Some conservation laws for a forced KdV equation

Nonlinear Analysis: Real World Applications 13 (2012) 2692–2700

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Some conservation laws for a forced KdV equation M.L. Gandarias ∗ , M.S. Bruzón Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, Cádiz, 11510, Spain

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Article history: Received 13 February 2012 Accepted 23 March 2012

The Korteweg–de Vries equation with a forcing term is established by recent studies as a simple mathematical model by which the physics of a shallow layer of fluid subject to external forcing is described. It serves as an analytical model of tsunami generation by sub-marine landslides. In this paper, we derive the classical Lie symmetries admitted by the forced KdV equation. Looking for travelling wave solutions we find that the forced KdV equation has abundant exact solutions that can be expressed in terms of the Jacobi elliptic functions. Hence the Korteweg–de Vries equation with a forcing term has plenty of periodic waves and solitary waves. These solutions are derived from the solutions of a simple nonlinear ordinary differential equation. The first author has introduced the concept of weak self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by Ibragimov in 2006. In a previous paper we found a class of weak self-adjoint forced Korteweg–de Vries equations which are neither self-adjoint nor quasi self-adjoint. By using a general theorem on conservation laws proved by Ibragimov, the Lie symmetries and the new concept of weak self-adjointness we find some conservation laws for some of these partial differential equations. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Self-adjointness Conservations laws Symmetries Partial differential equations

1. Introduction In [1] Pelinovsky proposed an analytical model of Tsunami generation by sub-marine landslides c ut + auux + buxxx + cux = − zx (x, t ) 2

(1)

with a=

3c 2h

,

b=

ch2 6

a ̸= 0, b ̸= 0, where u = u(x, t ) refers to the elevation of the free water surface, z = z (x, t ) represents the solid bottom, h is assumed to be the constant mean water depth and c = (gh)1/2 is the long wave speed with g being the gravity acceleration. When the right-hand side term of Eq. (1) 2c zx (x, t ), which is called the forcing term, equals zero Eq. (1) is reduced to the classical KdV equation. Therefore, Eq. (1) is called a KdV equation with forcing term or the forced KdV (fKdV) equation. In recent years, several explicit asymptotic derivations for this generic model equation (1) have been carried out [2,3]. Mathematically, we should note that in the absence of the forcing term 2c zx (x, t ), the classical KdV equation is completely integrable [4–7] while the KdV equation with a forcing term is not known to be integrable. The compound KdV equation ut + auux + bu2 ux + uxxx = 0

∗

Corresponding author. E-mail address: [email protected] (M.L. Gandarias).

1468-1218/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2012.03.013

M.L. Gandarias, M.S. Bruzón / Nonlinear Analysis: Real World Applications 13 (2012) 2692–2700

