −2a2 dt + 2a2 k2 + 2d ea t + 2a2 t + k3
(30)
and by (18) we get that
2
2t
2 2a3 k1 cos(ax + b) − 2a2 dea t x − 2akea u=−
2a2 k1 sin(ax + b) − a2 dea t x2 − 2akea t x + −2a2 dt + 2a2 k2 + 2d ea 2
2
2t
is a new exact solution of Eq. (1). Case 2: Setting in v2 c = 0, solving the invariant surface condition we obtain the nonclassical symmetry reduction z=−
log sinh(ax + b) a2
− t,
v = k3 − 2 log −2d log sinh(ax + b) + a2 dx2 − 2ax − 2a2 h .
(31)
Substitution of (31) into (19) leads to the ODE d2 h dz 2
+ a2
dh dz
− a2 d = 0
which has solution h(z ) = k2 e−a
2z
+
a2 dz − d a2
+ k1 ,
from which we obtain that an exact solution for (19) is
2 v = k3 − 2 log −2a2 k2 ea t sinh(ax + b) + a2 dx2 − 2ax + 2a2 dt − 2a2 k1 + 2d
(32)
M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
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and by (18) we get that
2
2 −2a3 kea t cosh(ax + b) + 2a2 dx − 2a u=−
−2a2 kea2 t sinh(ax + b) + a2 dx2 − 2ax + 2a2 dt − 2a2 k1 + 2d
is a new exact solution of Eq. (1). Case 3: Setting in v3 c = 0, solving the invariant surface condition we obtain the nonclassical symmetry reduction z=
log cos(ax + b) a2
v = k3 − 2 log −2d log cos(ax + b) − a2 dx2 − 2akx + 2a2 h .
− t,
(33)
Substitution of (33) into (19) leads to the ODE (28) which has solution (29) from which we obtain that an exact solution for (19) is
2 2 2 v = −2 log 2a2 k1 cos(ax + b) − a2 dea t x2 − 2akea t x + −2a2 dt + 2a2 k2 + 2d ea t + 2a2 t + k3
(34)
and by (18) we get that
2
2 −2a3 k1 sin(ax + b) − 2a2 dea t x − 2akea u=−
2t
2a2 k1 cos(ax + b) − a2 dea t x2 − 2akea t x + −2a2 dt + 2a2 k2 + 2d ea 2
2
2t
is a new exact solution of Eq. (1). Case 4: Setting in v4 c = 0, solving the invariant surface condition we obtain the nonclassical symmetry reduction z=−
log cosh(ax + b) a2
− t,
v = k3 − 2 log −2d log cosh(ax + b) + a2 dx2 − 2akx + 2a2 h .
(35)
Substitution of (35) into (19) leads to the ODE d2 h dz 2
+ a2
dh dz
+ a2 d = 0
which has solution h(z ) = k2 e−a
2z
− dz + k1 +
d a2
,
from which we obtain that an exact solution for (19) is
2 v = k3 − 2 log 2a2 k2 ea t cosh(ax + b) + a2 dx2 − 2akx + 2a2 dt + 2a2 k1 + 2d
(36)
and by (18) we get that
2
2 2a3 k2 ea t sinh (ax + b) + 2a2 dx − 2ak u=−
2a2 k2 ea t cosh(ax + b) + a2 dx2 − 2akx + 2a2 dt + 4a2 dk1 2
(37)
is a new exact solution of Eq. (1). Case 5: From v5 , solving the invariant surface condition we obtain the nonclassical symmetry reduction z=
log sin(ax + b) 3a2
− t,
v=
2 log sin(ax + b) 3
h p − 2 log e 2 cos(ax + b) − 3 sin (ax + b) + k3 .
(38)
Substitution of (38) into (19) leads to the ODE 2
d2 h dz 2
+
dh
2
dz
− 10a2
dh dz
+ 16a4 = 0
(39)
which has solution
h( z ) =
2 2 6a2 log e3a z +3a k1 − 1 + 2a2 3a2 z + 3a2 k1
3a2
+ k2
(40)
from which we obtain that a solution for (19) is
k2 2 k2 2 2 2 v = − 2 log e 2 +4a k1 cos(ax + b) sin(ax + b) − e3a t + 2 +a k1 cos(ax + b) − e4a t − 8a2 t − k3
(41)
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and by (18) we get that
2 −ae
k2 2
+4a2 k1
u=−
k2 2
e
k2 k2 2 2 2 sin2 (ax + b) + ae3a t + 2 +a k1 sin(ax + b) + ae 2 +4a k1 cos2 (ax + b)
k2 2 2 2 cos(ax + b) sin(ax + b) − e3a t + 2 +a k1 cos(ax + b) − e4a t
+4a2 k1
is a new exact solution of Eq. (1). Case 6: From v6 , solving the invariant surface condition we obtain the nonclassical symmetry reduction z=−
log sinh(ax + b)
h p v = −2 log e 2 (cosh(ax + b))2/3 3 coth(ax + b) − 1 .
