CLASSICAL AND NONCLASSICAL SYMMETRIES AND ... - AIMS

Dynamical Systems, Differential Equations and Applications AIMS Proceedings, 2015

doi:10.3934/proc.2015.0151 pp. 151–158

CLASSICAL AND NONCLASSICAL SYMMETRIES AND EXACT SOLUTIONS FOR A GENERALIZED BENJAMIN EQUATION

´ n, M. L. Gandarias and J. C. Camacho M. S. Bruzo Departamento de Matem´ aticas Universidad de C´ adiz, PO.BOX 40 11510 Puerto Real, C´ adiz, Spain

Abstract. We apply the Lie-group formalism to deduce symmetries of a generalized Benjamin equation. We make an analysis of the symmetry reductions of the equation. In order to obtain travelling wave solutions we apply an indirect F-function method. We obtained in an unified way simultaneously many periodic wave solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions. We compare these solutions with the solutions derived by other authors by using different methods and we observe that we have obtained new solutions for this equation.

1. Introduction. In this paper we consider the generalized Benjamin equation, ∆ ≡ utt + a(up ux )x + βuxxxx = 0

(1)

where a, β and p are arbitrary constants different to zero. This kind of equation is one of the most important nonlinear partial differential equations, used in the analysis of long wave in shallow water [8]. By applying the extended tanh method N. Taghizadeh et al found exact solutions for equation (1) [12]. In [2] Belgacem et al applied the extended trial equation method and they obtained new analytical solutions in terms of logaritmic functions, rational, elliptic and Jacobi elliptic functions. Symmetry reductions and exact solutions have several different important applications in the context of differential equations. Since solutions of partial differential equations asymptotically tend to solutions of lower-dimensional equations obtained by symmetry reduction, some of these special solutions will illustrate important physical phenomena. In particular, exact solutions arising from symmetry methods can often be used effectively to study properties such as asymptotics and “blow-up”. Furthermore, explicit solutions (such as those found by symmetry methods) can play an important role in the design and testing of numerical integrators; these solutions provide an important practical check on the accuracy and reliability of such integrators, [6]. The application of Lie transformations group theory for the construction of solutions of nonlinear partial differential equations (PDEs) is one of the most active fields of research in the theory of nonlinear PDEs and applications. Classical symmetries of nonlinear PDEs may be used to reduce the number of independent variables of the PDEs; in particular we might reduce the PDEs to ODEs. The ODEs may also have symmetries that allow us to reduce the order of the equation, and we can integrate to find exact solutions [9, 10, 11]. Bluman and Cole [3], in their study of symmetry reductions of the linear heat equation, proposed the so-called nonclassical method of group-invariant solutions. Bruzón and Gandarias applied the Lie-group method and the nonclassical method to derive symmetries of a Boussinesq equation which generalize equation (1), [7]. 2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 76M60. Key words and phrases. Exact solutions, classical symmetries, nonclassical symmetries.

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152

´ M. S. BRUZON, M. L. GANDARIAS AND J. C. CAMACHO

The aim of this paper is to make an analysis of equation (1), by using classical symmetries and nonclassical symmetries, and to obtain new solutions. The structure of the work is as follows: In Sec. 2 we study the Lie symmetries of equation (1), we obtain the symmetry reduction, similarity variables and the reduced ODEs. In Sec. 3 we apply the nonclassical method. In Sec. 4 we derive some exact solutions. Finally, in Sec. 5 some conclusions are presented. 2. Classical Symmetries. To apply the Lie classical method to equation (1) we consider the one-parameter Lie group of infinitesimal transformations in (x, t, u) given by x∗

