1. Introduction
Nonlinear science runs through physics, mathematics, astronomy, and other scientific fields and is a bridge connecting mathematics and other disciplines [1–4]. It is another major development in natural science after quantum mechanics and relativity. Nonlinear differential equations transform physical phenomena into mathematical language, so it is of practical significance to study nonlinear equations. The concept of conservation law plays an important role in the study of differential equations and many applications. The mathematical concept of conservation law is derived from familiar representations of physical laws such as energy [5], momentum [6], and angular momentum [7]. In the jet problem [8], the conserved quantity plays an important role in the solution process, which is used to determine the unknown index in the similar solution that cannot be obtained under homogeneous boundary conditions. In addition, we can see that many results, such as the existence, uniqueness, and stability of rational solutions, are determined by conservation law [9–11].
The well-known method of constructing conservation laws is the Noether theorem [12], which will produce all the conservation laws of the partial differential equation system with a Lagrangian. Specifically, every nontrivial conservation law is generated by some variational symmetries, but not necessarily variational point symmetries. The application of the Noether theorem depends on the appropriate Lagrangian formulation. In fact, many equations do not have it, so we look for additional methods to find the conservation laws. The first method is that Anco and Bluman proposed an algorithm to obtain the field equations for any system of local conservation laws in [13, 14], regardless of whether the system has a Lagrangian formulation. The algorithm is based on the conservation law multiplier to obtain the conservation law of the equation system by using the formula. The second method is that Ibragimov introduced the concept of adjoint equations of differential equations and took the new differential equations composed of given equations and their adjoint equations as the research object [15]. Using the adjoint variational principle [16] proposed by Atherton et al., the Lagrangian formulation of the new differential equations was obtained, and the theorem of finding conservation laws was given. Furthermore, there are many ways to construct conservation laws, such as the variational approach [17], the partial Noether approach [18], and so on [19–21].
The outline of this paper is as follows: In section 2 , the method of directly constructing conservation laws is reviewed. Section 3 applies the method of directly constructing conservation laws to the Gardner equation, the Landau–Ginzburg–Higgs (LGH) equation, and the Hirota–Satsuma equation; the conservation laws are obtained. In section 4 , we discuss the nonlinear self-adjointness of the above three equations and obtain new conserved quantities by using the conservation theorem proposed by Ibragimov.
2. Direct construction method for conservation laws of partial differential equations
2.1. Gardner equation
We first consider the Gardner equation2.2 ) is the determining equation for adjoint symmetries ${G}^{{\prime} * }(\omega ){| }_{\epsilon }=0$, where ε will denote the solution space of the equation (2.1 ). Therefore, the adjoint of the equation (2.2 ) is given by2.1 ). All nontrivial conservation laws' conserved densities in this form can be constructed from multipliers Λ on the Gardner equation, analogous to integrating factors, where Λ depends only on t, x, u, ux and uxx. In particular, by moving off the Gardner equation solution space, we have
$\begin{eqnarray}G\equiv {u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}=0.\end{eqnarray}$
The equation is widely used in various branches of physics, such as plasma physics, fluid physics, and quantum field theory [22–24]. It also describes various wave phenomena in plasma and solid [33, 34]. Its symmetries with infinitesimal generator Xu = η [25, 26] satisfy the determining equation $\begin{eqnarray}\begin{array}{l}0={D}_{t}\eta +6{u}_{x}\eta +6{{uD}}_{x}\eta -12{{uu}}_{x}\eta \\ \quad -6{u}^{2}{D}_{x}\eta +{D}_{x}^{3}\eta \quad {\rm{when}}\quad G=0,\end{array}\end{eqnarray}$
where ${D}_{t}={\partial }_{t}+{u}_{t}{\partial }_{u}+{u}_{{tx}}{\partial }_{{u}_{x}}+{u}_{{tt}}{\partial }_{{u}_{t}}+\cdots $ and ${D}_{x}\,={\partial }_{x}+{u}_{x}{\partial }_{u}+{u}_{{xx}}{\partial }_{{u}_{x}}+{u}_{{tx}}{\partial }_{{u}_{t}}+\cdots $ are total derivative operators with respect to t and x. The adjoint of the symmetry equation ( $\begin{eqnarray}\begin{array}{l}0=-{D}_{t}\omega -6{{uD}}_{x}\omega +6{u}^{2}{D}_{x}\omega -{D}_{x}^{3}\omega \\ \quad {\rm{when}}\quad G=0,\end{array}\end{eqnarray}$
which is the determining equation for the adjoint symmetries ω of the Gardner equation. Next, we consider local conservation laws DtΦt + DxΦx = 0 on all solutions u(t, x) of the equation ( $\begin{eqnarray}\begin{array}{l}{D}_{t}{{\rm{\Phi }}}^{t}+{D}_{x}{{\rm{\Phi }}}^{x}=({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}){{\rm{\Lambda }}}_{0}\\ \,+{D}_{x}({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}){{\rm{\Lambda }}}_{1}+\cdots ,\end{array}\end{eqnarray}$
for some expression Λ0, Λ1, … with no dependence on ut or differential consequence. This yields the multiplier $\begin{eqnarray}\begin{array}{l}{D}_{t}{{\rm{\Phi }}}^{t}+{D}_{x}({{\rm{\Phi }}}^{x}-{\rm{\Gamma }})\\ \quad =({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}){\rm{\Lambda }},\\ \quad {\rm{\Lambda }}={{\rm{\Lambda }}}_{0}-{D}_{x}{{\rm{\Lambda }}}_{1}+\cdots ,\end{array}\end{eqnarray}$
where Γ = 0 when u is restricted to be a Gardner equation solution.The definition for multipliers Λ(t, x, u, ux, uxx) is that (ut + 6uux − 6u2ux + uxxx)Λ must be a divergence expression for all functions u(t, x). The determining condition is expressed by
$\begin{eqnarray}\begin{array}{rcl}0 & = & {E}_{u}(G{\rm{\Lambda }})={E}_{u}(({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}){\rm{\Lambda }})\\ & = & -{D}_{t}{\rm{\Lambda }}-6{{uD}}_{x}{\rm{\Lambda }}+6{u}^{2}{D}_{x}{\rm{\Lambda }}-{D}_{x}^{3}{\rm{\Lambda }}\\ & & +G{{\rm{\Lambda }}}_{u}-{D}_{x}(G{{\rm{\Lambda }}}_{{u}_{x}})+{D}_{x}^{2}(G{{\rm{\Lambda }}}_{{u}_{{xx}}}),\end{array}\end{eqnarray}$
where ${E}_{u}={\partial }_{u}-{D}_{t}{\partial }_{{u}_{t}}-{D}_{x}{\partial }_{{u}_{x}}+{D}_{t}{D}_{x}{\partial }_{{u}_{{tx}}}+{D}_{x}^{2}{\partial }_{{u}_{{xx}}}+\cdots $ is the standard Euler operator which annihilates divergence expressions.By calculation, we can get the general form of the multiplier of the Gardner equation as follows (see appendix A ):
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Lambda }} & = & {c}_{1}\left[(2{u}^{3}-3{u}^{2}+u-{u}_{{xx}})t-\displaystyle \frac{x}{6}+\displaystyle \frac{1}{3}{xu}\right]\\ & & +{c}_{2}[-2{u}^{3}+3{u}^{2}+{u}_{{xx}}]+{c}_{3}u+{c}_{4},\end{array}\end{eqnarray}$
