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Extended rotation and scaling groups for nonlinear evolution equations
Nonlinear Analysis 52 (2003) 1655 – 1673
www.elsevier.com/locate/na
Extended rotation and scaling groups for nonlinear evolution equations Changzheng Qua; b;∗ , P.G. Estevezb a Department
b Area
of Mathematics, Northwest University, Xi’an 710069, People’s Republic of China de Fisica Teorica, Facultad de Ciencias, Universidad de Salamanca, 37008 Salamanca, Spain Received 13 December 2001; accepted 27 March 2002
Keywords: Rotation group; Scaling group; Symmetry group; Exact solution; Nonlinear evolution equation
1. Introduction We consider the (1+1)-dimensional kth-order nonlinear evolution equation of the form ut = E(x; u; u1 ; : : : ; uk );
(1)
where ut =@u=@t and ui denotes the ith partial derivatives with respect to space variable x. E is a smooth function of the indicated variables. Solutions of (1) are supposed to be smooth. It is well known that exact solutions play an important role in nonlinear phenomena and dynamics. For example, in some circumstance, the asymptotic behavior of 9nite-mass solution of nonlinear di:usion equation with nonlinear source term is described by explicit self-similar solutions. To date, several well-developed methods relating to the symmetry group have been used to construct exact solutions of nonlinear partial di:erential equations (PDEs), these include the classical symmetry group method [3,20], the nonclassical symmetry group method [1,2,5,7,9,10,19,23], the generalized conditional symmetry approach [11,24–28], the direct method [4,6,8,18], the di:erential constraint method [21,22], the sign-invariant and invariant subspace approach [12,13,16], etc. Many interesting exact solutions of nonlinear evolution equations have been obtained by these methods. ∗ Corresponding author. Department of Mathematics, Northwest University, Xi’an 710069, People’s Republic of China. E-mail address: [email protected] (C. Qu).
0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 2 7 8 - X
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In [14,15], Galaktionov proposed a “nonlinear” extension to the ordinary scaling group, which is described by the invariance of the set S0 = {u : ux = (1=x)F(u)}. This approach has been used to construct exact solutions to equations of the form (1), and it is related to the sign-invariant and invariant-subspace methods [12,13,16]. In the end of the paper [15], he addressed that it is interesting to give a “nonlinear” extension for non-scaling groups. The purpose of this paper is to introduce some new “nonlinear” extensions to the rotation and scaling groups. First, we propose a “nonlinear” extension to the rotation group in R2 , which is described by the invariant set R0 = {u(x) : ux = xF(u)}. Second we introduce a new “nonlinear” extension to the scaling group, u which is characterized by the invariant set S˜0 = {u : ux = F(u)=x + F exp[(n − 1) 0 1=F(z) d z]}, where = 0 and n = 1 are two constants, if = 0, it is reduced to S0 introduced in [14,15]. Finally, we propose an extension to both S0 and R0 , which is characterized by the invariant set E0 = {u(x) : ux = f(x)F(u); f = af2 + b}. The paper is organized as follows. In Section 2, we introduce the invariant set R0 , and use it to construct some exact solutions of nonlinear evolution equations of the form (1). The invariant sets S˜0 and E0 are introduced, respectively, in Sections 3 and 4, which are also used to construct exact solutions to some nonlinear evolution equations. Section 5 is concluding remarks on this work.
2. The extended rotation group 2.1. The rotation group Eq. (1) is invariant under the rotation group if it is invariant under the Lie group of transformations x∗ = x cos + u sin ; u∗ = −x sin + u cos , namely it admits the following in9nitesimal generator: V0 = x
@ @ −u : @u @x
If an equation admit the rotation group, the corresponding solutions can be obtained by solving ux = −x=u together with Eq. (1). Indeed, the equation ux = −x=u is the invariant-surface condition of the rotation group. In other words, the set {u : ux =−x=u} is invariant under the rotation group. Exact solutions corresponding to the rotation group take the form u = (g(t) − x2 )1=2 . 2.2. Algebraic di9erentiation If Eq. (1) does not admits the rotation group. In this case, we begin our discussion of invariant sets and exact solutions by introducing the set of functions R0 = {u : ux = xF(u)};
(2)
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where F(u) is a C ∞ function to be determined from the invariance condition u(x; 0) ∈ R0 ⇒ u(x; t) ∈ R0
for t ∈ (0; 1]:
The structure of manifold (2) was used in the construction of the 9rst-order sign-invariant for radially symmetric parabolic equations (see [12, p. 1605]). When F = −1=u; the contact structure (2) characterizes the rotation group. A straightforward calculation gives the di:erentiation rule, that is Lemma 1. If u2k = u2k+1 =
k
k
j=0
x2j G2j , then
x2k−2j+1 G˜ 2k+1−2j
(3a)
j=0
with G˜ 2k+1 = FG2k ; + 2(k − j + 1)G2k−2j+2 ; G˜ 2k−2j+1 = FG2k−2j−2
j = 1; : : : ; k − 1;
G˜ 1 = FG0 : If u2k+1 =
k
j=0
u2k+2 =
x2j+1 G2j+1 , then
k+1
x2k−2j+2 G˜ 2k−2j+2
(3b)
j=0
with G˜ 2k+2 = FG2k+1 ; G˜ 2k−2j+2 = FG2k−2j+1 + (2k − 2j + 3)G2k−2j+3 ;
j = 1; : : : :k;
G˜ 0 = G1 : We present several special cases, which will be used later uxx = x2 FF + F; uxxx = x3 F(FF ) + 3xFF ; uxxxx = x4 F(F(FF ) ) + 6x2 F(FF ) + 3FF : Now we apply the approach to several typical nonlinear evolution equations.
(4)
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2.3. Quasilinear heat equations Consider the general reaction–di:usion equation of parabolic type ut = D(u)uxx + Q(u)ux2 + G(u) ≡ A1 (u);
(5)
where D ¿ 0, Q and G are smooth functions. Substituting formulae (4) into (5), the right-hand side of (5) becomes A1 (u) = x2 (DFF + QF 2 ) + DF + G: For u ∈ R0 , the integration to equation ux = xF(u) yields u
1 1 d z = x2 + g(t); F(z) 2
(6)
where g(t) is to be determined. It follows from (6) that ut = g (t)F(u):
(7)
Therefore Eq. (5) is equivalent to g (t) = x2 (DF + QF) + D +
G : F
Since its left-hand side does not depend on x, di:erentiating it with respect to x, we get G + 2(DF + QF) ≡ 0 x3 F(DF + QF) + x F D + F which implies D, F and G satisfying DF + QF = c; G = −2c; F D+ F
(8)
where c is an arbitrary constant. g(t) satis9es g = d − 2cg with solutions given by g=
d + c0 e−2c(t−t0 ) ; 2c
(9)
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Table 1 D(u)
F(u)
Q(u)
G(u)
um
−
um−1 − cu
um−1 − cu − du−1
um
uk ; k = −1; 1
cu−k − kum−1
−uk+m −
um
u
cu−1 − um−1
−um+1 − 2cu ln u + du
eu
uk ; k = 1
cu−k − ku−1 eu
−uk eu +
eu
u
cu−1 − u−1 eu
−ueu − 2u ln u + du
1 u2 + 1
uk ; k = 1
cu−k − ku−1 (1 + u2 )−1
−(1 + u2 )−1 uk +
1 u2 + 1
u
cu−1 − u−1 (1 + u2 )−1
−u(1 + u2 )−1 − 2cu ln u + du
1 u
2c u + duk 1−k 2c u + duk k −1 2c u + duk k −1
where t0 is an arbitrary constant. System (8) is under-determined, and is easily solved after we know two of them. We consider several special cases (1) D = um , F = −1=u; (2) D = um , F = u; (3) D = um , F = uk , k = −1; 1; (4) D = eu , F = uk , k = 1; (5) D = eu , F = u; (6) D = 1=(1 + u2 ), F = uk , k = 1; (7) D = 1=(1 + u2 ), F = u. The corresponding results are given in Table 1. 2.4. Radially symmetric N -dimensional nonlinear parabolic equation The radially symmetric quasilinear heat equation in RN reads n−1 ut = D(u) uxx + ux + Q(u)ux2 + G(u) ≡ A2 (u): x Using formulae (4), we have in R0 that A2 (u) = x2 (DFF + QF 2 ) + NDF + G and g(t) satis9es g (t) = x2 (DF + QF) + ND +
G : F
Noting the left-hand side is independent of x, we have DF + QF = c; G = −2c: F ND + F Similarly, we can obtain equations which admit the invariant set R0 .