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with b > 0 arises in many physical models such as plasma physics, solid physics, quantum theory and so on. In [8] the orbital stability of solitary waves with nonzero asymptotic value for the compound KdV equation has been considered. In [9] a multiple scale analysis on the modified nonlinear Schrödinger equation has been performed. In [10] an adjoint technique is applied to a shallow water model in order to estimate the influence of the model’s parameters on the solution. Among parameters, the bottom topography, initial conditions, boundary conditions on rigid boundaries, viscosity coefficients, Coriolis parameter and the amplitude of the wind stress tension are considered. In [11] the global attractor for a weakly damped forced shallow water equation has been considered. This equation for some special values of the parameters becomes equivalent to a fifth-order shallow water equation. Recently, Constantin and Johnson [12] have claimed that equations like Camassa–Holm and Degasperis–Procesi might be relevant to the modelling of tsunamis. In [13] a twocomponent Degasperis–Procesi system which arises in shallow water theory has been considered. The authors investigate the formation of singularities and persistence properties of solutions they have discussed the wave-breaking mechanism and have shown travelling wave solutions. In a previous work [3] Zhao and Guo studied a type of forced KdV equation by using Hirota’s bilinear method. In a recent work Salas has found analytic solutions to the forced KdV equation (1) via Hirota’s bilinear method for the special choice 2 z = − xf (t ) + constant c f (t ) being an arbitrary smooth function of t. In [14] (see also [15]) a general theorem on conservation laws for arbitrary differential equation which does not require the existence of Lagrangians has been proved. This new theorem is based on the concept of adjoint equations for nonlinear equations. In [16], conservation laws have been derived for some classes of thin film equations by using the concept of selfadjointness. There are many equations with physical significance that are not self-adjoint. Therefore one cannot eliminate the nonlocal variables from conservation laws of these equations by setting v = u, in [17] Ibragimov generalized the concept of self-adjoint equations by introducing the definition of quasi self-adjoint equations. Recently, some works have been done in this direction to get conservation laws for nonlinear wave equations [18], for a generalized Burgers equation [19] and for a modified Zakharov–Kuznetsov equation [20]. In [21] Yasar and Özer have derived conservation laws for one-layer shallow water wave systems and by using this conserved systems they have found potential symmetries for the plane flow case. In [22] the authors have proved that the Camassa–Holm equation is self-adjoint and they have constructed conservation laws for the generalized Camassa–Holm equation using its symmetries. In [23] the conservation laws for a (1 + n)-dimensional heat equation on curved surfaces have been constructed by using a partial Noether’s approach associated with partial Lagrangian [24]. In [25] conservation laws were derived for a nonlocal shallow water wave equation. In [26], by using the nonlocal conservation theorem method [15] and the partial Lagrangian approach [24] conservation laws for the modified KdV equation were presented. It was observed that only the nonlocal conservation theorem method lead to the nontrivial and infinite conservation laws. It happens that many equations having remarkable symmetry properties, such as the forced KdV equation, are neither self-adjoint nor quasi self-adjoint. In [27] one of the present authors has generalized the concept of quasi self-adjoint equations by introducing the concept of weak self-adjoint equations, in [28] the concept of weak self-adjointness has been successfully applied to determine conservation laws for a porous medium equation. In [29] the concept of weak selfadjointness has been applied to derive conservation laws of some Hamilton–Jacobi–Bellman equations arising in financial mathematics. In [30] Ibragimov has generalized this concept and has introduced the concept of nonlinear self-adjointness. By using these two recent developments Freire and Sampaio [31] have deter mined the nonlinear self-adjoint class of a generalized fifth order equation and by using Ibragimov theorem [15] the authors have established some local conservation laws. In [32] Johnpillai and Khalique have studied the conservation laws of some special forms of the nonlinear scalar evolution equation, the modified Korteweg–de Vries (mKdV) equation with time dependent variable coefficients of damping and dispersion ut + u2 ux + a(t )u + b(t )uxxx = 0. The authors use the new conservation theorem [15] and the partial Lagrangian approach [24]. In this work, we first study equation ut + auux + buxxx + cux = f (t )

(2)

from the point of view of the theory of symmetry reductions in partial differential equations. We obtain the classical symmetries admitted by (2) for an arbitrary f , then, we use the transformations groups to reduce the equations to ordinary differential equations. In order to derive further reductions of these equations and to search for further travelling-wave solutions of (2) we employ an auxiliary simple equation. Looking for travelling-wave solutions we find that Eq. (2) has abundant exact solutions that can be expressed in terms of the Jacobi elliptic functions and consequently Eq. (2) has abundant exact solutions that can be expressed in terms of trigonometric and hyperbolic functions. Hence, (2) has plenty of periodic waves and solitary waves. These periodic and solitary waves are derived from the solutions of a simple nonlinear ordinary differential equation [33] and were derived in [34] by using the Hirota bilinear method. Then we determine, for (2), the subclasses of equations which are self-adjoint, quasi self-adjoint and weak self-adjoint and derive, by using the new concept of weak self-adjointness [27] and the notation and techniques of [14], some nontrivial conservation laws for Eq. (2).

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2. Classical symmetries In this section we perform Lie symmetry analysis for Eq. (2). Let us consider a one-parameter Lie group of infinitesimal transformations in (x, t , u) given by x∗ = x + εξ (x, t , u) + O (ε 2 ), t ∗ = t + ετ (x, t , u) + O (ε 2 ),

(3)

u = u + εφ(x, t , u) + O (ε ), ∗

2

where ε is the group parameter. Then one requires that this transformation leaves invariant the set of solutions of the Eq. (2). This yields to the overdetermined, linear system of equations for the infinitesimals ξ (x, t , u), τ (x, t , u) and φ(x, t , u). The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form v = ξ ( x , t , u)

∂ ∂ ∂ + τ (x, t , u) + φ(x, t , u) . ∂x ∂t ∂u

(4)

Having determined the infinitesimals, the symmetry variables are found by solving the invariant surface condition

Φ≡ξ

∂u ∂u +τ − φ = 0. ∂x ∂t

(5)

By solving this system we get that

τ = τ (t )

ξ=

τ′

x + δ(t ) φ = α(t )u + β(x, t ), 3 where α(t ), β(x, t ), τ (t ) and f (t ) must satisfy the following equations

(6)

a βx + αt = 0,

−τ ft − τt f + α f + βx c + b βx x x + βt = 0, a (2 τt + 3 α) = 0,

(7)

−τt t x + 2 c τt − 3 δt + 3 a β = 0.

