1. Introduction
In the field of optical communications, nonlinear mathematical physical models that can describe the pulse propagation in optical fibers play an important role. The famous nonlinear Schrödinger (NLS) equation [1, 2], the short pulse (SP) equation [3, 4], and the Wadati–Konno–Ichikawa equation [5, 6] have been established respectively to describe different pulse propagations in optical fibers. Based on the SP equation, Feng proposed a new model.1 ) can describe the pulse propagation at the scale of attoseconds [8]. The integrability of equation (1 ) has been studied in [7]. N-soliton solutions, one-order rogue wave and soliton interaction patterns of equation (1 ) were investigated in [9]. The relationship between the complex coupled dispersionless equations and equation (1 ) has been found [10]. Periodic solutions, soliton solutions, higher-order breathers and rogue wave solutions for equation (1 ) have been obtained [11, 12]. Long-time asymptotic behaviour and the general form of two-soliton solutions have been studied via Riemann–Hilbert approach [13, 14]. More results about equation (1 ) can be seen in [15, 16].
$\begin{eqnarray}{q}_{xt}+q+\displaystyle \frac{1}{2}{\left({\left|q\right|}^{2}{q}_{x}\right)}_{x}=0,\end{eqnarray}$
with $q=q(x,t)$ being a complex-valued function, and the equation was called complex short pulse (CSP) equation [7]. Since $q=q(x,t)$ comprises amplitude and phase, it is more accurate to describe optical waves. The equation (When birefringence effects were considered, the coupled complex short pulse (CCSP) equations2 ) have been studied [7, 17]. Lie symmetries and exact solutions of equation (2 ) have been obtained [18].
$\begin{eqnarray}\begin{array}{l}{q}_{1xt}+{q}_{1}+\displaystyle \frac{1}{2}{\left(({\left|{q}_{1}\right|}^{2}+{\left|{q}_{2}\right|}^{2}){q}_{1x}\right)}_{x}=0,\\ {q}_{2xt}+{q}_{2}+\displaystyle \frac{1}{2}{\left(({\left|{q}_{1}\right|}^{2}+{\left|{q}_{2}\right|}^{2}){q}_{2x}\right)}_{x}=0,\end{array}\end{eqnarray}$
were proposed [7], with ${q}_{1}$ and ${q}_{2}$ being complex-valued functions concerning $x$ and $t.$ Integrable properties, vector soliton solutions for equation (As we know, conservation laws is very fundamental not only in physics, but also in mathematics [19, 20]. Many methods have been developed to find conservation laws [21–34]. Recently, the adjoint equation has been applied to derive conservation laws in the nonlinear self-adjointness method and the symmetry/adjoint symmetry pair (SA) method [24, 25]. To our best knowledge, explicit conservation laws of (1 ) and (2 ) have not been reported. This paper is concerned with the conservation laws of (1 ) and (2 ) by the SA method.
The framework of the rest is shown below. In section 2 , we change equations (1 ) and (2 ) to real equations, then perform Lie symmetry analysis for the corresponding real equations. In section 3 , conservation laws of the real systems is derived using the SA method. The last section is devoted to some conclusions and discussions.
2. Symmetry analysis of equations (1 ) and (2 )
By taking 1 ) is transformed to
$\begin{eqnarray*}q=u+{\rm{I}}v,\end{eqnarray*}$
where ${{\rm{I}}}^{2}=-1,\,$ $u$ and $v$ are real dependent variables with respect to $x$ and $t,$ equation ( $\begin{eqnarray}\begin{array}{l}{u}_{xt}+u+u{{u}_{x}}^{2}+v{v}_{x}{u}_{x}+\displaystyle \frac{1}{2}{u}^{2}{u}_{xx}+\displaystyle \frac{1}{2}{v}^{2}{u}_{xx}=0,\,\,\,\\ {v}_{xt}+v+u{u}_{x}{v}_{x}+v{{v}_{x}}^{2}+\displaystyle \frac{1}{2}{u}^{2}{v}_{xx}+\displaystyle \frac{1}{2}{v}^{2}{v}_{xx}=0.\,\,\,\end{array}\end{eqnarray}$