(10)
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2.5. Equations with gradient-dependent di9usivity Consider the quasilinear di:usion equation ut = D(u)ux2 uxx + Q(u)ux4 + G(u) ≡ A3 (u):
(11)
It has applications in geometry and mechanics of non-Newtonian liquids. In the invariant set R0 , we have A3 (u) = x4 (DF 3 F + QF 4 ) + x2 DF 3 + G: The invariant condition implies DF 2 F + QF 3 = c; F(DF 2 ) = −4c; G = −2DF: F
(12)
This system is under-determined. Letting D = um , we get the general solution F = u(1−m)=3 ; Q=
G=
m−2 m − 1; 4 u(d − 2 ln u); du(1−m)=3 −
3 u(m+5)=3 ; m+2
m = −2; m = −2:
Then the corresponding equations have solutions
2 x exp + g(t) ; m = −2; 2 u= 3=(m+2) m+2 2 m+2 ; m = −2; x + g(t) 6 3 where g(t) satis9es g (t) = −
m+2 2 g + d: 3
2.6. Fourth-order equations Consider the fourth-order equation of the form ut = !(u)uxxxx + A1 (u) ≡ A3 (u);
(13)
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where A1 (u) is the second-order operator given in (5). If !=−1, G=0, it is Kuramoto– Shvashinskii equation which arises from the theory of Lame propagation. Moreover, if Q = D , it becomes Cahn–Hilliard equation. Applying the set R0 to Eq. (13), its right-hand side becomes A3 (u) = x4 !(F(FF ) ) + x2 [6!(FF ) + DF + QF] + 3!F + D +
G : F
Then the invariant surface condition implies (!(F(FF ) ) ) = 0; 4!(F(FF ) ) + F[6!(FF ) + DF + QF] = 0; G = 0: 2[6!(FF ) + DF + QF] + F 3!F + D + F The integration gives !(F(FF ) ) = c; u
1 d z + d1 ; F(z) 2 u u 1 1 G 3!F + D + = 4c − 2d1 d z + d2 ; F F(z) d z F(z)
6!(FF ) + DF + QF = −4c
where c, d1 and d2 are arbitrary constants. Eq. (13) has solutions given by (6) with g(t) satisfying the Ricatti equation g = 4cg2 − 2d1 g + d2 : For Kuramoto–Shvashinskii equation, the coeNcients are given by D = −3F + 4c
u
Q = F −1
2 u 1 1 d z − 2c1 d z + d2 ; F(z) F(z)
u d1 − 4c
+ 2d1 F
u
u 2 1 1 d z + 3Fu2 − 4cF d z F(z) F(z)
1 d z − d2 F − 6(FF ) F(z)
;
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where di ; i = 1; 2, are arbitrary constants and F satis9es F(FF ) = −cu + d3 : It has the special solution F =u for c = −1 and d3 = 0, where D and Q take the form D = 4(ln u)2 − 2d1 ln u + d2 − 3; Q = u−1 [4(ln u)2 + (4 + 2d1 ) ln u + d1 − d2 − 3]: 3. The extended scaling group 3.1. Invariant set Assume that Eq. (1) is invariant under the Lie group of scaling transformations x∗ = e x;
t ∗ = e" t;
where is a parameter, then the corresponding in9nitesimal generator is X =x
@ @ + "t : @x @t
If the operator A(u) is a homogeneous, i.e., it satis9es the condition " 1 1 1 A(x; u; u1 ; : : : ; um ); A sx; u; u1 ; : : : ; m um = s s s then the equation admits the self-similar solution x u = (%); % = 1=" t
(14)
and the PDE is reduced to the ODE A(%; ; ; : : : ; m ) +
1 % = 0: "
Galaktionov [14,15] proposed an interesting “nonlinear” extension to the scaling group. He considered the equation in an invariant set S0 = {u(x) : ux = (1=x)F(u)}, where F is a C ∞ function to be determined from the invariant condition u(·; 0) ∈ S0 ⇒ u(·; t) ∈ S0
for t ∈ (0; 1]:
It is easy to see that the contact structure of the semilinear equation ux =
1 F(u); x
includes the scaling invariant (14).
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For equations do not admit the invariant set S0 , we introduce the following invariant set: u 1 s S1 = u ∈ C ∞ ; ux = F(u) + F(u) exp (n − 1) d z ; (15) x F(z) where , s and n = 1 are some constants. If = 0, the set S1 is reduced to the invariant set S0 . For the simplicity, we only consider the case F =u in the following applications, namely, the invariant set becomes s S2 = u ∈ C ∞ ; ux = u + un ; (16) x where has been scaled to 1. In the set S2 , we have the following formulae: 1 1 uxx = 2 (s2 − s)u + s(n + 1)un + nu2n−1 ; x x uxxx = uxxxx =
1 1 1 s1 u + 2 s2 un + s3 u2n−1 + s4 u3n−2 ; 3 x x x 1 1 1 1 l1 u + 3 l2 un + 2 l3 u2n−1 + l4 u3n−2 + l5 u4n−3 ; x4 x x x
(17)
where s1 = (s2 − s)(s − 2); s4 = n(2n − 1);
s2 = (n2 + n + 1)s2 − (n + 2)s;
l1 = (s − 3)s1 ;
l3 = ns2 + [(n − 1)s − 1]s3 ;
s3 = 3n2 s;
l2 = s1 + (ns − 2)s2 ;
l4 = (2n − 1)s3 + (3n − 2)ss4 ;
l5 = (3n − 2)s4 : If Eq. (1) is invariant under the set S2 , its solutions are given by
1=(1−n) (1 − n) u= x + g(t)xs(1−n) s(n − 1) + 1
(18)
with g(t) to be determined. This type of solutions have appeared in physical situations, such as the special “dipole” solution takes this form [17]. In the followings we consider some applications of the invariant set S2 to Eq. (1) and adduce exact solutions to some parabolic and KdV-like equations. 3.2. Quasilinear heat equations For the quasilinear heat equation of the form (5), using the di:erentiation rules, one computes on the set S2 1 1 A1 (u) = 2 [(s2 − s)uD + s2 u2 Q] + [(n + 1)sun D + 2sun+1 Q] x x + nu2n−1 D + u2n Q + G(u):
