The Lagrangian, Self-Adjointness, and Conserved Quantities for a Generalized Regularized Long-Wave Equation

and Applied Analysis 3 3. Self-Adjointness and Conservation Laws In this section, we will investigate the self-adjointness and conservation laws of (1) by Ibragimov’s theorem. 3.1. Self-Adjointness for (1) and (2). Let H = u t + u x + au m u x + bu xxt ; (20) then we have the following formal Lagrangian for (1): L 1 = w (u t + u x + au m u x + bu xxt ) , (21) where w = w(x, t) is a new function. Computing the variational derivative of this formal Lagrangian, we obtain the following adjoint equation of (1): H ∗ = −w t − w x − aw x u m − bw xxt = 0. (22) Assume thatH| w=φ(t,x,u) = λH, for a certain function λ. Then it is easy to obtain that φ = c 1 u + c 2 and λ = −φ u = −c 1 , where c 1 , c 2 are two arbitrary constants.This implies that (1) is quasiself-adjoint; especially, if c 1 = 1 and c 2 = 0; that is, φ = u, then (1) is strictly self-adjoint (see [16] for the definitions). Thus, we have demonstrated the following statement. Theorem3. Equation (1) is quasiself-adjoint with the substitution w = c 1 u + c 2 , where c 1 ̸ = 0, c 2 are two arbitrary constants. Especially, (1) is strictly self-adjoint with the substitutionw = u. Next, we discuss the self-adjointness of (2), and then by the nonlinear self-adjointness and Ibragimov’s theorem on conservation laws we construct some conserved quantities of (1). Similar to the above, we set E = V xt + V xx + aVm x V xx + bV xxxt ; (23) then the formal Lagrangian for (2) is L 2 = w (V xt + V xx + aVm x V xx + bV xxxt ) (24) and the adjoint equation of (1) is E ∗ = w xt + w xx + aw xx Vm x + amw x Vm−1 x V xx + bw xxxt = 0. (25) According to [16], (2) will be nonlinearly self-adjoint if there exists a differential substitution w = h (x, t, V, V x , V t , V xx , . . . , ) , h ̸ = 0, (26) involving a finite number of partial derivatives of V with respect to x and t, such that the equation E ∗󵄨 󵄨 󵄨 󵄨w=h = μ 0 E + μ 1 D x E + μ 2 D t E + μ 3 D 2 x E + ⋅ ⋅ ⋅ (27) holds identically in the variables t, x, u, u x , u t , . . ., where μ 0 , μ 1 , . . . are undetermined variable coefficients different from ∞ on the solutions of (2). The highest order of derivatives involved in u is called the order of the differential substitution (26). The calculation provides the following result. Theorem 4. Equation (2) is nonlinearly self-adjoint. The minimal order of the differential substitution (26) satisfying (27) is equal to one and is given by the function h = c 1 V x + c 2 V t + g(t), where c 1 and c 2 are two constants and g(t) is a differentiable function of t. 3.2. Conservation Laws of (1). Now, we study the conservation laws of (1). We will first construct the conservation laws of (2) and then reduce them to the ones of (1). From the classical Lie group theory [17], we assume that a Lie point symmetry of (2) is a vector field


Introduction
The generalized regularized long-wave (GRLW) equation was first put forward as a model for small-amplitude long waves on the surface of water in a channel by Peregrine [1,2], where is a positive integer, and and are two constants. It is an alternative description of nonlinear dispersive waves to the Korteweg de Vries equation [1] and has been used to describe phenomena with weak nonlinearity and dispersion waves, including nonlinear transverse waves in shallow water, ion-acoustic and magnetohydrodynamic waves in plasma and phonon packets in nonlinear crystals, pressure waves in liquid-gas bubble mixture, rotating flow down a tube, and lossless propagation of shallow water waves; see [3] and the references therein. Equation (1) has been studied extensively in the past both analytically and numerically by means of, for example, the Adomian decomposition method, the Fourier method, the finite difference method, the finite element method, the variational iteration method, the mesh-free method, the cubic B-spline collocation procedure, the sine-cosine method, and the double reduction method [4][5][6][7][8][9][10][11], and so forth. A few of the conserved quantities by using the 1soliton solution and solitary-wave solution of the ( , ) equation were calculated in [12]. In [11], the authors employed the Lie symmetry method and double reduction theory to obtain some exact solutions and derived the conservation laws for (1) via the so-called "partial Noether approach" after increasing its order. Noting that the GRLW equation is a third-order partial differential equation and it is difficult to obtain a variational formulation/Lagrangian for (1) by a classical approach, the authors converted the equation to the following fourth-order equation by assuming new dependent variable V to be the derivative of original dependent variable by setting = V as follows: The authors claimed that the conserved vectors constructed by partial Noether's theorem failed to satisfy the divergence relation and need to be adjusted to satisfy the divergence relationship. For the "partial Noether approach, " in [13] Sarlet had given some negative comments (see Section 4 in [13]). So we will give some conservation laws of (1) by other methods.
In this paper, we investigate the Lagrangian and the selfadjointness and construct some new conservation laws for the generalized regularized long-wave equation by Ibragimov's theorem [14]. We should point out that the exact Lagrangian of (1) obtained in Section 2 is a nonlocal Lagrangian with an auxiliary function Ψ. In Section 3, we show that (1) is strictly self-adjoint, (2) is nonlinearly self-adjoint, and the minimal order of the differential substitution of (2) is equal to one.