(8)

Solving this system we find that:

ξ =δ−

k1 x

3 k1 t

τ = k2 −

φ = k1 u +

c k1

+

δt

2 2 a a where δ(t ), f (t ), k1 and k2 must satisfy the following condition 3 a ft k1 t − 2 a ft k2 + 5 a f k1 + 2 δt t = 0.

(9)

For any f (t ) arbitrary, the symmetries that are admitted by (1) are v1 =

∂ , ∂t

v2 = −

x ∂

−

3t ∂

 c ∂ + u+ , 2 ∂t a ∂u

2 ∂x where, for any f (t ), δ(t ) must satisfy (9).

v4 = δ

∂ δt ∂ ++ ∂x a ∂u

2.1. Reductions and solutions Once determined the components ξ , τ and φ of the infinitesimal vector v, by using the invariant surface condition

ξ ux + τ ut − η = 0,

(10)

we reduce Eq. (2) into an ODE, the solutions of which lead to solutions of Eq. (2) invariant with respect to the group of transformations generated by v. In the following reductions we distinguish: Reduction 1. When k1 ̸= 0, we can set without losing generality k1 = 1, k2 = 0, and the symmetry operator (4) takes the form X =−

x ∂ 2 ∂x

−

3t ∂

 c ∂ + u+ . 2 ∂t a ∂u

(11)

From (9) we get f (t ) = kt −5/3 . The invariant surface condition yields the following similarity solution and similarity variable: u=

h x2

c

− , a

ζ =

x3

(12)

t

that substituted in (2) leads to the following ODE 5

27 b hζ ζ ζ ζ 3 − hζ ζ 2 − k ζ 3 + 3 a h hζ ζ + 24 b hζ ζ − 2 a h2 − 24 b h = 0.

(13)

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Reduction 2. When k1 = 0, we can set without losing generality k2 = 1, and the symmetry operator (4) takes the form

∂ ∂ δt ∂ + + . ∂x ∂t a ∂u From (9) we get f − aδt = 0 and from the invariant surface condition we get: X = δ(t )

δ

+ h(ζ ), ζ = x − λ(t )  where λ(t ) = δ(t )dt + k1 t that substituted in (2) leads to the following ODE u=

a

bh′′′ + (ah + c − k1 )h′ = 0.

(14)

(15)

(16)

After integrating once with respect to ζ we get bh′′ +

ah2 2

+ (c − k1 )h + c2 = 0.

(17)

In [33] Kudryashov introduced the simplest equation method in order to look for exact solutions of nonlinear differential equations. He pointed out that many of the so called ‘‘new travelling wave solutions’’ could be derived from the solutions of a simple nonlinear ordinary differential equation. It was pointed out in [33] that equation h′′ + 3h2 − ωh + c1 = 0,

(18)

where c1 is an integrating constant, is very important and the author exhibited many examples of nonlinear partial differential equations describing physical phenomena where Eq. (18) arises. Multiplying Eq. (18) by h′ and integrating once with respect to z we get the nonlinear equation

(h′ )2 + 2h3 − ωh2 + 2c1 h + c2 = 0. Assuming that α, β and γ (α ≥ β ≥ γ ) are roots of the algebraic equation

(19)

1

ωh2 + c1 h + c2 = 0, 2 we can write Eq. (19) as h3 −

(20)

(h′ )2 = −2(h − α)(h − β)(h − γ ).

(21)

Eq. (21) has the general solution h = (α − β)cn

 2

α−γ 2



β −α z, γ −α

+β

(22)

where cn(z ) is the elliptic cosine. The general solution of (21) was first found by Korteweg and de Vries. The general solution of (16) can be written in terms of the Jacobian elliptic functions, so there exist solutions such as h= h= h=

12 b k2 p cn2 (k z , p)

8 b k2 p − 4 b k2 − c1 + c

a 4 b k2 p + 4 b k2 + c1 − c a 4 b k2 p + 4 b k2 + c1 − c a 2

h=

−

− −

a 12 b k2 p sn2 (k z , p) a 12 b k2 p a cn2 (k z , p)