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It follows from (18) that ut =
1 s−a g (t)un ; 1−n
where a = s(n − 1). Then g(t) satis9es 1 g (t) = xa−2 [(s2 − s)u1−n D + s2 u2−n Q] + xa−1 [(n + 1)sD + 2suQ] 1−n + xa [nun−1 D + un Q + u−n G] ≡ xa−2 D1 + xa−1 D2 + xa D3 :
(19)
Since the left-hand side of this equation is independent of x, di:erentiating it with respect to x, we have xa−3 [suD1 + (a − 2)D1 ] + xa−2 [un D1 + suD2 + (a − 1)D2 ] + xa−1 [un D2 + suD3 + aD3 ] + xa un D3 = 0 which yields suD1 + (a − 2)D1 = 0;
(20a)
un D1 + suD2 + (a − 1)D2 = 0;
(20b)
un D2 + suD3 + aD3 = 0;
(20c)
D3 = 0:
(20d)
Integrating (20b)–(20d), we have D3 = c1 ; D2 =
ac1 1−n + c2 ; u n−1
D1 = −
ac1 c2 (1 − a) 1−n u u2−2n + + c3 ; 2 2(1 − n) 1−n
where ci ; i = 1; 2; 3 are integration constants. The substitution of the expressions for D1 , D2 and D3 into (20a) implies the constants satisfying (a − 2)c3 = 0; (a − 1)c2 = 0; (a + 2)c1 = 0:
(21)
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Two possibilities arise: Case 1: c3 = c1 = 0, c2 = 0. In this case, a = 1. D, Q and G satisfy (s2 − s)u1−n D + s2 u2−n Q = 0; (n + 1)sD + 2suQ = c2 ; nun−1 D + un Q + u−n G = 0 which has solution D=
c2 = D0 ; 3
1 Q = (n − 2)D0 u−1 ; 3 G = 2(1 − n)D0 u2n−1 : In this case, g(t) satis9es g = (1 − n)c2 and is given by g(t) = (1 − n)c2 (t − t0 ): Case 2: c1 = c2 = 0, c3 = 0. In this case, a = 2. D, Q and G satisfy (s2 − s)u1−n D + s2 u2−n Q = c3 ; (n + 1)sD + 2suQ = 0; nun−1 D + un Q + u−n G = 0 which gives D=
1−n c3 un−1 ; 4
Q=
n2 − 1 c3 un−2 ; 8
G=
(n − 1)2 c3 u3n−2 : 8
In this case, g(t) is given by g = (1 − n)c3 (t − t0 ):
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Remark 1. If we use the general invariant set S1 to Eq. (5), then u 1 s ux = + H; H = F exp (n − 1) d z ; x F(z) uxx =
1 2 1 (s FF − sF) + s(HF) + HH : x2 x
So A1 (u) and (19) become A1 (u) =
1 1 [D(sFF − sF) + s2 F 2 Q] + [sD(HF) + 2sFHQ] x x2 + DHH + QH 2 + G
and
2 1 D a−2 D 2 QF a−1 +x g (t) = x (sFF − sF) + s s (HF) + 2sFQ H H 1−n H G : + xa DH + QH + H 3.3. Radially symmetric N -dimensional parabolic equation Consider the radially symmetric N -dimensional parabolic equation (11). Applying the invariant set S2 to it, we get A2 (u) =
1 1 [(s2 − s)uD + s2 u2 Q + s(N − 1)uD] + [((n + 1)s x2 x + N − 1)un D + 2sun+1 Q] + nu2n−1 D + u2n Q + G(u):
Then g(t) satis9es 1 g (t) = xa−2 D˜ 1 + xa−1 D˜ 2 + xa D˜ 3 ; 1−n
(22)
where D˜ 1 = [s2 − s + s(N − 1)]u1−n D + s2 u2−n Q; D˜ 2 = [(n + 1)s + N − 1]D + 2suQ; D˜ 3 = nun−1 D + un Q + u−n G: The left-hand side of (22) is independent of x, di:erentiating it with respect to x, we obtain system (20), in which Di are replaced by D˜ i . Three cases arise: (1) c1 = c3 = 0,
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Table 2 Number
1
2
3
D(u)
1 c2 4−N
1 c1 u1−n (N − 1)(n − 1)
n−1 c3 un−1 N −5
Q(u)
(n − 1)N − 2n + 3 c2 u−1 N −4
2s + 3N − 5 c1 u−n 4(N − 1)
(1 − n)[(n − 1)N + n + 3] c3 un−2 4(N − 5)
G(u)
(n − 1)(N − 3) c2 u2n−1 4−N
N −3 c1 un 4(N − 1)
(n − 1)2 (N − 3) c3 u3n−2 4(N − 5)
g(t)
(1 − n)c2 (t − t0 )
n−1 (t − t0 )−1 c1
(1 − n)c3 (t − t0 )
c2 = 0, N = 4; (2) c2 = c3 = 0, c1 = 0; (3) c1 = c2 = 0, c3 = 0, N = 5. We list the results in Table 2. 3.4. KdV-type equation Consider the KdV-type equation of the form ut = D(u)uxxx + Q(u)uxx + H (u)ux3 + G(u)ux ≡ L(u);
(23)
where D, Q, H and G are smooth functions of u. The analysis is the same as for the second-order quasilinear parabolic equations. Substituting the di:erential rule (17) into (23), we have L(u) =
1 1 (s1 uD + s3 u3 H ) + 2 (s2 un D + (s2 − s)uQ + 3s2 un+2 H ) x3 x +
1 (s3 u2n−1 D + (n + 1)sun Q + 3su2n+1 H + suG) x
+ s4 u3n−2 + nu2n−1 Q + un G + u3n H: g(t) satis9es 1 g = xa−3 J1 + xa−2 J2 + xa−1 J3 + xa J4 ; 1−n
(24)
where J1 = u−n (s1 uD + s3 u3 H ); J2 = u−n (s2 un D + (s2 − s)uQ + 3s2 un+2 H ); J3 = u−n (s3 u2n−1 D + (n + 1)sun Q + 3su2n+1 H + suG); J4 = u−n (s4 u3n−2 D + nu2n−1 Q + un G + u3n H ):
(25)
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The left-hand side of (24) is independent of x implies Ji ; i = 1; 2; 3; 4, satisfying suJ1 + (a − 3)J1 = 0; un J1 + suJ2 + (a − 2)J2 = 0; un J2 + suJ3 + (a − 1)J3 = 0; un J3 + suJ4 + aJ4 = 0; J4 = 0:
(26)
Solving system (26), we get s sc1 c2 (a − 1) 2−2n a − 2 1 J1 = u3−3n − u + + c3 u1−n + c4 ; 3(n − 1)2 2 n − 1 (1 − n)2 n−1 J2 =
sc1 c2 (1 − a) 1−n u2−2n + + c3 ; u 1−n 2(1 − n)
J3 =
ac1 1−n + c2 ; u n−1
J4 = c 1 and the integration constants ci ; i = 1; 2; 3; 4, satisfy (a − 3)c4 = 0; (a − 2)c3 = 0; (a − 1)(a + 3)c2 = 0; (a + 2)(2a + 3)c1 = 0: Several cases arise: (1) c1 = c2 = c3 = 0; c4 = 0; (2) c1 = c2 = c4 = 0; c3 = 0; (3) c1 = c3 = c4 = 0; c2 = 0; a = 1; (4) c1 = c3 = c4 = 0; c2 = 0; a = −3; (5) c2 = c3 = c4 = 0; c1 = 0. We put the corresponding results in Tables 3 and 4.