− t,
3a2
(42)
Substitution of (42) into (19) leads to the ODE 2
d2 h dz 2
+
dh
2
dh
+ 10a2
dz
dz
+ 16a4 = 0
(43)
which has solution
h(z ) = k2 −
2 2 8a2 3a2 z + 3a2 k1 − 6a2 log e3a z +3a k1 − 1
(44)
3a2
from which we obtain that a solution for (19) is
k2 k2 2 2 2 2 v = 8a2 k1 − 2 log −e4a t + 2 cosh(ax + b) sinh(ax + b) + ea t + 2 +3a k1 cosh (ax + b) − e4a k1
(45)
and by (18) we get that k2 k2 k2 2 2 2 2 2 −ae4a t + 2 sinh2 (ax + b) + aea t + 2 +3a k1 sinh(ax + b) − ae4a t + 2 cosh2 (ax + b)
u=−
2 t + k2 2
k2 2 2 2 cosh(ax + b) sinh (ax + b) + ea t + 2 +3a k1 cosh(ax + b) − 3e4a k1
−e4a
is a new exact solution of Eq. (1) Case 7: From v7 , solving the invariant surface condition we obtain the nonclassical symmetry reduction
z=
log cos(ax + b)
− t,
3a2
v=−
h
6 log e 2 sin(ax + b) −
√ 3
cos(ax + b) − 2 log cos (ax + b) 3
.
(46)
Substitution of (46) into (19) leads to the ODE (39) which has solution (40) from which we obtain that a solution for (19) is
k2 2 k2 2 2 2 v = 8a2 t − 2 log e 2 +4a k1 cos(ax + b) sin(ax + b) − e3a t + 2 +a k1 sin(ax + b) − e4a t
(47)
and by (18) we get that
2 −ae
k2 2
+4a2 k1
u=− e
k2 2
sin2 (ax + b) + ae
+4a2 k1
k2 2
+4a2 k1
k2 2 2 cos2 (ax + b) − ae3a t + 2 +a k1 cos(ax + b)
k2 2 2 2 cos(ax + b) sin(ax + b) − e3a t + 2 +a k1 sin(ax + b) − e4a t
is a new exact solution of Eq. (1) Case 8: From v8 , solving the invariant surface condition we obtain the nonclassical symmetry reduction z=−
log cosh(ax + b) 3a2
h
− t,
v = −2 log
e 2 sinh(ax + b)
√ 3
cosh(ax + b)
! −1 .
(48)
Substitution of (46) into (19) leads to the ODE (43) which has solution (44) from which we obtain that a solution for (19) is
v = 8a2 k1 − 2 log
e
2a2 t +k2 +6a2 k1 2
−e
8a2 t +k2 2
2k 1
cosh(ax + b) sinh(ax + b) − e4a
(49)
and by (18) we get that k2 k2 k2 2 2 2 2 2 a cosh(ax + b) ea t + 2 +3a k1 − e4a t + 2 cosh(ax + b) − ae4a t + 2 sinh2 (ax + b)
u=−
k2 k2 2 2 2 2 ea t + 2 +3a k1 − e4a t + 2 cosh(ax + b) sinh(ax + b) − e4a k1
is a new exact solution of Eq. (1).
M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
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Case 9.a: From v9 , solving the invariant surface condition we obtain the nonclassical symmetry reduction log (cx + bc − 1)
x + − t, c2 c v = −2 log(V ) + 8 log c + 2 log 3 + 4 log 2 V = ((18 − 18bc ) k3 − 18ck3 x) log (cx + bc − 1) − 3c 3 k3 x3 + −9bc 3 − 9c 2 k3 x2 + −6b2 c 3 − 6bc 2 + 12c k3 + 2c 3 h x − 2c 3 k4 + 6k3 + 2bc 3 − 2c 2 h. z=
(50)
Substitution of (52) into (19) leads to the ODE d2 h
dh + c2 − 9c 2 k3 = 0 dz 2 dz which has solution h(z ) = k2 e−c
2z
+
9c 2 k3 z − 9k3
+ k1 c2 from which we obtain that a solution for (19) is v = −2 log (V ) + 2cx + 8 log c+ 2 log 3 + 4 log 2 V = −3c 3 k3 x3 + 9c 2 − 9bc 3 k3 x2 + −18c 3 k3 t + −6b2 c 3 + 12bc 2 − 24c k3 + 2c 3 k1 x ecx 2 + 18c 2 − 18bc 3 k3 t − 2c 3 k4 + (24 − 18bc ) k3 + 2bc 3 − 2c 2 k1 ecx + 2c 2 k2 ec t
(51)
and by (18) we get a new exact solution of Eq. (1). Case 9.b: From v9 , with c = 0 solving the invariant surface condition we obtain the nonclassical symmetry reduction x2
− bx − t , 4 bk3 x3 3b2 k3 x2 k4 x b3 k3 x h k3 x + + − + + . v = −2 log z=−
2
8
2
4
6
2
(52)
6
Substitution of (52) into (19) leads to the ODE d2 h
+ 9k3 = 0 dz 2 which has solution h( z ) = −
9k3 z 2
+ k2 z + k1 2 from which we obtain that a solution for (19) is k3 x4 bk3 x3 3k3 tx2 k2 x2 3bk3 tx k4 x b3 k3 x bk2 x 3k3 t 2 k2 t k1 v = −2 log − − − − − − + − − − + 16
4
4
12
2
6
2
6
4
6
6
(53)
and by (18) we get a new exact solution of Eq. (1). Case 10: From v10 , solving the invariant surface condition we obtain the nonclassical symmetry reduction x2
bx
− t, 3 v = 2 − log k3 x5 + 5bk3 x4 + 10b2 k3 x3 + 10b3 k3 x2 + 4b4 k3 + 4h x + 4k4 + 4bh + log 24 . z=−
6
−
(54)
Substitution of (54) into (19) leads to the ODE d2 h
+ 45k3 = 0 dz 2 which has solution h( z ) = −
45k3 z 2
+ k2 z + k1 2 from which we obtain that a solution for (19) is v = −2 log (V ) + 4 log 3 + 8 log 2 V = −9k3 x5 − 45bk3 x4 − 180k3 tx3 − 60b2 k3 x3 − 4k2 x3 − 540bk3 tx2 − 12bk2 x2 − 540k3 t 2 x −360b2 k3 tx − 24k2 tx + 24b4 k3 x − 8b2 k2 x + 24k1 x − 540bk3 t 2 − 24bk2 t + 24k4 + 24bk1 and by (18) we get a new exact solution of Eq. (1). In Fig. 1 we can see solution (37) with k2 = k1 = a = 1, b = d = 0, for t = 0, 1, 3.