=

x + ξ(x, t, u) + O(2 ),

(2)

t∗

=

t + τ (x, t, u) + O(2 ),

(3)

u

∗

=

2

u + η(x, t, u) + O( ),

(4)

where is the group parameter. We require that this transformation leaves invariant the set of solutions of equation (1). This yields to an overdetermined, linear system of equations for the infinitesimals ξ(x, t, u), τ (x, t, u) and η(x, t, u). The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form ∂ ∂ ∂ + τ (x, t, u) + η(x, t, u) . (5) ∂x ∂t ∂u The functions u = u(x, t), which are invariant under the infinitesimal transformations v, are, in essence, solutions to an equation arising as the “invariant surface condition”: v = ξ(x, t, u)

∂u ∂u − τ (x, t, u) = 0. (6) ∂x ∂t The symmetry variables are found by solving the invariant surface condition. The reduction transforms the PDE into ODEs. We consider the classical Lie group symmetry analysis of equation (1). The set of solutions of equation (1) is invariant under the transformation (2)-(4) provided that η(x, t, u) − ξ(x, t, u)

pr(4) v(∆) = 0

when

∆ = 0,

where pr(4) v is the fourth prolongation of the vector field (5). Hence we obtain the following twelve determining equations for the infinitesimals: τu = 0, τx = 0, ξu = 0, ηuu = 0, ξt = 0, (7) 2ηux − 3ξxx = 0, 2ηtu − τtt = 0, τt − 2ξx = 0, aηxx up + βηxxxx + ηtt = 0, 2aξx up+1 + apηup + 6βηuxx u − 4βξxxx u = 0,

EXACT SOLUTIONS FOR A GENERALIZED BENJAMIN EQUATION

153

ηu u + 2ξx u + pη − η = 0, 2aηux up+1 − aξxx up+1 + 2apηx up + 4βηuxxx u − βξxxxx u = 0. From system (2), ξ = k1 x + k2 , τ = 2k1 t + k3 and η = δ(x, t) −

2k1 where a, p, β and p

δ = δ(x, t) satisfy (p − 1) δ = 0, apδup = 0, (8) 2apδx up = 0, aδxx up + βδxxxx + δtt = 0. Theorem 1 Group Classification Theorem The Lie point symmetry group of GB equation (1) with p 6= 0, is determined by the operators of the group of space and time translations v1 = ∂x ,

v2 = ∂t ,

(9)

and a special dilatation ∂ ∂ 2 ∂ + 2t − u . ∂x ∂t p ∂u From v1 + v2 substituting the infinitesimals into the invariant surface condition (6) we obtain the similarity variable and the similarity solution v3 = x

z = µx − λt, u(x, t) = h(z).

(10)

Substituting (10) into (1) we obtain the reduced equation 2

β µ4 h0000 + λ2 h00 + a hp−1 (h0 ) µ2 p + a µ2 hp h00 = 0.

(11)

Integrating twice respect to z we obtain a nonlinear ordinary differential equation β µ4 h00 + λ2 h +

a µ2 p+1 h + c1 z + c2 = 0, p+1

(12)

where c1 and c2 are integrating constants. If p = 1 equation (11) is a µ2 2 h + c1 z + c2 = 0 2 which is solvable in terms of the first Painlevé equation [4]. In Section 4 we construct solutions of equation (12) by an indirect F function method. From v3 substituting the infinitesimals into the invariant surface condition (6) we obtain the similarity variable and the similarity solution βµ4 h00 + λ2 h +

1

z = x t− 2 , 1 u(x, t) = h(z) t− p .

(13)

Substituting (13) into (1) we obtain the reduced equation 2

4 p2 β h h0000 + 4 p2 a hp+1 h00 + p2 z 2 h h00 + (3 p + 4) p z h h0 + 4 p3 a hp (h0 ) +(p + 1) 4 h2 = 0.