where c1, c2, c3 and c4 are arbitrary constants. So for the Gardner equation, here are some multipliers $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Lambda }}}_{1} & = & (2{u}^{3}-3{u}^{2}+u-{u}_{{xx}})t-\displaystyle \frac{x}{6}+\displaystyle \frac{1}{3}{xu},\\ {{\rm{\Lambda }}}_{2} & = & -2{u}^{3}+3{u}^{2}+{u}_{{xx}},\\ {{\rm{\Lambda }}}_{3} & = & u,\quad {{\rm{\Lambda }}}_{4}=1.\end{array}\end{eqnarray}$
For the multiplier ${{\rm{\Lambda }}}_{1}=(2{u}^{3}-3{u}^{2}+u-{u}_{{xx}})t-\tfrac{x}{6}+\tfrac{1}{3}{xu}$, according to the definition of multiplier ${{\rm{\Lambda }}}_{1}G={D}_{t}({{\rm{\Phi }}}_{1}^{t})+{D}_{x}({{\rm{\Phi }}}_{1}^{x})$, we have the following conserved vector: $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{1}^{t} & = & t\left(\displaystyle \frac{1}{2}{u}^{4}-{u}^{3}+\displaystyle \frac{1}{2}{u}^{2}-{u}_{t}{u}_{x}+\displaystyle \frac{1}{2}{u}_{x}^{2}\right)\\ & & -\displaystyle \frac{x}{6}u+\displaystyle \frac{x}{6}{u}^{2},\\ {{\rm{\Phi }}}_{1}^{x} & = & t\left(-2{u}^{6}+6{u}^{5}-6{u}^{4}+2{u}^{3}+2{u}^{3}{u}_{{xx}}\right.\\ & & -\left.3{u}^{2}{u}_{{xx}}+{{uu}}_{{xx}}-\displaystyle \frac{1}{2}{u}_{x}^{2}\right)\\ & & -\displaystyle \frac{x}{2}{u}^{4}+{{xu}}^{3}-\displaystyle \frac{x}{2}{u}^{2}-\displaystyle \frac{x}{6}{u}_{{xx}}\\ & & +\displaystyle \frac{1}{6}{u}_{x}+\displaystyle \frac{x}{3}{{xuu}}_{{xx}}-\displaystyle \frac{x}{2}{u}_{x}^{2}-{{uu}}_{x}.\end{array}\end{eqnarray}$
Similarly, for the multiplier Λ2 = − 2u3 + 3u2 + uxx, Λ3 = u and Λ4 = 1, we have $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{2}^{t} & = & -\displaystyle \frac{1}{2}{u}^{4}+{u}^{3}-\displaystyle \frac{1}{2}{u}_{x}^{2},\\ {{\rm{\Phi }}}_{2}^{x} & = & 2{u}^{6}-6{u}^{5}+\displaystyle \frac{9}{2}{u}^{4}+\displaystyle \frac{1}{2}{u}_{{xx}}^{2}\\ & & +3{u}^{2}{u}_{{xx}}-2{u}^{3}{u}_{{xx}}+{u}_{x}{u}_{t},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{3}^{t} & = & \displaystyle \frac{1}{2}{u}^{2},\\ {{\rm{\Phi }}}_{3}^{x} & = & -\displaystyle \frac{3}{2}{u}^{4}+2{u}^{3}-\displaystyle \frac{1}{2}{u}_{x}^{2}+{{uu}}_{{xx}},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{4}^{t} & = & u,\\ {{\rm{\Phi }}}_{4}^{x} & = & 3{u}^{2}-2{u}^{3}+{u}_{{xx}}.\end{array}\end{eqnarray}$
2.2. Landau–Ginzburg–Higgs equation
Consider the LGH equation
$\begin{eqnarray}G={u}_{{tt}}-{u}_{{xx}}+{g}^{2}{u}^{N}=0,\end{eqnarray}$
where N is an integer and g denotes an arbitrary constant. The LGH equation was introduced by Lev Devidovich Landau and Vitaly Lazarevich Ginzburg in [27], and has a very wide range of applications in radially inhomogeneous plasmas with a constant phase relation of ion-cyclotron waves [28]. It proves the superconductivity and unidirectional wave propagation in nonlinear media [29].Symmetries of the LGH equation with infinitesimal generator Xu = η [25, 26] satisfy the determining equation2.14 ) is self-adjointed. For any local conservation laws2.18 ), we can obtain the general form of the multiplier of the Hirota–Satsuma equation as follows (see appendix B ) :
$\begin{eqnarray}\begin{array}{c}{D}_{t}^{2}\eta -{D}_{x}^{2}\eta +{{Ng}}^{2}{u}^{N-1}\eta =0\\ \,{\rm{when}}\,G=0.\end{array}\end{eqnarray}$
Obviously, equation ( $\begin{eqnarray}{D}_{t}{{\rm{\Phi }}}^{t}+{D}_{x}{{\rm{\Phi }}}^{x}=0,\end{eqnarray}$
by moving off the LGH equation solution space, we have $\begin{eqnarray}\begin{array}{l}{D}_{t}{{\rm{\Phi }}}^{t}+{D}_{x}{{\rm{\Phi }}}^{x}=({u}_{{tt}}-{u}_{{xx}}+{g}^{2}{u}^{N}){{\rm{\Lambda }}}_{0}\\ \,+{D}_{x}({u}_{{tt}}-{u}_{{xx}}+{g}^{2}{u}^{N}){{\rm{\Lambda }}}_{1}+\cdots ,\end{array}\end{eqnarray}$
for some expressions Λ0, Λ1, … with no dependence on utt and differential consequences. This yields the multiplier $\begin{eqnarray}\begin{array}{l}{D}_{t}{{\rm{\Phi }}}^{t}+{D}_{x}({{\rm{\Phi }}}^{x}-{\rm{\Gamma }})=({u}_{{tt}}-{u}_{{xx}}+{g}^{2}{u}^{N}),\\ \quad {\rm{\Lambda }}={{\rm{\Lambda }}}_{0}-{D}_{x}{{\rm{\Lambda }}}_{1}+\cdots ,\end{array}\end{eqnarray}$
where Γ = 0 when u is restricted to be a LGH equation solution. Multipliers Λ are defined by the condition that (utt − uxx + g2uN)Λ is a divergence expression for all functions u(t, x). We restrict attention to Λ of first-order, depending on t, x, u, ut, ux. This leads to the necessary and sufficient determining condition $\begin{eqnarray}\begin{array}{rcl}0 & = & {E}_{u}(({u}_{{tt}}-{u}_{{xx}}-{m}^{2}u+{g}^{2}{u}^{3}){\rm{\Lambda }})\\ & = & {D}_{t}^{2}{\rm{\Lambda }}-{D}_{x}^{2}{\rm{\Lambda }}+{{Ng}}^{2}{u}^{N-1}{\rm{\Lambda }}\\ & & +({u}_{{tt}}-{u}_{{xx}}+{g}^{2}{u}^{N}){{\rm{\Lambda }}}_{u}\\ & & -{D}_{x}({{\rm{\Lambda }}}_{{u}_{x}}({u}_{{tt}}-{u}_{{xx}}+{g}^{2}{u}^{N}))\\ & & -{D}_{t}({{\rm{\Lambda }}}_{{u}_{t}}({u}_{{tt}}-{u}_{{xx}}+{g}^{2}{u}^{N})).\end{array}\end{eqnarray}$
By solving equations ( $\begin{eqnarray}{\rm{\Lambda }}={a}_{1}({u}_{x}-{u}_{t})+{a}_{2}({u}_{t}+{u}_{x})+{c}_{1}{u}_{t},\end{eqnarray}$
where a1, a2 and c1 are arbitrary constants. For the LGH equation, here are some multipliers $\begin{eqnarray}{{\rm{\Lambda }}}_{1}={u}_{x}-{u}_{t},\quad {{\rm{\Lambda }}}_{2}={u}_{x}+{u}_{t},\quad {{\rm{\Lambda }}}_{3}={u}_{t}.\end{eqnarray}$
For the multiplier Λ1 = ut + ux, according to the definition of multiplier ${{\rm{\Lambda }}}_{1}G={D}_{t}({{\rm{\Phi }}}_{1}^{t})+{D}_{x}({{\rm{\Phi }}}_{1}^{x})$, we have the following conserved vector: $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{1}^{t} & = & {u}_{x}{u}_{t}-\displaystyle \frac{1}{2}{u}_{t}^{2}-\displaystyle \frac{1}{2}{u}_{x}^{2}-\displaystyle \frac{{g}^{2}}{N+1}{u}^{N+1},\\ {{\rm{\Phi }}}_{1}^{x} & = & {u}_{x}{u}_{t}-\displaystyle \frac{1}{2}{u}_{t}^{2}-\displaystyle \frac{1}{2}{u}_{x}^{2}+\displaystyle \frac{{g}^{2}}{N+1}{u}^{N+1}.\end{array}\end{eqnarray}$
Similarly, for the multiplier Λ2 = ux and Λ3 = ut, we have $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{2}^{t} & = & {u}_{x}{u}_{t}+\displaystyle \frac{1}{2}{u}_{t}^{2}+\displaystyle \frac{1}{2}{u}_{x}^{2}+\displaystyle \frac{{g}^{2}}{N+1}{u}^{N+1},\\ {{\rm{\Phi }}}_{2}^{x} & = & -{u}_{x}{u}_{t}-\displaystyle \frac{1}{2}{u}_{t}^{2}-\displaystyle \frac{1}{2}{u}_{x}^{2}+\displaystyle \frac{{g}^{2}}{N+1}{u}^{N+1},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{3}^{t} & = & \displaystyle \frac{1}{2}{u}_{t}^{2}+\displaystyle \frac{1}{2}{u}_{x}^{2}+\displaystyle \frac{{g}^{2}}{N+1}{u}^{N+1},\\ {{\rm{\Phi }}}_{3}^{x} & = & -{u}_{t}{u}_{x}.\end{array}\end{eqnarray}$
2.3. Hirota–Satsuma equation
Consider the Hirota–Satsuma (HS) equation
$\begin{eqnarray}\begin{array}{c}\left\{\begin{array}{c}{G}^{1}={u}_{t}-6{{puu}}_{x}+{{rvv}}_{x}-{{pu}}_{{xxx}}=0,\\ {G}^{2}={v}_{t}+3{{puv}}_{x}+{{pv}}_{{xxx}}=0,\end{array}\right.\end{array}\end{eqnarray}$
where p and r are arbitrary constants. The HS equation describes the interaction of two long waves with different dispersion relations. It is related to most types of weak dispersive long waves, internal waves, acoustic waves, and planetary waves in geophysical hydrodynamics [30–32].Next, we consider the linearized equation of equations (2.24 )2.25 ) is given by2.29 ), we can obtain the general form of the multiplier of the HS equation as follows (see appendix C ):2.24 ).