2

4 b k p + 4 b k + c1 − c

(23)

12 b k2

−

.

a a sn2 (k z , p) Any solution h to Eq. (16) will determine a travelling wave solution to the original Eq. (2) u=

12 b k2 p cn2 (k (x − λ(t )) , p) a

12 b k p sn (k (x − λ(t )) , p) 2

u=− u=− u=−

−

2

a 12 b k2 p a cn2 (k (x − λ(t )) , p) 12 b k2 p a sn2 (k (x − λ(t )) , p)

where δ(t ) = a



8 b k2 p − 4 b k2 − c1 + c

f (t )dt and λ(t ) =

+ + 

+

+

a 4 b k2 p + 4 b k2 + c1 − c

a 4 b k2 p + 4 b k2 + c1 − c a 4 b k2 p + 4 b k2 + c1 − c a

δ(t )dt + c1 t.

+

δ(t )

+

δ(t )

a a

δ(t )

+

a

,

δ(t ) a

(24)

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In the particular case p = 1 we obtain the soliton solutions u=

12 b k2 sech2 (k (x − λ(t ))) a

u=−

u=−

u=−

−

12 b k2 tanh2 (k (x − λ(t ))) a 12 b k

12 b k2 a tanh (k (x − λ(t ))) 2

+

+

+

a

8 b k2 + c1 − c

+

a

8 b k + c1 − c a 8 b k2 + c1 − c a

δ(t ) a

+

+

δ(t )

+

δ(t )

2

2

a sech2 (k (x − λ(t )))

4 b k2 − c1 + c

,

δ(t ) a (25)

a

a

.

These solutions were recently derived by using Hirota bilinear method in [34]. 2.2. Self-adjoint and quasi self-adjoint equations The following definitions of adjoint equations and self-adjoint equations are applicable to any system of linear and nonlinear differential equations, where the number of equations is equal to the number of dependent variables (see [14]), and contain the usual definitions for linear equations as a particular case. Since we will deal in our paper with scalar equations, we will formulate these definitions in the case of one dependent variable only. Definition. Consider an sth-order partial differential equation F (x, u, u(1) , . . . , u(s) ) = 0

(26)

with independent variables x = (x , . . . , x ) and a dependent variable u, where u(1) = {ui }, u(2) = {uij }, . . . , denote the sets of the partial derivatives of the first, second, etc. orders, ui = ∂ u/∂ xi , uij = ∂ 2 u/∂ xi ∂ xj . The adjoint equation to (26) is 1

n

F ∗ (x, u, v, u(1) , v(1) , . . . , u(s) , v(s) ) = 0,

(27)

with F ∗ (x, u, v, u(1) , v(1) , . . . , u(s) , v(s) ) =

δ(v F ) , δu

(28)

where ∞  δ ∂ ∂ = + (−1)s Di1 · · · Dis δu ∂ u s=1 ∂ ui1 ···is

(29)

denotes the variational derivatives (the Euler–Lagrange operator), and v is a new dependent variable. Here Di =

∂ ∂ ∂ + ui + uij + ··· ∂ xi ∂u ∂ uj

are the total differentiations. Definition. Eq. (26) is said to be self-adjoint if the equation obtained from the adjoint Eq. (27) by the substitution

v = u:

F ∗ (x, u, u, u(1) , u(1) , . . . , u(s) , u(s) ) = 0, is identical to the original Eq. (26). In other words, if F ∗ (x, u, u(1) , u(1) , . . . , u(s) , u(s) ) = λ(x, u, u(1) , . . .) F (x, u, u(1) , . . . , u(s) ).

(30)

Definition. Eq. (26) is said to be quasi self-adjoint if the equation obtained from the adjoint Eq. (27) by the substitution

v = h(u) with a certain function h(u) such that h′ (u) ̸= 0. F ∗ (x, u, u, u(1) , u(1) , . . . , u(s) , u(s) ) = 0, is identical to the original Eq. (26).

M.L. Gandarias, M.S. Bruzón / Nonlinear Analysis: Real World Applications 13 (2012) 2692–2700

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2.3. Weak self-adjoint equations Definition. Eq. (26) is said to be weak self-adjoint if the equation obtained from the adjoint Eq. (27) by the substitution v = h(x, t , u): F ∗ (x, u, u, u(1) , u(1) , . . . , u(s) , u(s) ) = 0, is identical to the original Eq. (26). 2.4. General theorem on conservation laws We use the following theorem on conservation laws proved in [14]. Theorem. Any Lie point, Lie–Bäcklund or nonlocal symmetry X = ξ i (x, u, u(1) , . . .)