4. The extended rotation-scaling group In this section, we propose an extension for both extended scaling (S0 ) and the rotation group (R0 ), which is characterized by E0 = {u : ux = f(x)F(u)};
(27)
where f(x) satis9es fx = af2 + b;
(28)
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Table 3 Number
D(u)
H (u)
1
1 + n − 2n2 c4 un−1 15(n + 5)
(n − 1)(n3 − n2 − 2n + 5) c4 un−3 15(n + 5)
2
1 c3 8
(n − 3)(2 − n) c3 u−2 16
3
2−n c2 u1−n 3(n − 1)(n − 3)
(2 − n)2 (2n − 3) c2 u−1−n 3(n − 1)(n − 3)
4
n+2 c2 u1−n 3(1 + 5n)(1 − n)
n2 (2n − 5) c2 u−1−n 3(1 − n)(1 + 5n)
5
4n2 − 11n + 34 c1 u2−2n 6(7n − 1)(n − 1)2
−
2n4 − 4n3 + 22n2 + 11n − 4 c1 u−2n 6(7n − 1)(n − 1)2
Table 4 Number
Q(u)
G(u) 1)2 (n
g(t) 1)2 (5
1
+ 2) (n − c4 u2n−2 3(n + 5)
(n −
2
3 (1 − n)c3 un−1 4
3(2 − 5n + 3n2 ) c3 u2n−2 16
(1 − n)c3 (t − t0 )
3
2n − 5 c2 n−3
2n2 − 7n + 4 c2 un−1 3−n
(1 − n)c2 (t − t0 )
4
(2n + 1) c2 3(5n + 1)
n(2n + 1) c2 un−1 3(5n + 1)
5
−
n2 − 20n − 8 c1 u1−n 6(7n − 1)(n − 1)2
−
− 6n − 15(n + 5)
2n2 )
c4 u3n−3
5n3 − 38n2 + 97n − 10 c1 6(7n − 1)(n − 1)
(1 − n)c4 (t − t0 )
1 (n − 1)c2−1 (t − t0 )−1 4
−1=2 c1 − (t − t ) 0 4(n − 1)2
where a; b = 0 are two constants. Note that for a = 0 and b = 0, the invariant set E0 reduces, respectively, to R0 and S0 , so it is viewed as a generalization for S0 and R0 . In the set E0 , we have the following formulae: uxx = f2 (FF + aF) + bF; uxxx = f3 [F(FF + aF) + 2aF(F + a)] + bfF(3F + 2a); uxxxx = f4 {F[F(FF + aF) + 2aF(F + a)] + 3aF[(FF + aF) + 2a(F + a)]} + f2 {3b[F(FF + aF) + 2aF(F + a)] + b(a + F)(3FF + 2aF) } + b2 F(3F + 2a):
(29)
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In the invariant set E0 , solutions are given by (i) ab ¿ 0. u
ds = ln| cos +a(x + x0 )| + g(t); F(s)
(30a)
where +2 = b=a. (ii) ab ¡ 0. In this case we obtain two solutions: u and
u
ds = −ln sinh[+a(x + x0 )] + g(t) F(s)
(30b)
ds = −ln cosh[+a(x + x0 ) + g(t)]; F(s)
(30c)
where +2 = −b=a. Now, we use the above formulae to the construction of exact solutions to the following nonlinear evolution equations. 4.1. Quasilinear heat equations Consider the quasilinear heat equation (5). Substituting (27)–(29) into (5), we have G g (t) = f2 (DF + QF + aD) + + bD: (31) F The left-hand side of (31) does not depend on x yields (DF + QF + aD) F + 2a(DF + QF + aD) = 0; G + bD F + 2b(DF + QF + aD) = 0: F The integration gives
DF + QF + aD = c0 exp −2a
u
ds ; F(s)
u G ds bc0 + c1 ; + bD = exp −2a F a F(s) where c0 and c1 are arbitrary constants. Hence g(t) satis9es the equation 1 bc0 2 exp −2a g = c0 f (1) + f(y) dy e−2ag + c1 : a 0
C. Qu, P.G. Estevez / Nonlinear Analysis 52 (2003) 1655 – 1673
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In particular, equation ut = uxx + G(u); admits solutions (30) with F satisfying FF + 2a(F + a) = 0 and G(u) is given by
u
G = −2bF ln F − 2abF
ds + cF; F(s)
where c is a constant. Remark 2. This approach is also available to equations with x-variable coeNcients. For example, the equation ut = uxx + X (x)G(u) also admits solutions (30), where F, G and X satisfy G G = 0; F F +. F + 2a F + a + . F F G G / F + 2b F + . + a = 0; F F X (x) = .f2 + /; where . and / are constants. 4.2. Fourth-order equations Using the invariant set E0 to the fourth-order equation (13), we have g = f4 J1 (u) + f2 J2 (u) + J3 (u); where J1 = !{[F(FF + aF) + 2a(FF + aF)] + 3a[(FF + aF) + 2a(F + a)]}; F +a J2 = ! 3b[(FF + aF) + 2a(F + a)] + [3bFF + 2abF] F + D(F + a) + QF; G : F The left-hand side of (32) does not depend on x yields the following system J3 = b2 (3F + 2a)! + bD + FJ1 + 4aJ1 = 0; J1 + FJ2 + 2aJ2 = 0; 2bJ2 + FJ3 = 0:
(32)
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The integration gives J1 = c1 exp −4a
u
ds ; F(s)
u u ds ds 2bc1 ; + c2 exp −2a J2 = exp −4a F(s) a F(s) u u bc ds ds b 2 c1 + 2 exp −2a + c3 ; J3 = 2 exp −4a a F(s) a F(s) where ci ; i = 1; 2; 3 are arbitrary constants, where g satis9es g = d1 e−4ag + d2 e−2ag + c3 ; where
1 2bc1 2 b2 c1 4 d1 = c1 f (1) + f(y) dy ; f (1) + 2 exp −4a a a 0 d2 =
c2 f12
bc2 + a
exp −2a
0
1
f(y) dy :
5. Conclusions In this paper, we have shown that the rotation group admits a “nonlinear” extension, it is characterized by an invariant set of a contact 9rst-order di:erential structure, which is parallel to the “nonlinear” extension for the scaling group. This extension has been used to construct exact solutions to some nonlinear evolution equations of second- and fourth order. We also proposed a more general “nonlinear” extension to the scaling group, it is characterized by an invariant set of a contact 9rst-order di:erential structure with nonhomogeneous term. As the nonhomogeneous term is zero, it is reduced to the invariant set introduced in [14,15]. A further extension to both extended scaling groups S0 and rotation group R0 was also proposed. As the consequence, these extensions have been used to the construction of some exact solutions of nonlinear evolution equations, which include the KdV-type equations. Acknowledgements This research has been supported in part by DGICYT under project PB98-0262. Qu’s research was supported in part by the Ministry of Education and Culture of Spain, the NNSF of China (Grant No. 19901027) and NSF of Shaan Xi Province.
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References [1] D.J. Arrigo, P. Broadbridge, J.M. Hill, Nonclassical symmetry for nonlinear di:usion equations, IMA. J. Appl. Math. 52 (1994) 1–24. [2] G.W. Bluman, J.D. Cole, The general similarity solution of the heat equation, J. Math. Mech. 18 (1969) 1025–1042. [3] G.W. Bluman, S. Kumei, Symmetries and Di:erential Equations, Springer, New York, 1989. [4] P.A. Clarkson, New exact solutions for the Boussinesq equation, European J. Appl. Math. 1 (1990) 279–300. [5] P.A. Clarkson, Nonclassical symmetry reductions of the Boussinesq equation, Chaos, Soliton Fract. 5 (1995) 2261–2301. [6] P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq equation, J. Math. Phys. 30 (1989) 2201–2213. [7] P.A. Clarkson, E.L. Mans9eld, Symmetry reductions and exact solutions of a class of nonlinear heat equations, Physica D 70 (1993) 250–288. [8] P.G. Estevez, The direct method and the singular manifold method for the Fitzhugh–Nagumo equation, Phys. Lett. A 171 (1992) 259–261. [9] P.G. Estevez, Nonclassical symmetries and the singular manifold method: the Burgers and the Burgers– Huxley equations, J. Phys. A: Math. Gen. 27 (1994) 2113–2127. [10] P.G. Estevez, P.R. Gordoa, Nonclassical symmetries and the singular manifold method: theory and examples, Stud. Appl. Math. 95 (1995) 73–113. [11] A.S. Fokas, Q.M. Liu, Nonlinear interaction of traveling waves of nonintegrable equations, Phys. Rev. Lett. 72 (1994) 3293–3296. [12] V.A. Galaktionov, Quasilinear heat equations with 9rst-order sign-invarians and new explicit solutions, Nonlinear Anal. Theory Methods Appl. 23 (1994) 1595–1621. [13] V.A. Galaktionov, Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities, Proc. Roy. Soc. Edinburgh 125A (1995) 225–246. [14] V.A. Galaktionov, Ordered invariant sets for nonlinear evolution equations of KdV-type, Comput. Math. Phys. 39 (1999) 1564–1570. [15] V.A. Galaktionov, Groups of scalings and invariant sets for higher-order nonlinear evolution equations, Di:erential Integral Equation 14 (2001) 913–924. [16] V.A. Galaktionov, S.A. Posashkov, New explicit solutions of quasilinear heat equations with general 9rst-order sign-invariants, Physica D 99 (1996) 217–236. [17] J.R. King, Exact solutions to some nonlinear di:usion equations, Physica D 63 (1993) 35–65. [18] S.Y. Lou, A note on the new similarity reductions of the Boussinesq equation, Phys. Lett. A 151 (1990) 133–135. [19] M.C. Nucci, Iterating the nonclassical method, Physica D 7 (1994) 124–134. [20] P. Olver, Applications of Lie Groups to Di:erential Equations, 2nd Edition, Springer, New York, 1993. [21] P.J. Olver, P. Rosenau, The construction of special solutions to partial di:erential equations, Phys. Lett. A 114 (1986) 107–112. [22] P.J. Olver, P. Rosenau, Group invariant solutions of di:erential equations, SIAM J. Appl. Math. 47 (1987) 263–275. [23] E. Pucci, Similarity reductions of partial di:erential equations, J. Phys. A: Math. Gen. 25 (1992) 2631–2640. [24] C.Z. Qu, Group classi9cation and generalized conditional symmetry reduction of the nonlinear di:usion– convection equation with a nonlinear source, Stud. Appl. Math. 99 (1997) 107–136. [25] C.Z. Qu, Exact solution to nonlinear di:usion equations obtained by a generalized conditional symmetry method, IMA J. Appl. Math. 62 (1999) 283–302. [26] C.Z. Qu, Classi9cation and reduction of some systems of quasilinear partial di:erential equations, Nonlinear Anal. Theory Methods Appl. 42 (2000) 301–327. [27] C.Z. Qu, S.L. Zhang, R.C. Liu, Separation of variables and exact solutions to quasilinear di:usion equations with nonlinear source, Physica D 144 (2000) 97–123. [28] R.Z. Zhdanov, Conditional Lie–BVacklund symmetry and reduction of evolution equation, J. Phys. A: Math. Gen. 28 (1995) 3841–3850.