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M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
Fig. 1. Solution (37) and the corresponding time evolution plot related at times t = 0, t = 1 and t = 3.
3. Concluding remarks In this work we have introduced new classes of symmetries for PDE’s. If a PDE can be written in a conserved form (6), then a related system (7) and a related integrated equation may be obtained. The ansatz to generate nonclassical solutions of the associated potential system could yield solutions of the original equation which are neither nonclassical solutions of the original equation nor solutions arising from classical potential symmetries. To explain these results, we have proved the existence of families of solutions of the Burgers’ equation which are not found by means of potential nor nonclassical symmetry reductions. Acknowledgments The support of DGICYT project MTM2006-05031, Junta de Andalucía group FQM–201 and project P06-FQM-01448 are gratefully acknowledged. We warmly thank the referees for their useful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, Berlin, 1989. Hill, Differential Equations and Group Methods, CRC Press, Boca Raton, 1992. P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin, 1986. L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. H. Stephani, Differential Equations: Their Solution Using Symmetries, Cambridge University Press, Cambridge, 1989. G.W. Bluman, S. Kumei, J. Math. Phys. 21 (1980) 1019. G.W. Bluman, S. Kumei, European J. Appl. Math. 1 (1990) 189–216. S. Kumei, G.W. Bluman, SIAM J. Appl. Math. 42 (1982) 1157–1173. G.W. Bluman, J. Cole, The general similarity solution of the heat equation, J. Math. Mech. 18 (1969) 1025. I.S. Krasil’shchik, A.M. Vinogradov, Acta Appl. Math. 2 (1984) 79–96. I.S. Krasil’shchik, A.M. Vinogradov, Acta Appl. Math. 15 (1989) 161–209. A.M. Vinogradov, Symmetries of Partial Differential Equations, Kluver, Dordrecht, 1989. I.Sh. Akhatov, R.K. Gazizov, N.H. Ibragimov, J. Soviet. Math. 55 (1991) 1401. G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806–811. G.W. Bluman, Z. Yan, European J. Appl. Math. 16 (1989) 235–266. 2005. G.W. Bluman, V. Shtelen, Mathematics is for Solving Problems, in: S.L.P. Cook, V. Roytburd, M. Tulin (Eds.), SIAM, pp. 105–118. G. Saccomandi, Potential symmetries and direct reduction methods of order two, J. Phys. A 30 (1997) 2211–2217. M.L. Gandarias, Nonclassical Potential symmetries of a porous medium equation, in: I Colloquium Lie Theory, Univ. Vigo, Vigo, 2000, pp. 37–45. D.J. Arrigo, P. Broadbridge, J.M. Hill, J. Math. Phys. 34 (1993) 4692–4703. E. Pucci, J. Phys. A: Math. Gen. 25 (1992) 2631. M.L. Gandarias, Nonclassical Potential symmetries of the Burgers equation, in: Symmetry in Nonlinear Mathematical Physics, Natl. Acad. Sci. Ukraine. Inst. Math., Kiev, 1997, pp. 130–137. [22] M.L. Gandarias, New Potential symmetries, in: CRM Proceedings and Lecture Notes, vol. 25, American Math. Society. Publ., Providence, RI, 2000, pp. 161–165. [23] W.X. Ma, Z.X. Zhou, Prog. Theoret. Phys. 96 (1996) 449–457. [24] Qin Maochang, J. Phys. A: Math. Gen. 40 (2007) 4541–4551.