∂ ∂ + η(x, u, u(1) , . . .) i ∂x ∂u

(31)

of Eqs. (26) provides a conservation law Di (C i ) = 0 for the simultaneous system (26), (27). The conserved vector is given by

∂L − Dj ∂ ui



∂L − Dk + Dj (W ) ∂ uij





C i = ξ iL + W



∂L ∂ uij



∂L ∂ uijk

 + Dj Dk



∂L ∂ uijk



 − ···

   ∂L + · · · + Dj Dk (W ) − ··· + ···, ∂ uijk

(32)

where W and L are defined as follows: W = η − ξ j uj ,

L = v F x, u, u(1) , . . . , u(s) .





(33)

The proof is based on the following operator identity (N.H. Ibragimov, 1979): X + Di (ξ i ) = W

δ + Di N i , δu

(34)

where X is the operator (31) taken in the prolonged form: X = ξi

∂ ∂ ∂ ∂ +η + ζi + ζi1 i2 + ···, ∂ xi ∂u ∂ ui ∂ ui1 i2

ζi = Di (η) − uj Di (ξ j ),

ζi1 i2 = Di2 (ζi1 ) − uji1 Di2 (ξ j ), . . . .

For the expression of the operator N i and a discussion of the identity (34) in the general case of several dependent variables we refer the reader to [35, Section 8.4.4]. Let us single out some self-adjoint equations from the equations of the form (1). The result is given by the following statement. Theorem. Eq. (1) is quasi self-adjoint if and only if f (t ) = 0. Proof. Eq. (28) yields ut + auux + buxxx + cux − f (t ) = 0 F∗ =

δ [v(ut + auux + buxxx + cux − f (t ))] = −b vx x x − a u vx − c vx − vt . δu

(35) (36)

Setting v = h(u) in (36) we have F ∗ = −b hu ux x x − 3 b hu u ux ux x − b hu u u (ux )3 − a hu u ux − c hu ux − hu ut using Eq. (30) yields:

− b ux x x λ − a u ux λ − c ux λ − ut λ + f λ − b hu ux x x − 3 b hu u ux ux x − b hu u u (ux )3 − a hu u ux − c hu ux − hu ut = 0.

(37)

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From (37) it follows that the following conditions must be satisfied:

λ + hu = 0 −3b huu = 0 −bhu u u = 0 f hu = 0. Therefore (37) shows that the latter equation is satisfied if and only if f (t ) = 0 whence h(u) = k1 u + k2 ,

f (t ) = 0. 

2.5. The class of weak self-adjoint equations Let us single out some weak self-adjoint equations from the equations of the form (1). The result is given by the following statement. Theorem. Eq. (1) is weak self-adjoint for any arbitrary function f (t ). Proof. Eq. (28) yields F∗ =

δ [v(ut + auux + buxxx + cux − f (t ))] = −b vx x x − a u vx − c vx − vt . δu

(38)

Hence the adjoint equation to (2) is

− b vx x x − a u vx − c vx − vt = 0.

(39)

Setting v = h(u, t ) and substituting

v = h(t , u),

v t = h u ut + h t ,

vx = hu ux ,

vxx = hu uxx + huu u2x

into the adjoint Eq. (39) we have

− b hu ux x x − 3 b hu u ux ux x − b hu u u (ux )3 − a hu u ux − c hu ux − hu ut − ht = 0.

(40)

Using Eq. (30) yields:

− b ux x x λ − a u ux λ − c ux λ − ut λ + f λ − b hu ux x x − 3 b hu u ux ux x − b hu u u (ux )3 − a hu u ux − c hu ux − hu ut − ht = 0.

(41)

From (41) it follows that the following conditions must be satisfied:

λ + hu = 0 3b huu = 0 bhuuu = 0 f (t )hu + ht = 0. Therefore (42) shows that the latter equation is satisfied for any arbitrary f (t ) with h(u, t ) = u −



f (t )dt .

(42)

Namely, the adjoint Eq. (39) becomes equivalent to the original Eq. (1) upon the substitution (42) where f (t ) : arbitrary.  2.6. Conservation laws for a subclass of weak self-adjoint of forced KdV equations Let us apply Section 2.4 to the weak self-adjoint equation: ut + auux + buxxx + cux − f (t ) = 0.