www.elsevier.com/locate/na
Extended rotation and scaling groups for nonlinear evolution equations Changzheng Qua; b;∗ , P.G. Estevezb a Department
b Area
of Mathematics, Northwest University, Xi’an 710069, People’s Republic of China de Fisica Teorica, Facultad de Ciencias, Universidad de Salamanca, 37008 Salamanca, Spain Received 13 December 2001; accepted 27 March 2002
Keywords: Rotation group; Scaling group; Symmetry group; Exact solution; Nonlinear evolution equation
1. Introduction We consider the (1+1)-dimensional kth-order nonlinear evolution equation of the form ut = E(x; u; u1 ; : : : ; uk );
(1)
where ut =@u=@t and ui denotes the ith partial derivatives with respect to space variable x. E is a smooth function of the indicated variables. Solutions of (1) are supposed to be smooth. It is well known that exact solutions play an important role in nonlinear phenomena and dynamics. For example, in some circumstance, the asymptotic behavior of 9nite-mass solution of nonlinear di:usion equation with nonlinear source term is described by explicit self-similar solutions. To date, several well-developed methods relating to the symmetry group have been used to construct exact solutions of nonlinear partial di:erential equations (PDEs), these include the classical symmetry group method [3,20], the nonclassical symmetry group method [1,2,5,7,9,10,19,23], the generalized conditional symmetry approach [11,24–28], the direct method [4,6,8,18], the di:erential constraint method [21,22], the sign-invariant and invariant subspace approach [12,13,16], etc. Many interesting exact solutions of nonlinear evolution equations have been obtained by these methods. ∗ Corresponding author. Department of Mathematics, Northwest University, Xi’an 710069, People’s Republic of China. E-mail address: [email protected] (C. Qu).
0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 2 7 8 - X
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In [14,15], Galaktionov proposed a “nonlinear” extension to the ordinary scaling group, which is described by the invariance of the set S0 = {u : ux = (1=x)F(u)}. This approach has been used to construct exact solutions to equations of the form (1), and it is related to the sign-invariant and invariant-subspace methods [12,13,16]. In the end of the paper [15], he addressed that it is interesting to give a “nonlinear” extension for non-scaling groups. The purpose of this paper is to introduce some new “nonlinear” extensions to the rotation and scaling groups. First, we propose a “nonlinear” extension to the rotation group in R2 , which is described by the invariant set R0 = {u(x) : ux = xF(u)}. Second we introduce a new “nonlinear” extension to the scaling group, u which is characterized by the invariant set S˜0 = {u : ux = F(u)=x + F exp[(n − 1) 0 1=F(z) d z]}, where = 0 and n = 1 are two constants, if = 0, it is reduced to S0 introduced in [14,15]. Finally, we propose an extension to both S0 and R0 , which is characterized by the invariant set E0 = {u(x) : ux = f(x)F(u); f = af2 + b}. The paper is organized as follows. In Section 2, we introduce the invariant set R0 , and use it to construct some exact solutions of nonlinear evolution equations of the form (1). The invariant sets S˜0 and E0 are introduced, respectively, in Sections 3 and 4, which are also used to construct exact solutions to some nonlinear evolution equations. Section 5 is concluding remarks on this work.
2. The extended rotation group 2.1. The rotation group Eq. (1) is invariant under the rotation group if it is invariant under the Lie group of transformations x∗ = x cos + u sin ; u∗ = −x sin + u cos , namely it admits the following in9nitesimal generator: V0 = x
@ @ −u : @u @x
If an equation admit the rotation group, the corresponding solutions can be obtained by solving ux = −x=u together with Eq. (1). Indeed, the equation ux = −x=u is the invariant-surface condition of the rotation group. In other words, the set {u : ux =−x=u} is invariant under the rotation group. Exact solutions corresponding to the rotation group take the form u = (g(t) − x2 )1=2 . 2.2. Algebraic di9erentiation If Eq. (1) does not admits the rotation group. In this case, we begin our discussion of invariant sets and exact solutions by introducing the set of functions R0 = {u : ux = xF(u)};
(2)
C. Qu, P.G. Estevez / Nonlinear Analysis 52 (2003) 1655 – 1673
1657
where F(u) is a C ∞ function to be determined from the invariance condition u(x; 0) ∈ R0 ⇒ u(x; t) ∈ R0
for t ∈ (0; 1]:
The structure of manifold (2) was used in the construction of the 9rst-order sign-invariant for radially symmetric parabolic equations (see [12, p. 1605]). When F = −1=u; the contact structure (2) characterizes the rotation group. A straightforward calculation gives the di:erentiation rule, that is Lemma 1. If u2k = u2k+1 =
k
k
j=0
x2j G2j , then
x2k−2j+1 G˜ 2k+1−2j
(3a)
j=0
with G˜ 2k+1 = FG2k ; + 2(k − j + 1)G2k−2j+2 ; G˜ 2k−2j+1 = FG2k−2j−2
j = 1; : : : ; k − 1;
G˜ 1 = FG0 : If u2k+1 =
k
j=0
u2k+2 =
x2j+1 G2j+1 , then
k+1
x2k−2j+2 G˜ 2k−2j+2
(3b)
j=0
with G˜ 2k+2 = FG2k+1 ; G˜ 2k−2j+2 = FG2k−2j+1 + (2k − 2j + 3)G2k−2j+3 ;
j = 1; : : : :k;
G˜ 0 = G1 : We present several special cases, which will be used later uxx = x2 FF + F; uxxx = x3 F(FF ) + 3xFF ; uxxxx = x4 F(F(FF ) ) + 6x2 F(FF ) + 3FF : Now we apply the approach to several typical nonlinear evolution equations.