$\begin{eqnarray}\begin{array}{rcl}{\left({{\mathscr{L}}}_{g}\right)}_{1}^{1}{V}^{1} & = & -6{{pu}}_{x}{V}^{1}-6{{puD}}_{x}{V}^{1}-{{pD}}_{x}^{3}{V}^{1},\\ {\left({{\mathscr{L}}}_{g}\right)}_{1}^{2}{V}^{1} & = & 3{{pv}}_{x}{V}^{1},\\ {\left({{\mathscr{L}}}_{g}\right)}_{2}^{1}{V}^{2} & = & 2{{rv}}_{x}{V}^{2}+{{rvD}}_{x}{V}^{2},\\ {\left({{\mathscr{L}}}_{g}\right)}_{2}^{2}{V}^{2} & = & 3{{puD}}_{x}{V}^{2}+{{pD}}_{x}^{3}{V}^{2}.\end{array}\end{eqnarray}$
And the adjoint system of system ( $\begin{eqnarray}\begin{array}{rcl}{\left({{\mathscr{L}}}_{g}^{* }\right)}_{1}^{1}{W}_{1} & = & 6{{puD}}_{x}{W}_{1}+{{pD}}_{x}^{3}{W}_{1},\\ {\left({{\mathscr{L}}}_{g}^{* }\right)}_{1}^{2}{W}_{2} & = & 3{{pv}}_{x}{W}_{2},\\ {\left({{\mathscr{L}}}_{g}^{* }\right)}_{2}^{1}{W}_{1} & = & -{{rvD}}_{x}{W}_{1},\\ {\left({{\mathscr{L}}}_{g}^{* }\right)}_{2}^{2}{W}_{2} & = & -3{{pu}}_{x}{W}_{2}-3{{puD}}_{x}{W}_{2}-{{pD}}_{x}^{3}{W}_{2},\end{array}\end{eqnarray}$
here ${{ \mathcal L }}_{g}$ and ${{ \mathcal L }}_{g}^{* }$ denote a linearization operator defined as follows: $\begin{eqnarray}\begin{array}{l}{\left({{\mathscr{L}}}_{{\rm{\Lambda }}}\right)}_{\sigma \rho }{V}^{\rho }=\displaystyle \frac{\partial {{\rm{\Lambda }}}_{\sigma }}{\partial {u}^{\rho }}{V}^{\rho }+\displaystyle \frac{\partial {{\rm{\Lambda }}}_{\sigma }}{\partial {u}_{i}^{\rho }}{D}_{i}{V}^{\rho }+\cdots \\ \quad +\displaystyle \frac{\partial {{\rm{\Lambda }}}_{\sigma }}{\partial {u}_{{i}_{1}\cdots {i}_{p}}^{\rho }}{D}_{{i}_{1}}\cdots {D}_{{i}_{p}}{V}^{\rho },\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{\left({{\mathscr{L}}}_{{\rm{\Lambda }}}^{* }\right)}_{\sigma \rho }{W}^{\rho }=\displaystyle \frac{\partial {{\rm{\Lambda }}}_{\rho }}{\partial {u}^{\sigma }}{W}^{\rho }-{D}_{i}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{\rho }}{\partial {u}_{i}^{\sigma }}{W}^{\rho }\right)+\cdots \\ \quad +{\left(-1\right)}^{p}{D}_{{i}_{1}}\cdots {D}_{{i}_{p}}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{\rho }}{\partial {u}_{{i}_{1}\cdots {i}_{p}}^{\sigma }}{W}^{\rho }\right).\end{array}\end{eqnarray}$
Let ${{\mathscr{D}}}_{t}={\partial }_{t}-({g}^{1}{\partial }_{u}+{g}^{2}{\partial }_{v}+\cdots )$, where g1 = − 6puux + rvvx − puxxx, and g2 = 3puvx + pvxxx. For multipliers of the form Λ(x, t, u, ux, uxx) and u means u, v we have $\begin{eqnarray}\begin{array}{rcl}0 & = & {E}_{{u}^{\sigma }}({u}_{t}{{\rm{\Lambda }}}_{1}+{g}^{1}{{\rm{\Lambda }}}_{1}+{v}_{t}{{\rm{\Lambda }}}_{2}+{g}^{2}{{\rm{\Lambda }}}_{2})\\ & = & -{D}_{t}{{\rm{\Lambda }}}_{\sigma }+{\left({{\mathscr{L}}}_{g}^{* }\right)}_{\sigma }^{\rho }{{\rm{\Lambda }}}_{\rho }+{\left({{\mathscr{L}}}_{g}^{* }\right)}_{\sigma \rho }({u}_{t}^{\rho }+{g}^{\rho }),\\ \sigma & = & 1,2.\end{array}\end{eqnarray}$
By solving equations ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Lambda }}}_{1} & = & {a}_{1}{u}_{{xx}}-\displaystyle \frac{r}{4p}{a}_{1}{v}^{2}+3{a}_{1}{u}^{2}+{k}_{1},\\ {{\rm{\Lambda }}}_{2} & = & -\displaystyle \frac{r}{2p}{a}_{1}{v}_{{xx}}-\displaystyle \frac{r}{2p}{a}_{1}{uv},\end{array}\end{eqnarray}$
where a1 and k1 are arbitrary constants. So for the HS equation, here are some multipliers $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Lambda }}}_{11} & = & 1,\,{{\rm{\Lambda }}}_{12}=0,\\ {{\rm{\Lambda }}}_{21} & = & {u}_{{xx}}-\displaystyle \frac{r}{4p}{v}^{2}+3{u}^{2},\\ {{\rm{\Lambda }}}_{22} & = & \displaystyle \frac{r}{2p}{v}_{{xx}}-\displaystyle \frac{r}{2p}{uv}.\end{array}\end{eqnarray}$
For the multiplier Λ1 = (1, 0), according to the definition of multiplier ${{\rm{\Lambda }}}_{i}{G}^{i}={D}_{t}({{\rm{\Phi }}}_{1}^{t})+{D}_{x}({{\rm{\Phi }}}_{1}^{x})$, we have the following conserved vector: $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{1}^{t} & = & u,\\ {{\rm{\Phi }}}_{1}^{x} & = & -3{{pu}}^{2}+\displaystyle \frac{r}{2}{v}^{2}-{{pu}}_{{xx}}.\end{array}\end{eqnarray}$
Similarly, for the multiplier ${{\rm{\Lambda }}}_{2}=\left({u}_{{xx}}-\tfrac{r}{4p}{v}^{2}+3{u}^{2},\tfrac{r}{2p}{v}_{{xx}}-\tfrac{r}{2p}{uv}\right)$, we also have the following conserved vector: $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{2}^{t} & = & \displaystyle \frac{1}{2}{{uu}}_{{xx}}-\displaystyle \frac{r}{12p}{{uv}}^{2}+{u}^{3}-\displaystyle \frac{r}{4p}{{vv}}_{{xx}}-\displaystyle \frac{r}{6p}{{uv}}^{2},\\ {{\rm{\Phi }}}_{2}^{x} & = & \displaystyle \frac{1}{24}(-216{p}^{2}{u}^{4}+36{{pru}}^{2}{v}^{2}-3{r}^{2}{v}^{4}\\ & & -312{p}^{2}{u}^{2}{u}_{{xx}}-32{{pruvv}}_{{xx}}\\ & & +34{{prv}}_{x}^{2}u+6{{pru}}_{{xx}}{v}^{2}-34{{prvv}}_{x}{u}_{x}\\ & & -24{p}^{2}{{uu}}_{{xxxx}}-24{p}^{2}{u}_{{xx}}^{2}\\ & & -12{{prvv}}_{{xxxx}}+12{{prv}}_{x}{v}_{{xxx}}-12{{prv}}_{{xx}}^{2}+{p}^{2}{u}_{x}{u}_{{xxx}}).\end{array}\end{eqnarray}$
Anco et al consider the construction of conservation laws in some special cases [35], and here we will consider the conservation laws constructed by the more general case of equations (3. Nonlinear self-adjointness and conservation laws
In this section, we consider the nonlinear self-adjointness and the conservation laws of equations (2.1 ) and (2.24 ) by the method proposed by Ibragimov [15]. We know that this method cannot produce any new conservation laws unless they are already derived from multipliers. In particular, the multiplier method is complete, that is to say, every nontrivial conservation law comes from a multiplier. This section is mainly to supplement the conservation law that may be missed when the multiplier method is used to find the conservation law.