β (n + 1) (p + 2) (−2) h(z) = a p2

p1

2

z− p

(14)


154

is a solution of the equation (14) which leads to the solution of equation (1) 1 β (p + 1) (p + 2) (−2) p − p2 u(x, t) = x . a p2 We remark that for the case corresponding to dilatation, by the classical Lie method we obtain stationary solutions. If p = 1 equation (14) is 1 7 h0000 + ahh00 + a(h0 )2 + z 2 h00 + zh0 + 2h = 0 4 4 and this equation is solvable in terms of the fourth Painlevé equation [4]. 3. Nonclassical Symmetries. The basic idea of the method is that the PDE (1) is augmented with the invariance surface condition ∂u ∂u +τ − η = 0. (15) Φ≡ξ ∂x ∂t which is associated to the vector field (5). By requiring that both, (1) and (15), are invariant under the transformation with infinitesimal generator (5), an overdetermined nonlinear system of equations for the infinitesimals ξ(x, t, u), τ (x, t, u) and η(x, t, u) is obtained. The number of determining equations arising in the nonclassical method is smaller than for the classical method, consequently the set of solutions is, in general, larger than for the classical method. However, the associated vector fields do not form a vector space. To obtain nonclassical symmetries of (1) we apply the algorithm described in [5] for calculating the determining equations. We can distinguish two different cases: In the case τ 6= 0, without loss of generality, we may set τ (x, t, u) = 1, and we obtain a set of nine determining equations for the infinitesimals ξ(x, t, u) and η(x, t, u). ξu = 0, ηuu = 0, 2 ηux − 3 ξxx = 0, 2 s u ξx + ηu p u + η p2 − η p = 0, (16) −4 β u ξxxx + 4 u ξ 2 ξx + 2 a up+1 ξx + 2 u ξ ξt + a η p up + 6 β ηuxx u = 0, −4 ηx ξ ξx + 4 η ηu ξx + 4 ηt ξx − 2 ηx ξt + a ηxx up + β ηxxxx +ηtt + 2 η ηut = 0, 2

−β u ξxxxx − a up+1 ξxx + 4 u ξ (ξx ) − 2 u ξt ξx − 8 ηu u ξ ξx − u ξtt −2 ηu u ξt − 2 ηut u ξ + 2 a ηux up+1 + 2 a ηx p up + 4 β ηuxxx u = 0. If p 6= 1, from (16), we obtain that symmetries admitted by (1) are ξ=

x + k2 , 2 t + 2 k1

τ = 1,

η=−

u , p (t + k1 )

(17)

that is a classical symmetry. By comparing these symmetries with the symmetries obtained by the classical method in Section 2 we can observe that the nonclassical method applied to (1)gives only rise to the classical symmetries. If p = 1, the symmetries admitted by (1) are ξ = f (t)x + g(t), η=−

τ = 1,

2 f ft x2 + 2 f 3 x2 + f gt x + ft g x + 4 f 2 g x + a f u + g gt + 2 f g 2 , a

(18)


155

where f and g must satisfy ft t + 2 f ft − 4 f 3 = 0 gt t + 2 f gt − 4 f 2 g = 0 This solution was obtained by Clarkson [4]. In the case τ = 0, without loss of generality, we may set ξ = 1 and we obtain three determining equation in the infinitesimal η: ηuu = 0, ηtu = 0, a (ηxx + 2 η ηux ) up+2 + a η (3 ηx + 2 η ηu ) p up+1 + a η 3 (p − 1) p up + (β ηxxxx + 4 β ηxu ηxx + 6 β ηxxu ηx + 4 β ηu ηxu ηx+ 4 β η ηxxxu 2

(19)

2

+6 β η ηu ηxxu + 8 β η (ηxu ) + 4 β η (ηu ) ηxu + ηtt u2 = 0.

If n = 1, a = β the solutions were obtained by Clarkson [4] η=

48 2 , u+ x + x0 (x + x0 )3

η = 2xΦ2 (t) + Φ1 (t), where Φ1 and Φ2 satisfy d2 Φ2 d2 Φ1 2 + 6Φ + 6Φ2 Φ1 = 0. = 0, 2 dt2 dt2 From (19) we obtain that nonclassical symmetries admitted by (1) are ξ = 1,

τ = 0,

η = ρ(x) u.