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Nonclassical potential symmetries for the Burgers equation M.L. Gandarias ∗ , M.S. Bruzon Departamento de Matematicas, Universidad de Cadiz, PO.BOX 40, 11510 Puerto Real, Cadiz, Spain
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Keywords: Potential symmetries Nonclassical potential symmetries Partial differential equations
abstract In this work classes of symmetries for partial differential equations which can be written in a conserved form are found. These nonclassical potential symmetries are realized as nonclassical symmetries of the associated potential system and are neither classical potential symmetries realized as Lie symmetries of a related auxiliary system nor nonclassical symmetries of the considered equation. Some of these symmetries are carried out for the Burgers equation. These symmetries are realized as the nonclassical symmetries of the associated potential system and yield many new nonlocal generators and new explicit solutions of the Burgers equation. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Symmetries admitted by a partial differential equation (PDE) are useful for finding invariant solutions. These solutions are obtained by using group invariants to reduce the number of independent variables. The fundamental basis of the technique is that, when a differential equation is invariant under a Lie group of transformations, a symmetry reduction exists which reduces the equation to a lower dimensional equation. The machinery of Lie group theory provides the systematic method to search for these special group-invariant solutions. For PDE’s with two independent variables, as the Burgers equation
ut +
u2 2
− ux
=0
(1)
x
a single group reduction transforms the PDE into ordinary differential equations (ODE’s), which are generally easier to solve than the original PDE. Most of the required theory and description of the method can be found in for example [1–5]. Local symmetries admitted by a nonlinear PDE are also useful to discover whether or not the equation can be linearized by an invertible mapping and construct an explicit linearization when one exists. A nonlinear scalar PDE is linearizable by an invertible contact (point) transformation if and only if it admits an infinite-parameter Lie group of contact transformations satisfying specific criteria [6,1,7,8]. There have been several generalizations of the classical Lie group method for symmetry reductions. Bluman and Cole [9] developed the nonclassical method to study the symmetry reductions of the heat equation. The basic idea of the method is to require that the N order PDE
∆ x, t , u, u(1) (x, t ), . . . , u(N ) (x, t ) = 0,
(2) (l)
where (x, t ) ∈ R are the independent variables, u ∈ R is the dependent variable and u (x, t ) denote the set of all partial derivatives of l order of u and the invariant surface condition 2
ξ ∗
∂u ∂u +τ − φ = 0, ∂x ∂t
Corresponding author. E-mail address: [email protected] (M.L. Gandarias).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.078
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which is associated with the vector field v = ξ (x, t , u)
∂ ∂ ∂ + τ (x, t , u) + φ(x, t , u) , ∂x ∂t ∂u
(4)
are both invariant under the transformation with infinitesimal generator (4). Since then, a great number of papers have been devoted to the study of nonclassical symmetries of nonlinear PDE’s in both one and several dimensions. An obvious limitation of group-theoretic methods based in local symmetries, in their utility for particular PDE’s, is that there are partial differential equations (PDE’s) of physical interest possessing few symmetries or none at all [3]. It turns out that PDE’s can admit nonlocal symmetries whose infinitesimal generators depend on integrals of the dependent variables in some specific manner. It also happens that if a nonlinear scalar PDE does not admit an infinite-parameter Lie group of contact transformations is not linearizable by an invertible contact transformation. However most of the interesting linearizations involve non-invertible transformations, such linearizations can be found by embedding given nonlinear PDE’s in auxiliary systems of PDE’s [1]. Krasil’shchik and Vinogradov [10–12] gave criteria which must be satisfied by nonlocal symmetries of a PDE when realized as local symmetries of a system of PDE’s which ‘covers’ the given PDE. Akhatov, Gazizov and Ibragimov [13] gave nontrivial examples of nonlocal symmetries generated by heuristic procedures. In [14] Bluman introduced a method to find a new class of symmetries for a PDE. Suppose a given scalar second order PDE F (x, t , u, ux , ut , uxx , uxt , utt ) = 0,
(5)
where the subscripts denote the partial derivatives of u, can be written as a conservation law D Dt
f ( x , t , u, ux , ut ) −
D Dx
g (x, t , u, ux , ut ) = 0,
for some functions f and g of the indicated arguments. Here
(6) D Dx
and
D Dt
are total derivative operators defined by
∂ ∂ ∂ ∂ + ux + uxx + uxt + ···, ∂x ∂u ∂ ux ∂ ut ∂ ∂ ∂ ∂ = + ut + uxt + utt + ···. Dt ∂t ∂u ∂ ux ∂ ut D
Dx D
=
Through the conservation law (6) one can introduce an auxiliary potential variable v and form an auxiliary potential system (system approach)
vx = f (x, t , u, ux , ut ), vt = g (x, t , u, ux , ut ).
(7)
For many physical equations one can eliminate u from the potential system (7) and form an auxiliary integrated or potential equation (integrated equation approach) G(x, t , v, vx , vt , vxx , vxt , vtt ) = 0,
(8)
for some function G of the indicated arguments. Any Lie group of point transformations w = ξ (x, t , u, v)
∂ ∂ ∂ ∂ + τ (x, t , u, v) + φ(x, t , u, v) + ψ(x, t , u, v) , ∂x ∂t ∂u ∂v
(9)
admitted by (7) yields a nonlocal symmetry potential symmetry of the given PDE (6) if and only if the following condition is satisfied
ξv2 + τv2 + φv2 6= 0.
(10)
We point out that if we consider a Lie group of point transformations u = ξ (x, t , v)
∂ ∂ ∂ + τ (x, t , v) + ψ(x, t , v) ∂x ∂t ∂v
(11)
admitted by (8) the condition
ξv2 + τv2 6= 0
(12)
is a sufficient but not necessary condition in order to yield nonlocal symmetries of (6). The nature of potential symmetries allows one to extend the uses of point symmetries to such nonlocal symmetries. In particular: 1. Invariant solutions of (7), yield solutions of (5) which are not invariant solutions for any local symmetry admitted by (5). 2. If (5) admits a potential symmetry leading to the linearization of (7), then (5) is linearized by a noninvertible mapping.
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M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
Bluman [1] gave theorems which give necessary and sufficient conditions under which nonlinear partial differential equations (scalar or systems) can be transformed to linear PDE’s by invertible mappings. In particular such an invertible mapping does not exist if: 1. a nonlinear scalar PDE does not admit an infinite-parameter Lie group of contact transformations; 2. a nonlinear system of PDE’s does not admit an infinite-parameter Lie group of point transformations. Suppose (5) cannot be linearized by an invertible mapping but an associated system (7), admits an infinite-parameter Lie group of point transformations which leads to its linearization by an invertible mapping; then (5) is linearized by a noninvertible mapping. In [15] two algorithms were proposed which extend the nonclassical method to a potential system (7) or a potential equation (8):
• Algorithm I Nonclassical potential system approach: The nonclassical method is applied to the associated potential system (7). Any Lie group of point transformations v = ξ (x, t , u, v)
∂ ∂ ∂ ∂ + τ (x, t , u, v) + φ(x, t , u, v) + ψ(x, t , u, v) , ∂x ∂t ∂u ∂v
(13)
admitted by (7) yields a nonlocal symmetry potential symmetry of the given PDE (6) if the following condition is satisfied
ξv2 + τv2 + φv2 6= 0.