Firstly, we introduce a general form of systems3.1 ) is given by3.1 ), with v = (v1,…,vm) being new dependent variables. We use $\tfrac{\delta }{\delta {u}^{\alpha }}$ for the Euler–Lagrange operator
$\begin{eqnarray}{F}_{\alpha }({\boldsymbol{x}},{\boldsymbol{u}},{{\boldsymbol{u}}}_{(1)},\ldots ,{{\boldsymbol{u}}}_{(r)})=0,\quad \alpha =1,2,\cdots ,m,\end{eqnarray}$
where x = (x1, x2, …,xn) denotes n independent variables, u = (u1, u2,…,um) represents m dependent variables, and u(r) is a class of the partial derivatives of r-th order of u. Then an adjoint system of ( $\begin{eqnarray}{F}_{\alpha }^{* }({\boldsymbol{x}},{\boldsymbol{u}},{\boldsymbol{v}},{{\boldsymbol{u}}}_{(1)},{{\boldsymbol{v}}}_{(1)},\ldots ,{{\boldsymbol{u}}}_{(r)},{{\boldsymbol{v}}}_{(r)})=0,\quad \alpha =1,2,\cdots ,m,\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{c}{F}_{\alpha }^{\ast }({\boldsymbol{x}},{\boldsymbol{u}},{\boldsymbol{v}},{{\boldsymbol{u}}}_{\left(1\right)},{{\boldsymbol{v}}}_{\left(1\right)},\ldots ,{{\boldsymbol{u}}}_{\left(r\right)},{{\boldsymbol{v}}}_{\left(r\right)})\equiv \displaystyle \frac{\delta { \mathcal L }}{\delta {u}^{\alpha }}=0,\\ \,\alpha =1,2,\cdots ,m.\end{array}\end{eqnarray}$
Here ${ \mathcal L }={\sum }_{\beta =1}^{m}{{\boldsymbol{v}}}^{\beta }{F}_{\beta }$ is a Lagrangian form of ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\delta }{\delta {u}^{\alpha }}=\displaystyle \frac{\partial }{\partial {u}^{\alpha }}-{D}_{{i}_{1}}\displaystyle \frac{\partial }{\partial {u}_{{i}_{1}}^{\alpha }}+{D}_{{i}_{1}}{D}_{{i}_{2}}\displaystyle \frac{\partial }{\partial {u}_{{i}_{1}{i}_{2}}^{\alpha }}+\cdots \\ \quad +{\left(-1\right)}^{s}{D}_{{i}_{1}}{D}_{{i}_{2}}\cdots {D}_{{i}_{s}}\displaystyle \frac{\partial }{\partial {u}_{{i}_{1}{i}_{2}\cdots {i}_{s}}^{\alpha }}.\end{array}\end{eqnarray}$
Next, the definition of nonlinear self-adjointness and the theorem of conservation laws are given in [15]. The systems of m differential equations (
$\begin{eqnarray}{v}^{\alpha }={\psi }^{\alpha }(x,u),\quad \alpha =1,\cdots ,m,\end{eqnarray}$
such that $\psi (x,u)\ne 0$. Let the system of differential equations (
$\begin{eqnarray}X={\xi }^{i}({\boldsymbol{x}},{\boldsymbol{u}},{{\boldsymbol{u}}}_{(1)},\ldots )\displaystyle \frac{\partial }{\partial {x}^{i}}+{\eta }^{j}({\boldsymbol{x}},{\boldsymbol{u}},{{\boldsymbol{u}}}_{(1)},\ldots )\displaystyle \frac{\partial }{\partial {u}^{j}},\end{eqnarray}$
as well as a nonlocal symmetry of equations ( $\begin{eqnarray}\begin{array}{rcl}{C}^{i} & = & {\xi }^{i}{ \mathcal L }+{W}^{\alpha }\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{i}^{\alpha }}-{D}_{j}\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{ij}}^{\alpha }}\right)+{D}_{j}{D}_{k}\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{ijk}}^{\alpha }}\right)-\cdots \right]\\ & & +{D}_{j}({W}^{\alpha })\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{ij}}^{\alpha }}-{D}_{k}(\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{ijk}}^{\alpha }})+\cdots \right]+{D}_{j}{D}_{k}({W}^{\alpha })\\ & & \times \ \left[\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{ijk}}^{\alpha }}-\cdots \right],\end{array}\end{eqnarray}$
where $\begin{eqnarray}{W}^{\alpha }={\eta }^{\alpha }-{\xi }^{j}{u}_{j}^{\alpha },\end{eqnarray}$
and ${ \mathcal L }$ is the formal Lagrangian for the system (The Gardner equation is nonlinearly self-adjoint.
Taking the multiplier $\mu =\tfrac{1}{u}$ so that for
$\begin{eqnarray}\displaystyle \frac{1}{u}({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}})=0.\end{eqnarray}$
Then, the adjoint system of equation ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\delta { \mathcal L }}{\delta u} & = & -\displaystyle \frac{\omega }{{u}^{2}}({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}})-{D}_{t}\left(\displaystyle \frac{\omega }{u}\right)\\ & & -{D}_{x}(6\omega )+6{u}_{x}\displaystyle \frac{\omega }{u}-12{u}_{x}\omega +{D}_{x}(6u\omega )-{D}_{x}^{3}\left(\displaystyle \frac{\omega }{u}\right),\end{array}\end{eqnarray}$
with the formal Lagrangian $\begin{eqnarray}{ \mathcal L }=\displaystyle \frac{\omega }{u}({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}).\end{eqnarray}$
Substituting $\omega =u$ into ( $\begin{eqnarray}\displaystyle \frac{\delta { \mathcal L }}{\delta u}=-\displaystyle \frac{1}{u}({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}).\end{eqnarray}$
By comparing ( $\begin{eqnarray}\begin{array}{l}{\,X}_{1}=\displaystyle \frac{\partial }{\partial t},\quad {X}_{2}=\displaystyle \frac{\partial }{\partial x},\\ \quad {X}_{3}=\left(t+\displaystyle \frac{x}{3}\right)\displaystyle \frac{\partial }{\partial x}+t\displaystyle \frac{\partial }{\partial t}+\left(-\displaystyle \frac{u}{3}+\displaystyle \frac{1}{6}\right)\displaystyle \frac{\partial }{\partial u}.\,\end{array}\end{eqnarray}$
The conservation laws of symmetry ( $\begin{eqnarray}[{D}_{t}{C}^{t}+{D}_{x}{C}^{x}]{| }_{(3.1)}=0,\end{eqnarray}$
where the conserved vector C = (Ct, Cx) is given by (For the generator X1, the corresponding Lie characteristic function is given by W = − ut. Thus, formulas (3.7 ) yield the following conserved vector:3.7 ) yield the following conserved vector3.7 ), we have the conservation vector of the Gardner equation as follows:
$\begin{eqnarray}\begin{array}{rcl}{C}^{t} & = & {\xi }^{t}{ \mathcal L }+W\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{t}}\right]\\ & = & v({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}})-{u}_{t}v,\\ {C}^{x} & = & {\xi }^{x}{ \mathcal L }+W\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{x}}+{D}_{x}^{2}\displaystyle \frac{\partial { \mathcal L }}{\partial {U}_{{xxx}}}\right]\\ & & +{D}_{x}W\left[-{D}_{x}\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{xxx}}}\right]+{D}_{x}^{2}W\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{xxx}}}\right]\\ & = & -{u}_{t}(6{uv}-6{u}^{2}v+{v}_{{xx}})+{u}_{{tx}}{v}_{x}-{u}_{{txx}}v.\end{array}\end{eqnarray}$
Similarly, for the generator X2, we have W = − ux. Thus, formulas ( $\begin{eqnarray}\begin{array}{rcl}{C}^{t} & = & -{u}_{x}v,\\ {C}^{x} & = & v({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}})\\ & & -{u}_{x}(6{uv}-6{u}^{2}v+{v}_{{xx}})+{u}_{{xx}}{v}_{x}-{u}_{{xxx}}v.\end{array}\end{eqnarray}$
For the generator X3, we have $W=-\tfrac{u}{3}+\tfrac{1}{6}\,-\left(t+\tfrac{x}{3}\right){u}_{x}-{{tu}}_{t}$. According to formula ( $\begin{eqnarray}\begin{array}{rcl}{C}^{t} & = & t({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}})v\\ & & -\left(\displaystyle \frac{u}{3}-\displaystyle \frac{1}{6}+\left(t+\displaystyle \frac{x}{3}\right){u}_{x}+{{tu}}_{t}\right)v,\\ {C}^{x} & = & \left(t+\displaystyle \frac{x}{3}\right)v({u}_{t}+6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}})\\ & & +\left(-\displaystyle \frac{u}{3}+\displaystyle \frac{1}{6}-\left(t+\displaystyle \frac{x}{3}\right){u}_{x}-{{tu}}_{t}\right)(6{uv}\\ & & -6{u}^{2}v+{v}_{{xx}})-\left(-\displaystyle \frac{2}{3}{u}_{x}-\left(t+\displaystyle \frac{x}{3}\right){u}_{{xx}}-{{tu}}_{{tx}}\right){v}_{x}\\ & & +\left(-{u}_{{xx}}-\left(t+\displaystyle \frac{x}{3}\right){u}_{{xxx}}-{{tu}}_{{txx}}\right)v.\end{array}\end{eqnarray}$
For the generators X1 and X2, the translational symmetry yields trivial conservation laws. Compared with the conservation law obtained by the multiplier method, there is in fact no new conservation law, but it may be useful for us to study the problem later.For the LGH equation, it is easy to prove that it is not nonlinearly self-adjoint. According to Theorem 3, it cannot be obtained by formula (3.7 ). For the HS equation, let r = 3p, and then equation (2.24 ) becomes the following form:
$\begin{eqnarray}\left\{\begin{array}{l}{G}^{1}={u}_{t}-6{{puu}}_{x}+3{{pvv}}_{x}-{{pu}}_{{xxx}}=0,\\ {G}^{2}={v}_{t}+3{{puv}}_{x}+{{pv}}_{{xxx}}=0.\end{array}\right.\end{eqnarray}$
The HS equation is nonlinearly self-adjoint.