(20)

where a, β and ρ = ρ(x) satisfy ρ00 + (3p + 2)ρ ρ0 + p(p + 1)ρ3 = 0, (21) ρ0000 + 4 ρ ρ000 + 10 ρ00 ρ0 + 6ρ2 ρ00 + 12ρ(ρ0 )2 + 4ρ3 ρ0 = 0. System (21) can be reduced into equation 2

30 p (ρx ) + 5 p ρ2 ρx − 10 ρ2 ρx − (6 p3 + 3 p2 − p + 2) ρ4 = 0. A solution of equation (22) is ρ =

k1 x

(22)

when

6 k1 2 p3 + 3 k1 2 p2 − k1 2 p + 5 k1 p − 30 p + 2 k1 2 − 10 k1 = 0. This solution satisfy system (21) when a (k1 p − 2) (k1 p + k1 − 1) = 0 and −4 b (k1 − 3) (k1 − 2) (k1 − 1) = 0. These infinitesimal generators lead to new reductions, which cannot be obtained by Lie classical symmetries and they do not appear in [7]. 4. Travelling wave solutions. After multiplying (12), with c1 = 0, by 2h0 and integrating once with respect to z we get 2

b µ4 (h0 ) + h2 λ2 +

2 a µ2 hp+2 + 2 c2 h = 0. (p + 1) (p + 2)

(23)

Let us assume that equation (23) has solution of the form h(z) = AF B (z), where A and B are parameters to be determined later.

(24)


156

Table 1. Coefficients of equations (25) and (26) with p = 1. i

r=1 r = − 2 βcµ24 A r = − 12aβAµ2

1 2

m = −(1 + k 2 ) m= m=

2

− 4 βλ µ4 2 − 4 βλ µ4

q = 2k 2

h

q = − 12aβAµ2 q = − 2 βcµ24 A

A[sn(z, k)]2 A[sn(z, k)]−2

Table 2. Coefficients of equations (25) and (26) with p = 2. i

r=1

3 4

m = −(1 + k 2 ) m=

r=

2 − 6aβAµ2

m=

2 − βλµ4 2 − βλµ4

q = 2k 2 2 − 6aβAµ2

q=

h A[sn(z, k)] A[sn(z, k)]−1

By substituting (24) into (23) we obtain p

(F 0 )2 =

−

F 2−B (p2 A F B λ2 +3 p A F B λ2 +2 A F B λ2 +2 a µ2 A F B (A F B ) +2 c2 p2 +6 c2 p+4 c2 ) β µ4 (p+1) (p+2) A B 2

.

(25)

In the following we will determine the exponents and coefficients of equations (25). So that equation (25) is solvable in terms of Jacobi elliptic function, that is equation (25) becomes (F 0 )2 = r + mF 2 + qF 4 ,

(26)

where r, p and q are constants. We may choose them properly such that the corresponding solution F of the ODE (26) is one of the Jacobi elliptic, combined Jacobi elliptic functions. If r = 1, m = −(1 + k 2 ), q = 2k 2 , then the solution is B

h1 = A [sn(z|k)]

(27)

or B cn(z|k) h2 = A [cd(z|k)] ≡ A , dn(z|k) where 0 ≤ k ≤ 1, is called modulus of Jacobi elliptic functions, and sn(z|k) is the Jacobi elliptic sine function, [1]. If r = 1 − k 2 , m = 2k 2 − 1, q = −2k 2 , the solution is B