(14)
• Algorithm II Nonclassical potential equation approach: The nonclassical method is applied to the associated potential Eq. (8). Any Lie group of point transformations X = ξ (x, t , v)
∂ ∂ ∂ + τ (x, t , v) + ψ(x, t , v) ∂x ∂t ∂v
(15)
admitted by (8) yields a nonlocal symmetry potential symmetry of the given PDE (6) if the following condition is satisfied
ξv2 + τv2 6= 0.
(16)
Algorithm I has been considered in Bluman and Shtelen [16] and Saccomandi [17], Algorithm II has been considered in [18] for a dissipative KdV equation, but neither of these papers exhibited nonclassical potential solutions. The nonclassical symmetries for the Burgers have been considered in [19,20]. The nonclassical potential symmetries for the Burgers equation have been derived in [21] as nonclassical symmetries of the integrated equation (Algorithm II). In [15] it was pointed out that often the nonclassical method when it is applied to the potential system (Algoritm I) yields a set of undetermined determining equations while the nonclassical method is much easier to apply to the potential equation (Algorithm II). However we point out that a great disadvantage of Algoritm II is that condition (16) is a sufficient but not necessary condition in order to see if a generator is a nonclassical potential generator or not. In [22] we proposed a modification to the nonclassical potential system approach, in a way in which is easy to apply and we can give a sufficient and necessary condition in order to see if a generator is a nonclassical potential generator. Note that if the generator considered is not a nonclassical potential generator then no new solutions are found, i.e. all such solutions can be obtained from the nonclassical method applied to the given PDE (6).
• Modified Algorithm I Modified nonclassical potential system approach: The nonclassical method is applied to the associated potential system (7). Any Lie group of point transformations v = ξ (x, t , v)
∂ ∂ ∂ ∂ + τ (x, t , v) + φ(x, t , u, v) + ψ(x, t , v) , ∂x ∂t ∂u ∂v
(17)
admitted by (7) yields a nonlocal symmetry potential symmetry of the given PDE (6) if condition (14) is satisfied. In [22] by using the Modified Algorithm I, some nonclassical potential symmetries for the Burgers’ equation were derived. In [23] the authors constructed a hierarchy of coupled integrable systems. They pointed out that it might be very difficult to find a reduction of the integrable hierarchy to scalar Burgers’ equations. It could be interesting and the goal of a future work to apply modified nonclassical potential system approach to the first two nonlinear systems of that hierarchy. The aim of this work is to obtain new nonclassical potential symmetries and reductions for the Burgers’ equation (1) by applying the modified potential system approach to (1). We note that another procedure could be to apply the nonclassical potential method to the heat equation ft − fxx = 0 and generate the solutions of the Burgers’ equation through the Cole–Hopf transformation u = −(ln f )x . In [23] the authors constructed a hierarchy of coupled integrable systems. They pointed out that it might be very difficult to find a reduction of the integrable hierarchy to scalar Burgers’ equations. It could be interesting and our goal for a future work to apply the modified nonclassical potential system approach to the first two nonlinear systems of that hierarchy.
M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
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2. Nonclassical potential symmetries for the Burgers equation Let (1) be Burgers’ equation, in order to find the potential symmetries we write the equation in the conserved form. From this conserved form the associated auxiliary system
v x = u,
1
v t = ux − u2
(18)
2 with potentials as additionals dependent variables is given. If (u(x), v(x)) satisfies (18) then u(x) solves (1) and v(x) solves 1
vt = vxx − (vx )2 .
(19) 2 To obtain nonclassical potential symmetries for the Burgers’ equation, we apply the nonclassical method to (18) and so we require (18) and the invariant surface condition
ξ
∂v ∂v +τ − ψ = 0, ∂x ∂t
(20)
to be invariant under the infinitesimal generator (13). In the case τ 6= 0, without loss of generality, we may set τ (x, t , u, v) = 1. The nonclassical method applied to (18) gives rise to two determining equations for the infinitesimals ξ (x, t , u, v), φ(x, t , u, v) and ψ(x, t , u, v). If we require that
∂ξ ∂ψ = =0 ∂u ∂u
(modified Algorithm I) then ξ = ξ (x, t , v), ψ = ψ(x, t , v) and
∂ξ 2 ∂ψ ∂ξ ∂ψ u + − u+ , ∂v ∂v ∂x ∂x where ξ = ξ (x, t , v) and ψ = ψ(x, t , v) are related by the following conditions φ=−
(21)
∂ 2ξ ∂ξ + = 0, ∂v 2 ∂v ∂ 2ψ ∂ψ ∂ 2ξ ∂ξ −2 2 − +4 + 4ξ = 0, ∂v ∂v ∂v∂ x ∂v ∂ψ ∂ 2ψ ∂ξ ∂ 2ξ ∂ξ ∂ξ − −2 −2 ψ + 2 + 2ξ + = 0, ∂x ∂v∂ x ∂v ∂x ∂x ∂t ∂ 2ψ ∂ψ ∂ξ + + 2 ψ = 0. 2 ∂x ∂t ∂x 2
Solving these equations we obtain
ξ = ξ (x, t ), ψ = α(x, t )ev/2 + β(x, t ),
(22)
where ξ , α , and β , are related by
∂β ∂ 2ξ ∂ξ ∂ξ + 2 − 2ξ − = 0, ∂x ∂x ∂x ∂t ∂ 2 α ∂α ∂ξ − 2 + +2 α = 0, ∂x ∂t ∂x ∂β ∂ξ ∂ 2β − −2 β + 2 = 0. ∂t ∂x ∂x
(23) (24) (25)
Although we are not able to solve this system in general we can obtain some solutions, setting ξ = ξ (x), α = α(x) and β = β(x), we obtain nontrivial solutions given in Table 1 where a, b, c, d and k are arbitrary constants. 3 3 gauss_b −2, − 13 , 23 , − ax k2 + gauss_a −2, − 13 , 23 , − ax k1 3c 3c F (x) = . 3 − ax3c − 1 From Table 1 we can derive sixteen nonlocal generators vi = ξi ∂x + ∂t + ψi ∂v + φi ∂u with i = 1, . . . , 16 where φi is given by (21), ψi is given by (22), ξi , αi and βi with i = 1, . . . , 16 are given in Table 1.