The Auxiliary Lagrangian
In this section, we will study the Lagrangian for the generalized regularized long-wave equation (1). First, we briefly present some notations to be used in what follows. Let = ( 1 , 2 , . . . , ) be independent variables, let = ( ) be dependent variables, let be the total differentiation operator, and let denote the Euler-Lagrange operator. It is well known that a partial differential equation with odd order does not often admit a Lagrangian. However, noting the structure of (1), we can rewrite the equation as thus we find an auxiliary Lagrangian of (1) by the semiinverse method [15]. Consequently some conserved quantities of (1) can be derived from Noether's theorem. According to (5), we introduce an auxiliary function Ψ defined as so that (5) is automatically satisfied. By the semi-inverse method [15], we construct a trial Lagrangian in the form where is an unknown function to be determined. The Euler-Lagrange equations of (8) are Substituting (6) and (7) into the above equations, we obtain that Thus, we can take as which implies that the Lagrangian of (1) reads which is subjected to (6). Therefore, we demonstrate the following result.

Self-Adjointness and Conservation Laws
In this section, we will investigate the self-adjointness and conservation laws of (1) by Ibragimov's theorem. (1) and (2). Let

Theorem 3. Equation
Next, we discuss the self-adjointness of (2), and then by the nonlinear self-adjointness and Ibragimov's theorem on conservation laws we construct some conserved quantities of (1).
Similar to the above, we set then the formal Lagrangian for (2) is and the adjoint equation of (1) is According to [16], (2) will be nonlinearly self-adjoint if there exists a differential substitution involving a finite number of partial derivatives of V with respect to and , such that the equation * holds identically in the variables , , , , , . . ., where 0 , 1 , . . . are undetermined variable coefficients different from ∞ on the solutions of (2). The highest order of derivatives involved in is called the order of the differential substitution (26). The calculation provides the following result. (2) is nonlinearly self-adjoint. The minimal order of the differential substitution (26) satisfying (27) is equal to one and is given by the function ℎ = 1 V + (1). Now, we study the conservation laws of (1). We will first construct the conservation laws of (2) and then reduce them to the ones of (1). From the classical Lie group theory [17], we assume that a Lie point symmetry of (2) is a vector field

Conservation Laws of
on R + × R × R such that (4) = 0 when = 0, where is given by (23). Taking into account (2), the operator (4) is given as follows: where The condition (4) | =0 = 0 yields determining equations. Solving these determining equations, we can obtain the symmetries of (5) as follows: with ( ) as an arbitrary function. Ibragimov's theorem on conservation laws yields the following conserved quantities of (2) for a general Lie symmetry = ( / ) + ( / ) + ( / V). Consider the following: where = − V − V is the Lie characteristic function and is given by the differential substitution = 1 V + 2 V + ( ). Let us construct the conserved vector corresponding to the space translation group with the generator In this case = −V . Taking account of the differential substitution = 1 V + 2 V + ( ) and V = , from (32), we obtain the conserved quantities of (2), which can be transformed to the ones of (1) as follows: which can be reduced to the form 1 = 1 2 2 (2 + + ) , Here we have used (1).
By the procedure analogous to that used above, we obtain the conserved quantities of (2) corresponding to the generators 2 and 3 , which can be reduced by the differential substitutions = 1 V + 2 V + ( ) and V = to the conserved quantities of (1) as follows: Here we have used (1).

Remark 5. In fact, (2) admits a Lagrangian
and has the following Noether symmetry generators: where ( ) is a differentiable function in . Noether's theorem on conservation laws [18] can yield the conserved quantities of (2), which are the same as the ones in [11]. Those were constructed by the so-called "partial Noether approach. "

Conclusions
In this paper, we investigate the Lagrangians and selfadjointnesses of the GRLW equation (1) and its transformed equation (2). We show that (1) has a nonlocal Lagrangian with an auxiliary function and is strictly self-adjoint; its transformed equation is nonlinearly self-adjoint and the minimal order of the differential substitution is equal to one. Then by Ibragimov's theorem on conservation laws we obtain some conserved qualities of the generalized regularized longwave equation.