B

h3 = A [cn(z|k)] , where cn(z|k) is the Jacobi elliptic cosine function, [1]. If r = k 2 − 1, m = 2 − k 2 , q = −2, the solution is B

h4 = A [dn(z|k)] , where dn(z|k) is the Jacobi elliptic function of the third kind, [1]. By comparing the exponents and the coefficients of equations (25) and (26) we can obtain exact solutions. In Tables 1 and 2 we give some examples. In Table 1, in the first line, we show for p = 1, the coefficients of equation (25) which admits solution (27). For B = 2, substituting r, m and q in line i = 1 and by solving the system we can deduce the parameters A, λ and µ for which equation (26) admits solution (27). Substituting r, m and q in line i = 2 and by solving the system we can deduce the parameters A, λ and µ for which equation (26) admits solution (27) with B = −2. In Table 2 we show for p = 2 in the first line the coefficients of equation (25) which admits solution (27). For B = 1, substituting r, m and q in line i = 3 and by solving the system we can deduce the parameters A, λ and µ for which equation (26) admits solution (27). For B = −1, substituting r, m and q in line i = 4 and by solving the system we can deduce the parameters A, λ and µ for which equation (26) admits solution (27).


157

From these solutions we can derive some explicit solutions with physical interest: soliton q s 5 solutions, compactons, kinks and antikinks. For example, for µ = λ = 2 , s = 12 , A = 1, B = 2, the solution sin2 (µx − λt) |x − t| ≤ 2π s , u(x, t) = 0 |x − t| > 2π s is a sine-type double compacton (that is solution which has two peaks, see Fig. 1) 10

5

0

-5

-10

1.0

0.5

0.0 -10

-5

0

5

10

Figure 1. Solution (27), with k = 0, for µ = λ = 2s , s = B = 2.

q

5 12 ,

A = 1 and

5. Conclusions. We prove that the Generalized Benjamin equation (1) admits a threeparameter symmetry group. From the characteristic equation, we find the similarity variables. We obtain the reduced form of the original nonlinear partial differential equation as a nonlinear ordinary differential equation. Some of these solutions are new solutions for equation (1) and they do not appear in [7]. Acknowledgments. The support of DGICYT project MTM2009-11875 with the participation of FEDER and Junta de Andaluc´ıa FQM-201 group is gratefully acknowledged. REFERENCES [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [2] F. M. Belgacem, H. Bulut, H. M. Baskonus and T. Akturk, Mathematical analysis of the Generalized Benjamin and Burger-Kdv Equations via the Extended Trial Equation Method, J. Assoc. of Arab Univ. Basic and Appl. Sc., (2013), preprint. [3] G. W. Bluman and J. Cole, The General Similarity Solution of the Heat Equation, Phys. J. Math. Mech. 18 (1969) 1025–1042. [4] P. A. Clarkson, Nonclassical Symmetry Reductions of the Boussinesq Equation. Chaos, Solitons & Fractal, 5 12 (1995) 2261–2301. [5] P. A. Clarkson and E. L. Mansfield, Algorithms for the nonclassical method of symmetry reductions. SIAM J. Appl. Math., 55 (1994), 1693–1719. [6] P. A. Clarkson and T. J. Priestley, Symmetries of a Generalised Boussinesq equation. Electronic Edition, 1996. [7] M.L. Gandarias and M.S. Bruz´ on, Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation, J. Nonlin. Math. Phys. 5 (1998) 8–12. [8] W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele and A. Meerpole, Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, J. Phys. A. Math. Gen., 19 (5) (1986), 607–628. [9] N. H. Ibragimov, Transformation groups applied to mathematical physics, Reidel–Dordrecht, (1985). [10] P. Olver, Applications of Lie groups to differential equations, Springer-Verlag, New York, (1993). [11] L. V. Ovsyannikov, Group analysis of differential equations, Academic, New York, (1982).

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[12] N. Taghizadeh, M. Mirzazadeh and S. R. Moosavi Noori, Exact Solutions of the Generalized Benjamin Equation and (3 + 1)- Dimensional Gkp Equation by the Extended Tanh Method, Appl. Appl. Math., 7 (2012), 175–187.

Received September 2014; revised December 2014. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]