The formal Lagrangian of the HS equation (
$\begin{eqnarray}\begin{array}{l}{ \mathcal L }={\omega }^{1}({u}_{t}-6{{puu}}_{x}+3{{pvv}}_{x}-{{pu}}_{{xxx}})\\ \quad +{\omega }^{2}({v}_{t}+3{{puv}}_{x}+{{pv}}_{{xxx}}).\end{array}\end{eqnarray}$
Then, one has $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\delta { \mathcal L }}{\delta u} & = & -{\omega }_{t}^{1}+6{pu}{\omega }_{x}^{1}+p{\omega }_{{xxx}}^{1}-3{{pv}}_{x}{\omega }^{2},\\ \displaystyle \frac{\delta { \mathcal L }}{\delta v} & = & -3{pv}{\omega }_{x}^{1}-{\omega }_{t}^{2}+3{{pu}}_{x}{\omega }^{2}+3{pu}{\omega }_{x}^{2}-p{\omega }_{{xxx}}^{2}.\end{array}\end{eqnarray}$
Substituting ${\omega }^{1}=u$ and ${\omega }^{2}=v$ into equation ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\delta { \mathcal L }}{\delta u} & = & -({u}_{t}-6{{puu}}_{x}+3{{pvv}}_{x}-{{pu}}_{{xxx}}),\\ \displaystyle \frac{\delta { \mathcal L }}{\delta v} & = & -({v}_{t}+3{{puv}}_{x}+{{pv}}_{{xxx}}).\end{array}\end{eqnarray}$
Hence, the special HS equation is nonlinearly self-adjoint. $\begin{eqnarray}\begin{array}{l}{X}_{1}=\displaystyle \frac{\partial }{\partial t},\quad {X}_{2}=\displaystyle \frac{\partial }{\partial x},\quad {X}_{3}=\displaystyle \frac{x}{3}\displaystyle \frac{\partial }{\partial x}\\ \quad +t\displaystyle \frac{\partial }{\partial t}-\displaystyle \frac{2}{3}u\displaystyle \frac{\partial }{\partial u}-\displaystyle \frac{2}{3}v\displaystyle \frac{\partial }{\partial v}.\end{array}\end{eqnarray}$
The corresponding conservation law is limited by $\begin{eqnarray}[{D}_{t}{C}^{t}+{D}_{x}{C}^{x}]{| }_{(4.21)}=0,\end{eqnarray}$
where C = (Ct, Cx) denotes the conserved vector of ( $\begin{eqnarray}\begin{array}{rcl}{C}^{t} & = & {\xi }^{t}{ \mathcal L }+{W}^{1}\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{t}}\right]+{W}^{2}\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {v}_{t}}\right]\\ & = & {\omega }^{1}({u}_{t}-6{{puu}}_{x}+3{{pvv}}_{x}-{{pu}}_{{xxx}})\\ & & +{\omega }^{2}({v}_{t}+3{{puv}}_{x}+{{pv}}_{{xxx}})-{u}_{t}{\omega }_{1}-{v}_{t}{\omega }_{2},\\ {C}^{x} & = & {\xi }^{x}{ \mathcal L }+{W}^{1}\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{x}}+{D}_{x}^{2}\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{xxx}}}\right]\\ & & +{D}_{x}{W}^{1}\left[-{D}_{x}\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{xxx}}}\right]+{D}_{x}^{2}{W}^{1}\displaystyle \frac{\partial { \mathcal L }}{\partial {u}_{{xxx}}}\\ & & +{W}^{2}\left[\displaystyle \frac{\partial { \mathcal L }}{\partial {v}_{x}}+{D}_{x}^{2}\displaystyle \frac{\partial { \mathcal L }}{\partial {v}_{{xxx}}}\right]\\ & & +{D}_{x}{W}^{2}\left[-{D}_{x}\displaystyle \frac{\partial { \mathcal L }}{\partial {v}_{{xxx}}}\right]+{D}_{x}^{2}{W}^{2}\displaystyle \frac{\partial { \mathcal L }}{\partial {v}_{{xxx}}}\\ & = & {u}_{t}(6{pu}{\omega }^{1}+{{pw}}_{{xx}}^{1})-{{pu}}_{{xt}}{\omega }_{x}^{1}+{{pu}}_{{xxt}}{\omega }^{1}\\ & & -{v}_{t}(3{pv}{\omega }^{1}-3{pu}{\omega }^{2}+p{\omega }_{{xx}}^{2})\\ & & +{{pv}}_{{xt}}{\omega }_{x}^{2}-{{pv}}_{{xxt}}{\omega }^{2}.\end{array}\end{eqnarray}$
For the generator X2, we have W1 = − ux, W2 = − vx. Thus, formulas ( $\begin{eqnarray}\begin{array}{rcl}{C}^{t} & = & -{u}_{x}{\omega }^{1}-{v}_{x}{\omega }^{2},\\ {C}^{x} & = & {\omega }^{1}({u}_{t}-6{{puu}}_{x}+3{{pvv}}_{x}-{{pu}}_{{xxx}})+{\omega }^{2}\\ & & \times ({v}_{t}+3{{puv}}_{x}+{{pv}}_{{xxx}})+{u}_{x}(6{pu}{\omega }^{1}+{{pw}}_{{xx}}^{1})\\ & & -{{pu}}_{{xx}}{\omega }_{x}^{1}+{{pu}}_{{xxx}}{\omega }^{1}-{v}_{x}(3{pv}{\omega }^{1}-3{pu}{\omega }^{2}+p{\omega }_{{xx}}^{2})\\ & & +{{pv}}_{{xx}}{\omega }_{x}^{2}-{{pv}}_{{xxx}}{\omega }^{2}.\end{array}\end{eqnarray}$
For the generator X3, we have ${W}^{1}=-\left(\tfrac{2}{3}u+\tfrac{x}{3}{u}_{x}+\tfrac{2t}{3}{u}_{t}\right),{W}^{2}=-\left(\tfrac{2}{3}v+\tfrac{x}{3}{v}_{x}+\tfrac{2t}{3}{v}_{t}\right)$. Thus, formulas ( $\begin{eqnarray}\begin{array}{rcl}{C}^{t} & = & t[{\omega }^{1}({u}_{t}-6{{puu}}_{x}+3{{pvv}}_{x}-{{pu}}_{{xxx}})\\ & & +{\omega }^{2}({v}_{t}+3{{puv}}_{x}+{{pv}}_{{xxx}})]\\ & & -\left(\displaystyle \frac{2}{3}u+\displaystyle \frac{x}{3}{u}_{x}+\displaystyle \frac{2t}{3}{u}_{t}\right){\omega }^{1}\\ & & -\left(\displaystyle \frac{2}{3}v+\displaystyle \frac{x}{3}{v}_{x}+\displaystyle \frac{2t}{3}{v}_{t}\right){\omega }^{2},\\ {C}^{x} & = & \displaystyle \frac{x}{3}[{\omega }^{1}({u}_{t}-6{{puu}}_{x}+3{{pvv}}_{x}-{{pu}}_{{xxx}})\\ & & +{\omega }^{2}({v}_{t}+3{{puv}}_{x}+{{pv}}_{{xxx}})]\\ & & -\left(\displaystyle \frac{2}{3}u+\displaystyle \frac{x}{3}{u}_{x}+\displaystyle \frac{2t}{3}{u}_{t}\right)(-6{pu}{\omega }^{1}+p{\omega }_{{xx}}^{1})\\ & & -\ p\left({u}_{x}+\displaystyle \frac{x}{3}{u}_{x}+\displaystyle \frac{2}{3}{{tu}}_{{tx}}\right){\omega }_{x}^{1}\\ & & -\left(\displaystyle \frac{4}{3}{u}_{{xx}}+\displaystyle \frac{x}{3}{u}_{{xxx}}+\displaystyle \frac{2}{3}{{tu}}_{{txx}}\right){\omega }^{1}-(\displaystyle \frac{2}{3}v+\displaystyle \frac{x}{3}{v}_{x}\\ & & +\displaystyle \frac{2t}{3}{v}_{t})(3{pv}{\omega }^{1}-3{pu}{\omega }^{2}+p{\omega }_{{xx}}^{2})\\ & & +p\left({v}_{x}+\displaystyle \frac{x}{3}{v}_{{xx}}+\displaystyle \frac{2t}{3}{v}_{{tx}}\right){\omega }_{x}^{2}-p\left(\displaystyle \frac{4}{3}{v}_{{xx}}\right.\\ & & +\left.\displaystyle \frac{x}{3}{v}_{{xxx}}+\displaystyle \frac{2t}{3}{v}_{{txx}}\right){\omega }^{2}.\end{array}\end{eqnarray}$