(4)
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2.3. Quasilinear heat equations Consider the general reaction–di:usion equation of parabolic type ut = D(u)uxx + Q(u)ux2 + G(u) ≡ A1 (u);
(5)
where D ¿ 0, Q and G are smooth functions. Substituting formulae (4) into (5), the right-hand side of (5) becomes A1 (u) = x2 (DFF + QF 2 ) + DF + G: For u ∈ R0 , the integration to equation ux = xF(u) yields u
1 1 d z = x2 + g(t); F(z) 2
(6)
where g(t) is to be determined. It follows from (6) that ut = g (t)F(u):
(7)
Therefore Eq. (5) is equivalent to g (t) = x2 (DF + QF) + D +
G : F
Since its left-hand side does not depend on x, di:erentiating it with respect to x, we get G + 2(DF + QF) ≡ 0 x3 F(DF + QF) + x F D + F which implies D, F and G satisfying DF + QF = c; G = −2c; F D+ F
(8)
where c is an arbitrary constant. g(t) satis9es g = d − 2cg with solutions given by g=
d + c0 e−2c(t−t0 ) ; 2c
(9)
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Table 1 D(u)
F(u)
Q(u)
G(u)
um
−
um−1 − cu
um−1 − cu − du−1
um
uk ; k = −1; 1
cu−k − kum−1
−uk+m −
um
u
cu−1 − um−1
−um+1 − 2cu ln u + du
eu
uk ; k = 1
cu−k − ku−1 eu
−uk eu +
eu
u
cu−1 − u−1 eu
−ueu − 2u ln u + du
1 u2 + 1
uk ; k = 1
cu−k − ku−1 (1 + u2 )−1
−(1 + u2 )−1 uk +
1 u2 + 1
u
cu−1 − u−1 (1 + u2 )−1
−u(1 + u2 )−1 − 2cu ln u + du
1 u
2c u + duk 1−k 2c u + duk k −1 2c u + duk k −1
where t0 is an arbitrary constant. System (8) is under-determined, and is easily solved after we know two of them. We consider several special cases (1) D = um , F = −1=u; (2) D = um , F = u; (3) D = um , F = uk , k = −1; 1; (4) D = eu , F = uk , k = 1; (5) D = eu , F = u; (6) D = 1=(1 + u2 ), F = uk , k = 1; (7) D = 1=(1 + u2 ), F = u. The corresponding results are given in Table 1. 2.4. Radially symmetric N -dimensional nonlinear parabolic equation The radially symmetric quasilinear heat equation in RN reads n−1 ut = D(u) uxx + ux + Q(u)ux2 + G(u) ≡ A2 (u): x Using formulae (4), we have in R0 that A2 (u) = x2 (DFF + QF 2 ) + NDF + G and g(t) satis9es g (t) = x2 (DF + QF) + ND +
G : F
Noting the left-hand side is independent of x, we have DF + QF = c; G = −2c: F ND + F Similarly, we can obtain equations which admit the invariant set R0 .
(10)
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2.5. Equations with gradient-dependent di9usivity Consider the quasilinear di:usion equation ut = D(u)ux2 uxx + Q(u)ux4 + G(u) ≡ A3 (u):
(11)
It has applications in geometry and mechanics of non-Newtonian liquids. In the invariant set R0 , we have A3 (u) = x4 (DF 3 F + QF 4 ) + x2 DF 3 + G: The invariant condition implies DF 2 F + QF 3 = c; F(DF 2 ) = −4c; G = −2DF: F
(12)
This system is under-determined. Letting D = um , we get the general solution F = u(1−m)=3 ; Q=
G=
m−2 m − 1; 4 u(d − 2 ln u); du(1−m)=3 −
3 u(m+5)=3 ; m+2
m = −2; m = −2:
Then the corresponding equations have solutions
2 x exp + g(t) ; m = −2; 2 u= 3=(m+2) m+2 2 m+2 ; m = −2; x + g(t) 6 3 where g(t) satis9es g (t) = −
m+2 2 g + d: 3
2.6. Fourth-order equations Consider the fourth-order equation of the form ut = !(u)uxxxx + A1 (u) ≡ A3 (u);
(13)
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1661
where A1 (u) is the second-order operator given in (5). If !=−1, G=0, it is Kuramoto– Shvashinskii equation which arises from the theory of Lame propagation. Moreover, if Q = D , it becomes Cahn–Hilliard equation. Applying the set R0 to Eq. (13), its right-hand side becomes A3 (u) = x4 !(F(FF ) ) + x2 [6!(FF ) + DF + QF] + 3!F + D +
G : F
Then the invariant surface condition implies (!(F(FF ) ) ) = 0; 4!(F(FF ) ) + F[6!(FF ) + DF + QF] = 0; G = 0: 2[6!(FF ) + DF + QF] + F 3!F + D + F The integration gives !(F(FF ) ) = c; u
1 d z + d1 ; F(z) 2 u u 1 1 G 3!F + D + = 4c − 2d1 d z + d2 ; F F(z) d z F(z)
6!(FF ) + DF + QF = −4c
where c, d1 and d2 are arbitrary constants. Eq. (13) has solutions given by (6) with g(t) satisfying the Ricatti equation g = 4cg2 − 2d1 g + d2 : For Kuramoto–Shvashinskii equation, the coeNcients are given by D = −3F + 4c
u
Q = F −1
2 u 1 1 d z − 2c1 d z + d2 ; F(z) F(z)
u d1 − 4c
+ 2d1 F
u
u 2 1 1 d z + 3Fu2 − 4cF d z F(z) F(z)
1 d z − d2 F − 6(FF ) F(z)
;
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where di ; i = 1; 2, are arbitrary constants and F satis9es F(FF ) = −cu + d3 : It has the special solution F =u for c = −1 and d3 = 0, where D and Q take the form D = 4(ln u)2 − 2d1 ln u + d2 − 3; Q = u−1 [4(ln u)2 + (4 + 2d1 ) ln u + d1 − d2 − 3]: 3. The extended scaling group 3.1. Invariant set Assume that Eq. (1) is invariant under the Lie group of scaling transformations x∗ = e x;
t ∗ = e" t;
where is a parameter, then the corresponding in9nitesimal generator is X =x
@ @ + "t : @x @t
If the operator A(u) is a homogeneous, i.e., it satis9es the condition " 1 1 1 A(x; u; u1 ; : : : ; um ); A sx; u; u1 ; : : : ; m um = s s s then the equation admits the self-similar solution x u = (%); % = 1=" t
(14)
and the PDE is reduced to the ODE A(%; ; ; : : : ; m ) +
1 % = 0: "
Galaktionov [14,15] proposed an interesting “nonlinear” extension to the scaling group. He considered the equation in an invariant set S0 = {u(x) : ux = (1=x)F(u)}, where F is a C ∞ function to be determined from the invariant condition u(·; 0) ∈ S0 ⇒ u(·; t) ∈ S0
for t ∈ (0; 1]:
It is easy to see that the contact structure of the semilinear equation ux =
1 F(u); x
includes the scaling invariant (14).