(43)

In this case we have





L = ut + auux + buxxx + cux − f (t ) v. We will write generators of point transformation group admitted by Eq. (43) in the form X = ξ1

∂ ∂ ∂ + ξ2 + η ∂t ∂x ∂u

(44)

M.L. Gandarias, M.S. Bruzón / Nonlinear Analysis: Real World Applications 13 (2012) 2692–2700

2699

by setting t = x1 , x = x2 . The conservation law will be written Dt (C 1 ) + Dx (C 2 ) = 0.

(45)

1 Let us find the conservation law provided by the following general symmetry of Eq. (2) with f (t ) : arbitrary:

      ∂ c 3k1 t ∂ δt ∂ + k2 − + k1 u + + . (46) 2 ∂x 2 ∂t a a ∂u        3k t In this case we have W = k1 (u + ac ) + δat − δ − k1 2x ux − k2 − 21 ut , h(t , u) = u − g (t ), g (t ) = f (t )dt and 

X = −k1

x

+δ

Eq. (32) yield the conservation law (45) with g k1 u c k1 u δt u 3 f g k1 t c g k1 δt g − f k2 u − + + − + f g k2 − − + Dx (B1 ) 2 2 a a 2 a a 3 b k1 u ux x 3 b f k1 t ux x b g k 1 ux x b c k 1 ux x b δ t ux x = − + + − b f k 2 ux x − + + 2 2 2 2 a a 3 b k1 (ux )2 a k 1 u3 3 a f k1 t u2 a f k 2 u2 a g k 1 u2 5 c k1 u2 δ t u2

C1 = C2

3 k1 u2

− +

B1 =

+

3 f k1 t u

4 f g k1 x

4 3 c f k1 t u 2

k1 u2 x 4

+

+

2

+ b g k 2 ux x + −

4

+

− c f k2 u −

g k1 u x

3 c k1 t u2

2

2 3 b k 1 t u ux x

−

3 b k1 t (ux )

+

4 c g k1 u

4 a g k 2 u2 2

2

2

−

+

−

2

2

+

c k1 u a

+

−

c δt u

+ b k2 u ux x −

b k2 (ux )2

c k2 u2 2

2

−

−

δ u2 2

−

4 c g k1 2

−

a a 3 b g k1 t ux x

a k1 t u3

2

+

2 3 c g k1 t u 2

a k 2 u3 3

+

+

4

+δf g −

−

c δt g a

2

− Dt (B1 )

3 a g k1 t u2 4

+ c g k2 u − δ g u.

We simplify the conserved vector by transferring the terms of the form Dx (. . .) from C 1 to C 2 and obtain g k1 u c k1 u δt u 3 f g k1 t c g k1 δt g − f k2 u − + + − + f g k2 − − 2 2 a a 2 a a 3 b k1 u ux x 3 b f k1 t ux x b g k 1 ux x b c k 1 ux x b δ t ux x = − + + − b f k 2 ux x − + + 2 2 2 2 a a 2 3 2 2 2 2 a k1 u 3 a f k1 t u a f k2 u a g k1 u 5 c k1 u δ t u2 3 b k1 (ux )

C1 = C2

3 k1 u2

+

3 f k1 t u

4 f g k1 x

− +

where g (t ) =



4 3 c f k1 t u 2

+

2

+

− c f k2 u −

4 c g k1 u 2

−

2

+

c k1 u a

2

+

−

c δt u a

4 c g k1 2

−

a

+

4

+δf g −

+

c δt g

2

a

f (t )dt and δ(t ), f (t ), k1 and k2 must satisfy condition (9).

3. Conclusions The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov in [17]. In [27] one of the present authors has generalized the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. In [36] we found a class of weak self-adjoint forced KdV equations. In this paper we have derived the classical Lie symmetries admitted by the forced KdV equation. By using a general theorem on conservation laws proved in [14], the Lie symmetries and the new concept of weak self-adjointness [27] we found some conservation laws for some of these partial differential equations. Acknowledgements We warmly thank the referees for their very interesting suggestions. The support of DGICYT project MTM2009-11875, and Junta de Andalucía group FQM-201 is gratefully acknowledged. References [1] E. Pelinovsky, in: A.C. Yalciner, et al. (Eds.), Submarine Landslides and Tsunamis, Kluwer Academic Publisher, Netherlands, 2003. [2] S.R. Clarke, R. Grimshaw, Resonantly generated internal waves in a contraction, Journal of Fluid Mechanics 274 (1994) 139–161.

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