According to the above calculation, we can find that the Gardner equation has infinite polynomial conservation laws, while the HS equation has only three conservation laws, whether using the above method or other methods.4. Conclusion and discussion
In this paper, we use two methods to calculate the conservation laws, the direct construction of conservation laws and the conservation theorem method proposed by Ibragimov, and construct the conservation laws of the Gardner equation, the LGH equation, and the Hirota–Satsuma equation. The conservation theorem method is more widely used than the Noether theorem. It does not require the equation system to have a Lagrangian function, but it does require it to be nonlinearly self-adjoint. The direct construction conservation law method has no requirement for the equation system, but the calculation of its multiplier is more complicated. Compared with the Noether theorem, the multiplier method is complete. In fact, every conservation law is generated by multipliers. In contrast, the use of symmetric, adjoint symmetric formulas, partial Lagrangians, and other methods is usually incomplete. They either do not apply to every partial differential equations system, or do not necessarily produce all conservation laws. The multiplier method generates all conservation laws for any partial differential equations system, regardless of whether the system has a Lagrangian function.
Funding
This work was funded by the National Natural Science Foundation of China (Grant Nos. 12371256 & 11971475).
Appendix A
By comparing the coefficient G and DxG of equation (2.6 ), it is found that the system consists of the adjoint symmetry determining equations on ΛA3 ), we find that the highest order derivative term is ${u}_{{xxx}}{{\rm{\Lambda }}}_{{u}_{{xx}}{u}_{{xx}}}$, and hence ${u}_{{xxx}}{{\rm{\Lambda }}}_{{u}_{{xx}}{u}_{{xx}}}=0$, so thatA3 ), we can getA1 ), we can get the following equationsA4 ) and (A5 ), we can obtain the general form of Λ as follows:
$\begin{eqnarray}-{{\mathscr{D}}}_{t}{\rm{\Lambda }}-6{{uD}}_{x}{\rm{\Lambda }}+6{u}^{2}{D}_{x}{\rm{\Lambda }}-{D}_{x}^{3}{\rm{\Lambda }}=0,\end{eqnarray}$
and extra determining equations on Λ, $\begin{eqnarray}-{D}_{x}{{\rm{\Lambda }}}_{{u}_{x}}+{D}_{x}^{2}{{\rm{\Lambda }}}_{{u}_{{xx}}}=0,\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Lambda }}}_{{u}_{x}}-{D}_{x}{{\rm{\Lambda }}}_{{u}_{{xx}}}=0,\end{eqnarray}$
where ${{\mathscr{D}}}_{t}={\partial }_{t}-(6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}){\partial }_{u}-{\left(6{{uu}}_{x}-6{u}^{2}{u}_{x}+{u}_{{xxx}}\right)}_{x}{\partial }_{{u}_{x}}+\cdots $ is the total derivative operator, which expresses t derivatives of u through the Gardner equation. By expanding equation ( $\begin{eqnarray}{\rm{\Lambda }}=a(t,x,u,{u}_{x}){u}_{{xx}}+b(t,x,u,{u}_{x}).\end{eqnarray}$
Substitute Λ in the above formula into ( $\begin{eqnarray}{b}_{{u}_{x}}-{a}_{u}{u}_{x}-{a}_{x}=0.\end{eqnarray}$
By comparing the coefficients uxxxx, uxxx and so on in equation ( $\begin{eqnarray*}\begin{array}{l}3{{\rm{\Lambda }}}_{{{xu}}_{x}}+3{u}_{{xx}}{{\rm{\Lambda }}}_{{u}_{x}{u}_{{xx}}}+3{u}_{x}{{\rm{\Lambda }}}_{{{uu}}_{{xx}}}=0,\\ 6{u}_{x}{{\rm{\Lambda }}}_{{{xuu}}_{{xx}}}+3{u}_{x}^{2}{{\rm{\Lambda }}}_{{{uuu}}_{{xx}}}+3{{\rm{\Lambda }}}_{{{xu}}_{x}}+3{{\rm{\Lambda }}}_{{{xxu}}_{x}}+3{u}_{{xx}}{{\rm{\Lambda }}}_{{{uu}}_{{xx}}}=0,\\ {{\rm{\Lambda }}}_{t}-(6{{uu}}_{x}-6{u}^{2}{u}_{x}){{\rm{\Lambda }}}_{u}-(18{u}_{x}{u}_{{xx}}-12{u}_{x}^{3}\\ -36{{uu}}_{x}{u}_{{xx}}){{\rm{\Lambda }}}_{{u}_{{xx}}}+(6u-6{u}^{2})({{\rm{\Lambda }}}_{x}+{u}_{x}{{\rm{\Lambda }}}_{u})\\ -(6{u}_{x}^{2}-12{{uu}}_{x}^{2}){{\rm{\Lambda }}}_{{u}_{x}}+3{u}_{x}{u}_{{xx}}{{\rm{\Lambda }}}_{{uu}}+3{u}_{x}{{\rm{\Lambda }}}_{{xxu}}\\ +3{u}_{{xx}}{{\rm{\Lambda }}}_{{xu}}+3{u}_{x}^{2}{{\rm{\Lambda }}}_{{xuu}}+{u}_{x}^{3}{{\rm{\Lambda }}}_{{uuu}}+{{\rm{\Lambda }}}_{{xxx}}=0.\end{array}\end{eqnarray*}$
By solving the above equations and equations ( $\begin{eqnarray}\begin{array}{l}{\rm{\Lambda }}={c}_{1}[(2{u}^{3}-3{u}^{2}+u-{u}_{{xx}})t\\ \quad -\displaystyle \frac{x}{6}+\displaystyle \frac{1}{3}{xu}]+{c}_{2}[-2{u}^{3}+3{u}^{2}+{u}_{{xx}}]+{c}_{3}u+{c}_{4},\end{array}\end{eqnarray}$
where c1, c2, c3 and c4 are arbitrary constants.Appendix B
Equation (2.18 ) leads to a split system of two determining equations for Λ(t, x, u, ut, ux), consisting of2.14 ) on Λ, andB2 ), we can getB1 ), comparison coefficient uxx, we can getB5 ) and (B6 ), we haveB1 ), we haveB4 ), (B5 ), (B7 ) and (B8 ), we can get the general form of multiplier isB9 ) into (B1 ), we can getB10 ) separates into
$\begin{eqnarray}{{\mathscr{D}}}_{t}^{2}{\rm{\Lambda }}-{D}_{x}^{2}{\rm{\Lambda }}+{{Ng}}^{2}{u}^{N-1}{\rm{\Lambda }}=0,\end{eqnarray}$
which is the symmetry determining equation ( $\begin{eqnarray}2{{\rm{\Lambda }}}_{u}+{{\mathscr{D}}}_{t}{{\rm{\Lambda }}}_{{u}_{t}}-{D}_{x}{{\rm{\Lambda }}}_{{u}_{x}}=0,\end{eqnarray}$
which is an extra determining equation on Λ. Here ${D}_{t}={\partial }_{t}+{u}_{t}{\partial }_{u}+{u}_{{tx}}{\partial }_{{u}_{x}}+({u}_{{tt}}-{u}_{{xx}}+{g}^{2}{u}^{N}){\partial }_{{u}_{t}}+\cdots $ is the total derivative operator which expresses t derivatives through the LGH equation. And from equation ( $\begin{eqnarray}\begin{array}{l}2{{\rm{\Lambda }}}_{u}+{{\rm{\Lambda }}}_{{{tu}}_{t}}+{u}_{t}{{\rm{\Lambda }}}_{{{uu}}_{t}}+({u}_{{xx}}-{g}^{2}{u}^{N}){{\rm{\Lambda }}}_{{u}_{t}{u}_{t}}\\ \quad -{{\rm{\Lambda }}}_{{{xu}}_{x}}-{u}_{x}{{\rm{\Lambda }}}_{{{uu}}_{x}}-{u}_{{xx}}{{\rm{\Lambda }}}_{{u}_{x}{u}_{x}}=0.\end{array}\end{eqnarray}$