C. Qu, P.G. Estevez / Nonlinear Analysis 52 (2003) 1655 – 1673
1663
For equations do not admit the invariant set S0 , we introduce the following invariant set: u 1 s S1 = u ∈ C ∞ ; ux = F(u) + F(u) exp (n − 1) d z ; (15) x F(z) where , s and n = 1 are some constants. If = 0, the set S1 is reduced to the invariant set S0 . For the simplicity, we only consider the case F =u in the following applications, namely, the invariant set becomes s S2 = u ∈ C ∞ ; ux = u + un ; (16) x where has been scaled to 1. In the set S2 , we have the following formulae: 1 1 uxx = 2 (s2 − s)u + s(n + 1)un + nu2n−1 ; x x uxxx = uxxxx =
1 1 1 s1 u + 2 s2 un + s3 u2n−1 + s4 u3n−2 ; 3 x x x 1 1 1 1 l1 u + 3 l2 un + 2 l3 u2n−1 + l4 u3n−2 + l5 u4n−3 ; x4 x x x
(17)
where s1 = (s2 − s)(s − 2); s4 = n(2n − 1);
s2 = (n2 + n + 1)s2 − (n + 2)s;
l1 = (s − 3)s1 ;
l3 = ns2 + [(n − 1)s − 1]s3 ;
s3 = 3n2 s;
l2 = s1 + (ns − 2)s2 ;
l4 = (2n − 1)s3 + (3n − 2)ss4 ;
l5 = (3n − 2)s4 : If Eq. (1) is invariant under the set S2 , its solutions are given by
1=(1−n) (1 − n) u= x + g(t)xs(1−n) s(n − 1) + 1
(18)
with g(t) to be determined. This type of solutions have appeared in physical situations, such as the special “dipole” solution takes this form [17]. In the followings we consider some applications of the invariant set S2 to Eq. (1) and adduce exact solutions to some parabolic and KdV-like equations. 3.2. Quasilinear heat equations For the quasilinear heat equation of the form (5), using the di:erentiation rules, one computes on the set S2 1 1 A1 (u) = 2 [(s2 − s)uD + s2 u2 Q] + [(n + 1)sun D + 2sun+1 Q] x x + nu2n−1 D + u2n Q + G(u):
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It follows from (18) that ut =
1 s−a g (t)un ; 1−n
where a = s(n − 1). Then g(t) satis9es 1 g (t) = xa−2 [(s2 − s)u1−n D + s2 u2−n Q] + xa−1 [(n + 1)sD + 2suQ] 1−n + xa [nun−1 D + un Q + u−n G] ≡ xa−2 D1 + xa−1 D2 + xa D3 :
(19)
Since the left-hand side of this equation is independent of x, di:erentiating it with respect to x, we have xa−3 [suD1 + (a − 2)D1 ] + xa−2 [un D1 + suD2 + (a − 1)D2 ] + xa−1 [un D2 + suD3 + aD3 ] + xa un D3 = 0 which yields suD1 + (a − 2)D1 = 0;
(20a)
un D1 + suD2 + (a − 1)D2 = 0;
(20b)
un D2 + suD3 + aD3 = 0;
(20c)
D3 = 0:
(20d)
Integrating (20b)–(20d), we have D3 = c1 ; D2 =
ac1 1−n + c2 ; u n−1
D1 = −
ac1 c2 (1 − a) 1−n u u2−2n + + c3 ; 2 2(1 − n) 1−n
where ci ; i = 1; 2; 3 are integration constants. The substitution of the expressions for D1 , D2 and D3 into (20a) implies the constants satisfying (a − 2)c3 = 0; (a − 1)c2 = 0; (a + 2)c1 = 0:
(21)
C. Qu, P.G. Estevez / Nonlinear Analysis 52 (2003) 1655 – 1673
Two possibilities arise: Case 1: c3 = c1 = 0, c2 = 0. In this case, a = 1. D, Q and G satisfy (s2 − s)u1−n D + s2 u2−n Q = 0; (n + 1)sD + 2suQ = c2 ; nun−1 D + un Q + u−n G = 0 which has solution D=
c2 = D0 ; 3
1 Q = (n − 2)D0 u−1 ; 3 G = 2(1 − n)D0 u2n−1 : In this case, g(t) satis9es g = (1 − n)c2 and is given by g(t) = (1 − n)c2 (t − t0 ): Case 2: c1 = c2 = 0, c3 = 0. In this case, a = 2. D, Q and G satisfy (s2 − s)u1−n D + s2 u2−n Q = c3 ; (n + 1)sD + 2suQ = 0; nun−1 D + un Q + u−n G = 0 which gives D=
1−n c3 un−1 ; 4
Q=
n2 − 1 c3 un−2 ; 8
G=
(n − 1)2 c3 u3n−2 : 8
In this case, g(t) is given by g = (1 − n)c3 (t − t0 ):
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Remark 1. If we use the general invariant set S1 to Eq. (5), then u 1 s ux = + H; H = F exp (n − 1) d z ; x F(z) uxx =
1 2 1 (s FF − sF) + s(HF) + HH : x2 x
So A1 (u) and (19) become A1 (u) =
1 1 [D(sFF − sF) + s2 F 2 Q] + [sD(HF) + 2sFHQ] x x2 + DHH + QH 2 + G
and
2 1 D a−2 D 2 QF a−1 +x g (t) = x (sFF − sF) + s s (HF) + 2sFQ H H 1−n H G : + xa DH + QH + H 3.3. Radially symmetric N -dimensional parabolic equation Consider the radially symmetric N -dimensional parabolic equation (11). Applying the invariant set S2 to it, we get A2 (u) =
1 1 [(s2 − s)uD + s2 u2 Q + s(N − 1)uD] + [((n + 1)s x2 x + N − 1)un D + 2sun+1 Q] + nu2n−1 D + u2n Q + G(u):
Then g(t) satis9es 1 g (t) = xa−2 D˜ 1 + xa−1 D˜ 2 + xa D˜ 3 ; 1−n
(22)
where D˜ 1 = [s2 − s + s(N − 1)]u1−n D + s2 u2−n Q; D˜ 2 = [(n + 1)s + N − 1]D + 2suQ; D˜ 3 = nun−1 D + un Q + u−n G: The left-hand side of (22) is independent of x, di:erentiating it with respect to x, we obtain system (20), in which Di are replaced by D˜ i . Three cases arise: (1) c1 = c3 = 0,
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Table 2 Number
1
2
3
D(u)
1 c2 4−N
1 c1 u1−n (N − 1)(n − 1)
n−1 c3 un−1 N −5
Q(u)
(n − 1)N − 2n + 3 c2 u−1 N −4
2s + 3N − 5 c1 u−n 4(N − 1)
(1 − n)[(n − 1)N + n + 3] c3 un−2 4(N − 5)
G(u)
(n − 1)(N − 3) c2 u2n−1 4−N
N −3 c1 un 4(N − 1)
(n − 1)2 (N − 3) c3 u3n−2 4(N − 5)
g(t)
(1 − n)c2 (t − t0 )
n−1 (t − t0 )−1 c1
(1 − n)c3 (t − t0 )
c2 = 0, N = 4; (2) c2 = c3 = 0, c1 = 0; (3) c1 = c2 = 0, c3 = 0, N = 5. We list the results in Table 2. 3.4. KdV-type equation Consider the KdV-type equation of the form ut = D(u)uxxx + Q(u)uxx + H (u)ux3 + G(u)ux ≡ L(u);
(23)
where D, Q, H and G are smooth functions of u. The analysis is the same as for the second-order quasilinear parabolic equations. Substituting the di:erential rule (17) into (23), we have L(u) =
1 1 (s1 uD + s3 u3 H ) + 2 (s2 un D + (s2 − s)uQ + 3s2 un+2 H ) x3 x +
1 (s3 u2n−1 D + (n + 1)sun Q + 3su2n+1 H + suG) x
+ s4 u3n−2 + nu2n−1 Q + un G + u3n H: g(t) satis9es 1 g = xa−3 J1 + xa−2 J2 + xa−1 J3 + xa J4 ; 1−n
(24)
where J1 = u−n (s1 uD + s3 u3 H ); J2 = u−n (s2 un D + (s2 − s)uQ + 3s2 un+2 H ); J3 = u−n (s3 u2n−1 D + (n + 1)sun Q + 3su2n+1 H + suG); J4 = u−n (s4 u3n−2 D + nu2n−1 Q + un G + u3n H ):
(25)
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The left-hand side of (24) is independent of x implies Ji ; i = 1; 2; 3; 4, satisfying suJ1 + (a − 3)J1 = 0; un J1 + suJ2 + (a − 2)J2 = 0; un J2 + suJ3 + (a − 1)J3 = 0; un J3 + suJ4 + aJ4 = 0; J4 = 0:
(26)
Solving system (26), we get s sc1 c2 (a − 1) 2−2n a − 2 1 J1 = u3−3n − u + + c3 u1−n + c4 ; 3(n − 1)2 2 n − 1 (1 − n)2 n−1 J2 =
sc1 c2 (1 − a) 1−n u2−2n + + c3 ; u 1−n 2(1 − n)
J3 =
ac1 1−n + c2 ; u n−1
J4 = c 1 and the integration constants ci ; i = 1; 2; 3; 4, satisfy (a − 3)c4 = 0; (a − 2)c3 = 0; (a − 1)(a + 3)c2 = 0; (a + 2)(2a + 3)c1 = 0: Several cases arise: (1) c1 = c2 = c3 = 0; c4 = 0; (2) c1 = c2 = c4 = 0; c3 = 0; (3) c1 = c3 = c4 = 0; c2 = 0; a = 1; (4) c1 = c3 = c4 = 0; c2 = 0; a = −3; (5) c2 = c3 = c4 = 0; c1 = 0. We put the corresponding results in Tables 3 and 4.