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M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
Table 1
ξi
βi
αi
1 2 3 4 5
a tan(ax + b) + c −a tanh(ax + b) + c −a cot(ax + b) + c −a coth(ax + b) + c 3a tan(ax + b)
2ac tan(ax + b), −2ac tanh(ax + b) −2ac cot(ax + b) −2ac coth(ax + b) 2a2 3 tan2 (ax + b) + 1
6
−3a tanh(ax + b) −3a cot(ax + b) −3a coth(ax + b) − x+1 b + c − x+3 b
2a2 3 tanh2 (ax + b) − 1
(adx + k) tan(ax + b) + d −(adx + k) tanh(ax + b) + d −(adx + k) cot(ax + b) + d −(adx + k) coth(ax + b) + d k 3 tan2 (ax + b) + 1 k 3 tanh2 (ax + b) − 1 k 3 cot2 (ax + b) − 1 k 3 coth2 (ax + b) − 1 k 4 k3 (x + b)2 − 3(x+ b) k k3 (x + b)3 − (x+4b)2
i
7 8 9 10
2a2 3 cot2 (ax + b) − 1
2a2 3 coth2 (ax + b) − 1
− x2c , +b 6 , 2 (x+b)
− a23x(3ax+)3ac a tan (a(x + 2bt + c )) −a tanh (a(x + 2bt + c )) −a cot (a(x + 2bt + c )) −a coth (a(x + 2bt + c )) 1 − x+2bt +c 2
11 12 13 14 15 16
6a2 x a2 x3 +3ac
,
F (x )
keb(x+bt ) a tan (a(x + 2bt + c )) keb(x+bt ) a tanh (a(x + 2bt + c )) keb(x+bt ) a cot (a(x + 2bt + c )) keb(x+bt ) a coth (a(x + 2bt + c )) 1 keb(x+bt ) x+2bt +c
2ab tan (a(x + 2bt + c )) − 2b2 , −2ab tanh (a(x + 2bt + c )) − 2b2 , 2ab cot (a(x + 2bt + c )) − 2b2 , 2ab coth (a(x + 2bt + c )) − 2b2 , 1 eb(x+bt ) x+2bt +c
Generators v1 , v10 , with c = 0, were derived in [21] by considering algorithm II and in [22] by considering modified Algorithm I. Generators v5 , v6 , and v8 , were derived in [24]. Case 1: Setting in v1 c = 0, solving the invariant surface condition we obtain the nonclassical symmetry reduction z=
log sin(ax + b) a2
− t,
v = k3 − 2 log −2d log sin(ax + b) − a2 dx2 − 2akx + 2a2 h .
(27)
Substitution of (27) into (19) leads to the ODE d2 h dz 2
− a2
dh dz
+ a2 d = 0
(28)
which has solution h(z ) = k1 ea
2z
+
a2 dz + d a2
+ k2 ,
(29)
from which we obtain that an exact solution for (19) is
2 2 2 v = −2 log 2a2 k1 sin(ax + b) − a2 dea t x2 − 2akea t x + −2a2 dt + 2a2 k2 + 2d ea t + 2a2 t + k3
(30)
and by (18) we get that
2
2t
2 2a3 k1 cos(ax + b) − 2a2 dea t x − 2akea u=−
2a2 k1 sin(ax + b) − a2 dea t x2 − 2akea t x + −2a2 dt + 2a2 k2 + 2d ea 2
2
2t
is a new exact solution of Eq. (1). Case 2: Setting in v2 c = 0, solving the invariant surface condition we obtain the nonclassical symmetry reduction z=−
log sinh(ax + b) a2
− t,
v = k3 − 2 log −2d log sinh(ax + b) + a2 dx2 − 2ax − 2a2 h .
(31)
Substitution of (31) into (19) leads to the ODE d2 h dz 2
+ a2
dh dz
− a2 d = 0
which has solution h(z ) = k2 e−a
2z
+
a2 dz − d a2
+ k1 ,
from which we obtain that an exact solution for (19) is
2 v = k3 − 2 log −2a2 k2 ea t sinh(ax + b) + a2 dx2 − 2ax + 2a2 dt − 2a2 k1 + 2d
(32)
M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
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and by (18) we get that
2
2 −2a3 kea t cosh(ax + b) + 2a2 dx − 2a u=−
−2a2 kea2 t sinh(ax + b) + a2 dx2 − 2ax + 2a2 dt − 2a2 k1 + 2d
is a new exact solution of Eq. (1). Case 3: Setting in v3 c = 0, solving the invariant surface condition we obtain the nonclassical symmetry reduction z=
log cos(ax + b) a2
v = k3 − 2 log −2d log cos(ax + b) − a2 dx2 − 2akx + 2a2 h .