Since Λ does not contain uxx, we have $\begin{eqnarray}{{\rm{\Lambda }}}_{{u}_{t}{u}_{t}}={{\rm{\Lambda }}}_{{u}_{x}{u}_{x}},\end{eqnarray}$
$\begin{eqnarray}2{{\rm{\Lambda }}}_{u}+{{\rm{\Lambda }}}_{{{tu}}_{t}}+{u}_{t}{{\rm{\Lambda }}}_{{{uu}}_{t}}-{g}^{2}{u}^{N}{{\rm{\Lambda }}}_{{u}_{t}{u}_{t}}-{{\rm{\Lambda }}}_{{{xu}}_{x}}-{u}_{x}{{\rm{\Lambda }}}_{{{uu}}_{x}}=0.\end{eqnarray}$
We next consider equation ( $\begin{eqnarray}-2{u}_{x}{{\rm{\Lambda }}}_{{{uu}}_{x}}-2{{\rm{\Lambda }}}_{{{xu}}_{x}}+2{{\rm{\Lambda }}}_{{{tu}}_{t}}+2{u}_{t}{{\rm{\Lambda }}}_{{{uu}}_{t}}-2{g}^{2}{u}^{N}{{\rm{\Lambda }}}_{{u}_{t}{u}_{t}}=0.\end{eqnarray}$
Combining equations ( $\begin{eqnarray}4{{\rm{\Lambda }}}_{u}=0.\end{eqnarray}$
Compare the coefficients of uxt in equation ( $\begin{eqnarray}-2{{\rm{\Lambda }}}_{{{xu}}_{t}}+2{{\rm{\Lambda }}}_{{{tu}}_{x}}-2{g}^{2}{u}^{N}{{\rm{\Lambda }}}_{{u}_{t}{u}_{x}}=0.\end{eqnarray}$
By solving equations ( $\begin{eqnarray}\begin{array}{l}{\rm{\Lambda }}=({u}_{x}-{u}_{t}){F}_{1}(t-x)\\ \quad +({u}_{t}+{u}_{x}){F}_{2}(t+x)+{c}_{1}{u}_{t}+{F}_{3}(x,t),\end{array}\end{eqnarray}$
where F1(x + t) is an arbitrary function of x + t, F2(t) is an arbitrary function of t, F3 is only related to x, t. Substitute equations ( $\begin{eqnarray}\begin{array}{rcl} & & -({u}_{t}+{u}_{x}){F}_{1{tt}}+({u}_{t}+{u}_{x}){F}_{2{tt}}+{F}_{3{tt}}\\ & & +(-{u}_{t}+{u}_{x}){F}_{1{xx}}-({u}_{t}+{u}_{x}){F}_{2{xx}}\\ & & -{F}_{3{xx}}+((-{u}_{t}+{u}_{x}){F}_{1}+({u}_{t}+{u}_{x}){F}_{2}\\ & & +{c}_{1}{u}_{t}+{F}_{3}){{Ng}}^{2}{u}^{N-1}=0,\end{array}\end{eqnarray}$
Equation ( $\begin{eqnarray}{F}_{2}^{{\prime\prime} }=0,\quad {F}_{1}^{{\prime} }=0,\quad {F}_{3}=0,\end{eqnarray}$
so we can obtain F1 = a1 = constant, F2 = a2 = constant and F3 = 0. Therefore, we can get the specific form of multipliers $\begin{eqnarray}{\rm{\Lambda }}={a}_{1}({u}_{x}-{u}_{t})+{a}_{2}({u}_{t}+{u}_{x})+{c}_{1}{u}_{t},\end{eqnarray}$
where a1, a2 and c1 are arbitrary constants.Appendix C
Substitute equations (2.26 ) into (2.29 ) when σ is 1, 2, respectively. We have2.29 ) are given by2.29 )C6 ). Its highest-order term is ${u}_{{xxx}}{{\rm{\Lambda }}}_{1{u}_{{xx}}{u}_{{xx}}}$ and ${v}_{{xxx}}{{\rm{\Lambda }}}_{2{v}_{{xx}}{v}_{{xx}}}$, hence ${{\rm{\Lambda }}}_{1{u}_{{xx}}{u}_{{xx}}}=0$ and ${{\rm{\Lambda }}}_{2{v}_{{xx}}{v}_{{xx}}}=0$. Similarly, we have ${{\rm{\Lambda }}}_{1{v}_{{xx}}{v}_{{xx}}}=0$ and ${u}_{{xxx}}{{\rm{\Lambda }}}_{1{u}_{{xx}}{u}_{{xx}}}$ from equation (C7 ). This means that Λ1 and Λ2 are linear with respect to uxx and vxx, combining equation (C5 ), we have the general form of Λ1 and Λ2C6 ), after some cancellations,C3 ) and (C4 ), we haveC12 ) separates intoC11 ). Similarly, we have ${a}_{1{u}_{x}}={b}_{1x}={b}_{1u}={b}_{1v}={b}_{2{v}_{x}}=0$. That is to say, a1, b1 and b2 are independent of x, u, ux, equation (C10 ) becomesC3 ), we haveC3 ) with the coefficient uxxx in equation (C4 ), we haveC3 ) and (C4 ), we can obtain the following equations:C17 )–(C24 ), we can get
$\begin{eqnarray}\begin{array}{rcl} & & -{D}_{t}{{\rm{\Lambda }}}_{1}+6{{puD}}_{x}{{\rm{\Lambda }}}_{1}+{{pD}}_{x}{x}^{3}{{\rm{\Lambda }}}_{1}+3{{pv}}_{x}{{\rm{\Lambda }}}_{2}\\ & & +\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial u}{G}^{1}-{D}_{x}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {u}_{x}}{G}^{1}\right)\right.\\ & & \left.+{D}_{x}^{2}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {u}_{{xx}}}{G}^{1}\right)\right)+\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial u}{G}^{2}-{D}_{x}\right.\\ & & \times \left.\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {u}_{x}}{G}^{2}\right)+{D}_{x}^{2}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {u}_{{xx}}}{G}^{2}\right)\right)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl} & & -{D}_{t}{{\rm{\Lambda }}}_{2}-{{rvD}}_{x}{{\rm{\Lambda }}}_{1}-3p{{\rm{\Lambda }}}_{2}-3{{puD}}_{x}{{\rm{\Lambda }}}_{2}-{{pD}}_{x}^{3}{{\rm{\Lambda }}}_{2}\\ & & +\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial v}{G}^{1}-{D}_{x}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {v}_{x}}{G}^{1}\right)\right.\\ & & \left.+{D}_{x}^{2}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {v}_{{xx}}}{G}^{1}\right)\right)+\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial v}{G}^{2}-{D}_{x}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {v}_{x}}{G}^{2}\right)\right.\\ & & \left.+{D}_{x}^{2}\left(\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {v}_{{xx}}}{G}^{2}\right)\right)=0.\end{array}\end{eqnarray}$
Consequently, the non-leading terms in equation ( $\begin{eqnarray}-{{\mathscr{D}}}_{t}{{\rm{\Lambda }}}_{1}+6{{puD}}_{x}{{\rm{\Lambda }}}_{1}+{{pD}}_{x}{x}^{3}{{\rm{\Lambda }}}_{1}+3{{pv}}_{x}{{\rm{\Lambda }}}_{2}=0,\end{eqnarray}$
$\begin{eqnarray}-{{\mathscr{D}}}_{t}{{\rm{\Lambda }}}_{2}-{{rvD}}_{x}{{\rm{\Lambda }}}_{1}-3p{{\rm{\Lambda }}}_{2}-3{{puD}}_{x}{{\rm{\Lambda }}}_{2}-{{pD}}_{x}^{3}{{\rm{\Lambda }}}_{2}=0.\end{eqnarray}$