4. The extended rotation-scaling group In this section, we propose an extension for both extended scaling (S0 ) and the rotation group (R0 ), which is characterized by E0 = {u : ux = f(x)F(u)};
(27)
where f(x) satis9es fx = af2 + b;
(28)
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Table 3 Number
D(u)
H (u)
1
1 + n − 2n2 c4 un−1 15(n + 5)
(n − 1)(n3 − n2 − 2n + 5) c4 un−3 15(n + 5)
2
1 c3 8
(n − 3)(2 − n) c3 u−2 16
3
2−n c2 u1−n 3(n − 1)(n − 3)
(2 − n)2 (2n − 3) c2 u−1−n 3(n − 1)(n − 3)
4
n+2 c2 u1−n 3(1 + 5n)(1 − n)
n2 (2n − 5) c2 u−1−n 3(1 − n)(1 + 5n)
5
4n2 − 11n + 34 c1 u2−2n 6(7n − 1)(n − 1)2
−
2n4 − 4n3 + 22n2 + 11n − 4 c1 u−2n 6(7n − 1)(n − 1)2
Table 4 Number
Q(u)
G(u) 1)2 (n
g(t) 1)2 (5
1
+ 2) (n − c4 u2n−2 3(n + 5)
(n −
2
3 (1 − n)c3 un−1 4
3(2 − 5n + 3n2 ) c3 u2n−2 16
(1 − n)c3 (t − t0 )
3
2n − 5 c2 n−3
2n2 − 7n + 4 c2 un−1 3−n
(1 − n)c2 (t − t0 )
4
(2n + 1) c2 3(5n + 1)
n(2n + 1) c2 un−1 3(5n + 1)
5
−
n2 − 20n − 8 c1 u1−n 6(7n − 1)(n − 1)2
−
− 6n − 15(n + 5)
2n2 )
c4 u3n−3
5n3 − 38n2 + 97n − 10 c1 6(7n − 1)(n − 1)
(1 − n)c4 (t − t0 )
1 (n − 1)c2−1 (t − t0 )−1 4
−1=2 c1 − (t − t ) 0 4(n − 1)2
where a; b = 0 are two constants. Note that for a = 0 and b = 0, the invariant set E0 reduces, respectively, to R0 and S0 , so it is viewed as a generalization for S0 and R0 . In the set E0 , we have the following formulae: uxx = f2 (FF + aF) + bF; uxxx = f3 [F(FF + aF) + 2aF(F + a)] + bfF(3F + 2a); uxxxx = f4 {F[F(FF + aF) + 2aF(F + a)] + 3aF[(FF + aF) + 2a(F + a)]} + f2 {3b[F(FF + aF) + 2aF(F + a)] + b(a + F)(3FF + 2aF) } + b2 F(3F + 2a):
(29)
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In the invariant set E0 , solutions are given by (i) ab ¿ 0. u
ds = ln| cos +a(x + x0 )| + g(t); F(s)
(30a)
where +2 = b=a. (ii) ab ¡ 0. In this case we obtain two solutions: u and
u
ds = −ln sinh[+a(x + x0 )] + g(t) F(s)
(30b)
ds = −ln cosh[+a(x + x0 ) + g(t)]; F(s)
(30c)
where +2 = −b=a. Now, we use the above formulae to the construction of exact solutions to the following nonlinear evolution equations. 4.1. Quasilinear heat equations Consider the quasilinear heat equation (5). Substituting (27)–(29) into (5), we have G g (t) = f2 (DF + QF + aD) + + bD: (31) F The left-hand side of (31) does not depend on x yields (DF + QF + aD) F + 2a(DF + QF + aD) = 0; G + bD F + 2b(DF + QF + aD) = 0: F The integration gives
DF + QF + aD = c0 exp −2a
u
ds ; F(s)
u G ds bc0 + c1 ; + bD = exp −2a F a F(s) where c0 and c1 are arbitrary constants. Hence g(t) satis9es the equation 1 bc0 2 exp −2a g = c0 f (1) + f(y) dy e−2ag + c1 : a 0
C. Qu, P.G. Estevez / Nonlinear Analysis 52 (2003) 1655 – 1673
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In particular, equation ut = uxx + G(u); admits solutions (30) with F satisfying FF + 2a(F + a) = 0 and G(u) is given by
u
G = −2bF ln F − 2abF
ds + cF; F(s)
where c is a constant. Remark 2. This approach is also available to equations with x-variable coeNcients. For example, the equation ut = uxx + X (x)G(u) also admits solutions (30), where F, G and X satisfy G G = 0; F F +. F + 2a F + a + . F F G G / F + 2b F + . + a = 0; F F X (x) = .f2 + /; where . and / are constants. 4.2. Fourth-order equations Using the invariant set E0 to the fourth-order equation (13), we have g = f4 J1 (u) + f2 J2 (u) + J3 (u); where J1 = !{[F(FF + aF) + 2a(FF + aF)] + 3a[(FF + aF) + 2a(F + a)]}; F +a J2 = ! 3b[(FF + aF) + 2a(F + a)] + [3bFF + 2abF] F + D(F + a) + QF; G : F The left-hand side of (32) does not depend on x yields the following system J3 = b2 (3F + 2a)! + bD + FJ1 + 4aJ1 = 0; J1 + FJ2 + 2aJ2 = 0; 2bJ2 + FJ3 = 0:
(32)
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The integration gives J1 = c1 exp −4a
u
ds ; F(s)
u u ds ds 2bc1 ; + c2 exp −2a J2 = exp −4a F(s) a F(s) u u bc ds ds b 2 c1 + 2 exp −2a + c3 ; J3 = 2 exp −4a a F(s) a F(s) where ci ; i = 1; 2; 3 are arbitrary constants, where g satis9es g = d1 e−4ag + d2 e−2ag + c3 ; where
1 2bc1 2 b2 c1 4 d1 = c1 f (1) + f(y) dy ; f (1) + 2 exp −4a a a 0 d2 =
c2 f12
bc2 + a
exp −2a
0
1
f(y) dy :
5. Conclusions In this paper, we have shown that the rotation group admits a “nonlinear” extension, it is characterized by an invariant set of a contact 9rst-order di:erential structure, which is parallel to the “nonlinear” extension for the scaling group. This extension has been used to construct exact solutions to some nonlinear evolution equations of second- and fourth order. We also proposed a more general “nonlinear” extension to the scaling group, it is characterized by an invariant set of a contact 9rst-order di:erential structure with nonhomogeneous term. As the nonhomogeneous term is zero, it is reduced to the invariant set introduced in [14,15]. A further extension to both extended scaling groups S0 and rotation group R0 was also proposed. As the consequence, these extensions have been used to the construction of some exact solutions of nonlinear evolution equations, which include the KdV-type equations. Acknowledgements This research has been supported in part by DGICYT under project PB98-0262. Qu’s research was supported in part by the Ministry of Education and Culture of Spain, the NNSF of China (Grant No. 19901027) and NSF of Shaan Xi Province.
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