− t,
(33)
Substitution of (33) into (19) leads to the ODE (28) which has solution (29) from which we obtain that an exact solution for (19) is
2 2 2 v = −2 log 2a2 k1 cos(ax + b) − a2 dea t x2 − 2akea t x + −2a2 dt + 2a2 k2 + 2d ea t + 2a2 t + k3
(34)
and by (18) we get that
2
2 −2a3 k1 sin(ax + b) − 2a2 dea t x − 2akea u=−
2t
2a2 k1 cos(ax + b) − a2 dea t x2 − 2akea t x + −2a2 dt + 2a2 k2 + 2d ea 2
2
2t
is a new exact solution of Eq. (1). Case 4: Setting in v4 c = 0, solving the invariant surface condition we obtain the nonclassical symmetry reduction z=−
log cosh(ax + b) a2
− t,
v = k3 − 2 log −2d log cosh(ax + b) + a2 dx2 − 2akx + 2a2 h .
(35)
Substitution of (35) into (19) leads to the ODE d2 h dz 2
+ a2
dh dz
+ a2 d = 0
which has solution h(z ) = k2 e−a
2z
− dz + k1 +
d a2
,
from which we obtain that an exact solution for (19) is
2 v = k3 − 2 log 2a2 k2 ea t cosh(ax + b) + a2 dx2 − 2akx + 2a2 dt + 2a2 k1 + 2d
(36)
and by (18) we get that
2
2 2a3 k2 ea t sinh (ax + b) + 2a2 dx − 2ak u=−
2a2 k2 ea t cosh(ax + b) + a2 dx2 − 2akx + 2a2 dt + 4a2 dk1 2
(37)
is a new exact solution of Eq. (1). Case 5: From v5 , solving the invariant surface condition we obtain the nonclassical symmetry reduction z=
log sin(ax + b) 3a2
− t,
v=
2 log sin(ax + b) 3
h p − 2 log e 2 cos(ax + b) − 3 sin (ax + b) + k3 .
(38)
Substitution of (38) into (19) leads to the ODE 2
d2 h dz 2
+
dh
2
dz
− 10a2
dh dz
+ 16a4 = 0
(39)
which has solution
h( z ) =
2 2 6a2 log e3a z +3a k1 − 1 + 2a2 3a2 z + 3a2 k1
3a2
+ k2
(40)
from which we obtain that a solution for (19) is
k2 2 k2 2 2 2 v = − 2 log e 2 +4a k1 cos(ax + b) sin(ax + b) − e3a t + 2 +a k1 cos(ax + b) − e4a t − 8a2 t − k3
(41)
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and by (18) we get that
2 −ae
k2 2
+4a2 k1
u=−
k2 2
e
k2 k2 2 2 2 sin2 (ax + b) + ae3a t + 2 +a k1 sin(ax + b) + ae 2 +4a k1 cos2 (ax + b)
k2 2 2 2 cos(ax + b) sin(ax + b) − e3a t + 2 +a k1 cos(ax + b) − e4a t
+4a2 k1
is a new exact solution of Eq. (1). Case 6: From v6 , solving the invariant surface condition we obtain the nonclassical symmetry reduction z=−
log sinh(ax + b)
h p v = −2 log e 2 (cosh(ax + b))2/3 3 coth(ax + b) − 1 .
− t,
3a2
(42)
Substitution of (42) into (19) leads to the ODE 2
d2 h dz 2
+
dh
2
dh
+ 10a2
dz
dz
+ 16a4 = 0
(43)
which has solution
h(z ) = k2 −
2 2 8a2 3a2 z + 3a2 k1 − 6a2 log e3a z +3a k1 − 1
(44)
3a2
from which we obtain that a solution for (19) is
k2 k2 2 2 2 2 v = 8a2 k1 − 2 log −e4a t + 2 cosh(ax + b) sinh(ax + b) + ea t + 2 +3a k1 cosh (ax + b) − e4a k1
(45)
and by (18) we get that k2 k2 k2 2 2 2 2 2 −ae4a t + 2 sinh2 (ax + b) + aea t + 2 +3a k1 sinh(ax + b) − ae4a t + 2 cosh2 (ax + b)
u=−
2 t + k2 2
k2 2 2 2 cosh(ax + b) sinh (ax + b) + ea t + 2 +3a k1 cosh(ax + b) − 3e4a k1
−e4a
is a new exact solution of Eq. (1) Case 7: From v7 , solving the invariant surface condition we obtain the nonclassical symmetry reduction
z=
log cos(ax + b)
− t,
3a2
v=−
h
6 log e 2 sin(ax + b) −
√ 3
cos(ax + b) − 2 log cos (ax + b) 3
.
(46)
Substitution of (46) into (19) leads to the ODE (39) which has solution (40) from which we obtain that a solution for (19) is
k2 2 k2 2 2 2 v = 8a2 t − 2 log e 2 +4a k1 cos(ax + b) sin(ax + b) − e3a t + 2 +a k1 sin(ax + b) − e4a t
(47)
and by (18) we get that
2 −ae
k2 2
+4a2 k1
u=− e
k2 2
sin2 (ax + b) + ae
+4a2 k1
k2 2
+4a2 k1
k2 2 2 cos2 (ax + b) − ae3a t + 2 +a k1 cos(ax + b)
k2 2 2 2 cos(ax + b) sin(ax + b) − e3a t + 2 +a k1 sin(ax + b) − e4a t
is a new exact solution of Eq. (1) Case 8: From v8 , solving the invariant surface condition we obtain the nonclassical symmetry reduction z=−
log cosh(ax + b) 3a2
h
− t,
v = −2 log
e 2 sinh(ax + b)
√ 3
cosh(ax + b)
! −1 .