Comparing the coefficients of derivatives of G and G, we can get the leading term in equation ( $\begin{eqnarray}-\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {v}_{{xx}}}+\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {u}_{{xx}}}=0,\end{eqnarray}$
$\begin{eqnarray}2\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {u}_{x}}-2{D}_{x}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {u}_{{xx}}}=0,\quad 2\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {v}_{x}}-2{D}_{x}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {v}_{{xx}}}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {v}_{x}}+\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {u}_{x}}-2{D}_{x}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {u}_{{xx}}}=0,\\ \quad \times \displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {u}_{x}}+\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {v}_{x}}-2{D}_{x}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {v}_{{xx}}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}-{D}_{x}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {u}_{x}}+{D}_{x}^{2}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial {u}_{{xx}}}=0,\quad -{D}_{x}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {v}_{x}}+{D}_{x}^{2}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {v}_{{xx}}}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial v}+\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial u}-{D}_{x}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {u}_{x}}+{D}_{x}^{2}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {u}_{{xx}}}=0,\\ \quad -\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial u}+\displaystyle \frac{\partial {{\rm{\Lambda }}}_{1}}{\partial v}-{D}_{x}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {v}_{x}}+{D}_{x}^{2}\displaystyle \frac{\partial {{\rm{\Lambda }}}_{2}}{\partial {v}_{{xx}}}=0.\end{array}\end{eqnarray}$
Consider equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Lambda }}}_{1} & = & {a}_{1}(t,x,{\boldsymbol{u}},{{\boldsymbol{u}}}_{x}){u}_{{xx}}+{b}_{1}(t,x,{\boldsymbol{u}},{{\bf{u}}}_{x}){v}_{{xx}}\\ & & +{c}_{1}(t,x,{\boldsymbol{u}},{{\boldsymbol{u}}}_{x}),\\ {{\rm{\Lambda }}}_{2} & = & {b}_{1}(t,x,{\boldsymbol{u}},{{\boldsymbol{u}}}_{x}){u}_{{xx}}+{b}_{2}(t,x,{\boldsymbol{u}},{{\boldsymbol{u}}}_{x}){v}_{{xx}}\\ & & +{c}_{2}(t,x,{\boldsymbol{u}},{{\boldsymbol{u}}}_{x}),\end{array}\end{eqnarray}$
Then the remaining terms in equation ( $\begin{eqnarray}\begin{array}{l}{b}_{1{u}_{x}}{v}_{{xx}}+{c}_{1{u}_{x}}-({a}_{1x}+{a}_{1u}{u}_{x}+{a}_{1v}{v}_{x}+{a}_{1{v}_{x}}{v}_{{xx}})=0,\\ {b}_{1{v}_{x}}{v}_{{xx}}+{c}_{2{v}_{x}}-({b}_{2x}+{b}_{2u}{u}_{x}+{b}_{2v}{v}_{x}+{b}_{2{u}_{x}}{u}_{{xx}})=0.\end{array}\end{eqnarray}$
According to the coefficients of uxxxx and vxxxx of equations ( $\begin{eqnarray}\begin{array}{l}{a}_{1{v}_{x}}{u}_{{xx}}+{b}_{1{v}_{x}}{v}_{{xx}}+{c}_{1{v}_{x}}=0,\\ {b}_{1{u}_{x}}{u}_{{xx}}+{b}_{2{u}_{x}}{u}_{{xx}}+{c}_{2{u}_{x}}=0.\end{array}\end{eqnarray}$
Clearly, equation ( $\begin{eqnarray}{a}_{1{v}_{x}}={b}_{1{v}_{x}}={c}_{1{v}_{x}}=0,\quad {b}_{1{u}_{x}}={b}_{2{u}_{x}}={c}_{2{u}_{x}}=0.\end{eqnarray}$
Hence a1x = a1u = a1v = 0, b2x = b2u = b2v = 0 from equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Lambda }}}_{1} & = & {a}_{1}(t){u}_{{xx}}+{b}_{1}(t){v}_{{xx}}+{c}_{1}(t,x,{\boldsymbol{u}}),\\ {{\rm{\Lambda }}}_{2} & = & {b}_{1}(t){u}_{{xx}}+{b}_{2}(t){v}_{{xx}}+{c}_{2}(t,x,{\boldsymbol{u}}),\end{array}\end{eqnarray}$
Next, we solve the concrete forms of a1, b1, b2, c1, c2. The coefficient of uxxxxx from equation ( $\begin{eqnarray}2{{pb}}_{1}(t)=0,\end{eqnarray}$
hence b1(t) = 0. Comparing the coefficient vxxx in equation ( $\begin{eqnarray}2{{pc}}_{1v}+{{rva}}_{1}(t)=0,\quad 2{{pc}}_{2u}+{{rva}}_{1}(t)=0.\end{eqnarray}$
It yields that $\begin{eqnarray}\begin{array}{rcl}{c}_{1} & = & -\displaystyle \frac{r}{4p}{v}^{2}{a}_{1}(t)+{k}_{1}(x,t,u),\\ {c}_{2} & = & -\displaystyle \frac{r}{2p}{{uva}}_{1}(t)+{l}_{1}(x,t,v).\end{array}\end{eqnarray}$
Furthermore, by comparing the coefficients of ux, uxx, and constant terms in equations ( $\begin{eqnarray}-a{{\prime} }_{1}(t)-18{a}_{1}(t){{pu}}_{x}+3{{pu}}_{x}{c}_{1{uu}}+3{{pc}}_{1{xu}}=0,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{r}{2}{a}_{1}(t)+{{pb}}_{2}(t)=0,\end{eqnarray}$
$\begin{eqnarray}b{{\prime} }_{2}(t)-3{{pb}}_{2}(t)+3{{pc}}_{2{uv}}=0,\end{eqnarray}$
$\begin{eqnarray}9{{puc}}_{2u}+{{rvc}}_{1u}+3{{pc}}_{2}=0,\end{eqnarray}$
$\begin{eqnarray}-c{{\prime} }_{1}(t)+6{{puc}}_{1x}+{{pc}}_{1{xxx}}=0,\end{eqnarray}$
$\begin{eqnarray}{c}_{2t}+{{rvc}}_{1x}+3{{puc}}_{2x}+{{pc}}_{2{xxx}}=0,\end{eqnarray}$
$\begin{eqnarray}{c}_{1{xv}}={c}_{2{xu}}={c}_{2{vv}}=0.\end{eqnarray}$
Simultaneous equations ( $\begin{eqnarray}\begin{array}{rcl}{c}_{1} & = & -\displaystyle \frac{r}{4p}{a}_{1}{v}^{2}+3{a}_{1}{u}^{2}+{k}_{1},\\ {c}_{2} & = & -\displaystyle \frac{r}{2p}{a}_{1}{uv},\quad {b}_{2}=-\displaystyle \frac{r}{2p}{a}_{1},\end{array}\end{eqnarray}$
where a1 and k1 are arbitrary constants. Therefore, we can get the specific form of multipliers $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Lambda }}}_{1} & = & {a}_{1}{u}_{{xx}}-\displaystyle \frac{r}{4p}{a}_{1}{v}^{2}+3{a}_{1}{u}^{2}+{k}_{1},\\ {{\rm{\Lambda }}}_{2} & = & -\displaystyle \frac{r}{2p}{a}_{1}{v}_{{xx}}-\displaystyle \frac{r}{2p}{a}_{1}{uv},\end{array}\end{eqnarray}$
where a1 and k1 are arbitrary constants.