(48)
Substitution of (46) into (19) leads to the ODE (43) which has solution (44) from which we obtain that a solution for (19) is
v = 8a2 k1 − 2 log
e
2a2 t +k2 +6a2 k1 2
−e
8a2 t +k2 2
2k 1
cosh(ax + b) sinh(ax + b) − e4a
(49)
and by (18) we get that k2 k2 k2 2 2 2 2 2 a cosh(ax + b) ea t + 2 +3a k1 − e4a t + 2 cosh(ax + b) − ae4a t + 2 sinh2 (ax + b)
u=−
k2 k2 2 2 2 2 ea t + 2 +3a k1 − e4a t + 2 cosh(ax + b) sinh(ax + b) − e4a k1
is a new exact solution of Eq. (1).
M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
e1833
Case 9.a: From v9 , solving the invariant surface condition we obtain the nonclassical symmetry reduction log (cx + bc − 1)
x + − t, c2 c v = −2 log(V ) + 8 log c + 2 log 3 + 4 log 2 V = ((18 − 18bc ) k3 − 18ck3 x) log (cx + bc − 1) − 3c 3 k3 x3 + −9bc 3 − 9c 2 k3 x2 + −6b2 c 3 − 6bc 2 + 12c k3 + 2c 3 h x − 2c 3 k4 + 6k3 + 2bc 3 − 2c 2 h. z=
(50)
Substitution of (52) into (19) leads to the ODE d2 h
dh + c2 − 9c 2 k3 = 0 dz 2 dz which has solution h(z ) = k2 e−c
2z
+
9c 2 k3 z − 9k3
+ k1 c2 from which we obtain that a solution for (19) is v = −2 log (V ) + 2cx + 8 log c+ 2 log 3 + 4 log 2 V = −3c 3 k3 x3 + 9c 2 − 9bc 3 k3 x2 + −18c 3 k3 t + −6b2 c 3 + 12bc 2 − 24c k3 + 2c 3 k1 x ecx 2 + 18c 2 − 18bc 3 k3 t − 2c 3 k4 + (24 − 18bc ) k3 + 2bc 3 − 2c 2 k1 ecx + 2c 2 k2 ec t
(51)
and by (18) we get a new exact solution of Eq. (1). Case 9.b: From v9 , with c = 0 solving the invariant surface condition we obtain the nonclassical symmetry reduction x2
− bx − t , 4 bk3 x3 3b2 k3 x2 k4 x b3 k3 x h k3 x + + − + + . v = −2 log z=−
2
8
2
4
6
2
(52)
6
Substitution of (52) into (19) leads to the ODE d2 h
+ 9k3 = 0 dz 2 which has solution h( z ) = −
9k3 z 2
+ k2 z + k1 2 from which we obtain that a solution for (19) is k3 x4 bk3 x3 3k3 tx2 k2 x2 3bk3 tx k4 x b3 k3 x bk2 x 3k3 t 2 k2 t k1 v = −2 log − − − − − − + − − − + 16
4
4
12
2
6
2
6
4
6
6
(53)
and by (18) we get a new exact solution of Eq. (1). Case 10: From v10 , solving the invariant surface condition we obtain the nonclassical symmetry reduction x2
bx
− t, 3 v = 2 − log k3 x5 + 5bk3 x4 + 10b2 k3 x3 + 10b3 k3 x2 + 4b4 k3 + 4h x + 4k4 + 4bh + log 24 . z=−
6
−
(54)
Substitution of (54) into (19) leads to the ODE d2 h
+ 45k3 = 0 dz 2 which has solution h( z ) = −
45k3 z 2
+ k2 z + k1 2 from which we obtain that a solution for (19) is v = −2 log (V ) + 4 log 3 + 8 log 2 V = −9k3 x5 − 45bk3 x4 − 180k3 tx3 − 60b2 k3 x3 − 4k2 x3 − 540bk3 tx2 − 12bk2 x2 − 540k3 t 2 x −360b2 k3 tx − 24k2 tx + 24b4 k3 x − 8b2 k2 x + 24k1 x − 540bk3 t 2 − 24bk2 t + 24k4 + 24bk1 and by (18) we get a new exact solution of Eq. (1). In Fig. 1 we can see solution (37) with k2 = k1 = a = 1, b = d = 0, for t = 0, 1, 3.
(55)
e1834
M.L. Gandarias, M.S. Bruzon / Nonlinear Analysis 71 (2009) e1826–e1834
Fig. 1. Solution (37) and the corresponding time evolution plot related at times t = 0, t = 1 and t = 3.
3. Concluding remarks In this work we have introduced new classes of symmetries for PDE’s. If a PDE can be written in a conserved form (6), then a related system (7) and a related integrated equation may be obtained. The ansatz to generate nonclassical solutions of the associated potential system could yield solutions of the original equation which are neither nonclassical solutions of the original equation nor solutions arising from classical potential symmetries. To explain these results, we have proved the existence of families of solutions of the Burgers’ equation which are not found by means of potential nor nonclassical symmetry reductions. Acknowledgments The support of DGICYT project MTM2006-05031, Junta de Andalucía group FQM–201 and project P06-FQM-01448 are gratefully acknowledged. We warmly thank the referees for their useful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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