Using the transformation $u=\displaystyle \frac{1}{2}\,(\hat{u}+\hat{v}),v\,=\displaystyle \frac{1}{2{\rm{I}}}\,(\hat{u}-\hat{v})$ and denoting $\hat{u}=u,\hat{v}=v$ (see (4.20) in [28]), equation (3 ) are transformed to4 ) instead of equation (1 ). By taking2 ) are changed to
$\begin{eqnarray}\begin{array}{l}{{\rm{\Delta }}}_{1}={u}_{xt}+u+\displaystyle \frac{1}{2}{(uv{u}_{x})}_{x}=0,\,\,\,\\ {{\rm{\Delta }}}_{2}={v}_{xt}+v+\displaystyle \frac{1}{2}{(uv{v}_{x})}_{x}=0.\,\,\,\end{array}\end{eqnarray}$
In the following, we will investigate equation ( $\begin{eqnarray*}{q}_{1}=u+{\rm{I}}v,\,\,{q}_{2}=p+{\rm{I}}q.\end{eqnarray*}$
Equation ( $\begin{eqnarray}\begin{array}{l}{G}_{1}=u{u}_{x}{v}_{x}+v{{v}_{x}}^{2}+p{p}_{x}{v}_{x}+q{q}_{x}{v}_{x}+v+{v}_{xt}\\ \,\,+\,\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){v}_{xx}=0,\\ {G}_{2}=u{{u}_{x}}^{2}+v{v}_{x}{u}_{x}+p{p}_{x}{u}_{x}+q{q}_{x}{u}_{x}+u+{u}_{xt}\\ \,\,+\,\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){u}_{xx}=0,\\ {G}_{3}=u{u}_{x}{q}_{x}+v{v}_{x}{q}_{x}+p{p}_{x}{q}_{x}+q{{q}_{x}}^{2}+q+{q}_{xt}\\ \,\,+\,\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){q}_{xx}=0,\\ {G}_{4}=u{u}_{x}{p}_{x}+v{v}_{x}{p}_{x}+p{{p}_{x}}^{2}+q{q}_{x}{p}_{x}+p+{p}_{xt}\\ \,\,+\,\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){p}_{xx}=0,\end{array}\end{eqnarray}$
where $u,v,\,p$ and $q$ are real dependent variables of $x$ and $t.$As stated by Noether, Lie symmetries have close relationships with conservation laws. Next, we will study Lie symmetries for equations (4 ) and (5 ) by the classical Lie symmetry approach. Suppose that the Lie symmetry of equation (4 ) is expressed as follows:6 ) is determined by4 ) can be obtained as follows
$\begin{eqnarray}V=X\displaystyle \frac{\partial }{\partial x}+T\displaystyle \frac{\partial }{\partial t}+U\displaystyle \frac{\partial }{\partial u}+V\displaystyle \frac{\partial }{\partial v},\end{eqnarray}$
where the coefficients $X,T,U$ and $V$ are undetermined functions with respect to $x,t,u$ and $v.$ According to procedures of the classical Lie symmetry approach, ( $\begin{eqnarray}\left\{\begin{array}{c}{\left.p{r}^{(2)}V({{\rm{\Delta }}}_{1})\right|}_{{{\rm{\Delta }}}_{1}=0,{{\rm{\Delta }}}_{2}=0}=0,\\ {\left.p{r}^{(2)}V({{\rm{\Delta }}}_{2})\right|}_{{{\rm{\Delta }}}_{1}=0,{{\rm{\Delta }}}_{2}=0}=0,\end{array}\right.\end{eqnarray}$
and $p{r}^{(2)}V$ denotes the second prolongation of $V,$ $\begin{eqnarray}\begin{array}{l}p{r}^{(2)}V=U\displaystyle \frac{\partial }{\partial u}+V\displaystyle \frac{\partial }{\partial v}+{\phi }^{x}\displaystyle \frac{\partial }{\partial {u}_{x}}+{\psi }^{x}\displaystyle \frac{\partial }{\partial {v}_{x}}\\ \,\,\,\,+\,{\phi }^{xx}\displaystyle \frac{\partial }{\partial {u}_{xx}}+{\phi }^{xt}\displaystyle \frac{\partial }{\partial {u}_{xt}}+{\psi }^{xx}\displaystyle \frac{\partial }{\partial {v}_{xx}}+{\psi }^{xt}\displaystyle \frac{\partial }{\partial {v}_{xt}},\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}{\phi }^{x}={D}_{x}(U-X{u}_{x}-T{u}_{t})+X{u}_{xx}+T{u}_{xt},\\ {\phi }^{xx}={D}_{xx}(U-X{u}_{x}-T{u}_{t})+X{u}_{xxx}+T{u}_{txx},\\ {\phi }^{xt}={D}_{xt}(U-X{u}_{x}-T{u}_{t})+X{u}_{xxt}+T{u}_{xtt},\\ {\psi }^{x}={D}_{x}(V-X{v}_{x}-T{v}_{t})+X{v}_{xx}+T{v}_{xt},\\ {\psi }^{xx}={D}_{xx}(V-X{v}_{x}-T{v}_{t})+X{v}_{xxx}+T{v}_{xxt},\\ {\psi }^{xt}={D}_{xt}(V-X{v}_{x}-T{v}_{t})+X{v}_{xxt}+T{v}_{xtt},\end{array}\end{eqnarray}$
where ${D}_{x}$ and ${D}_{t}$ are total differential operators. After calculations, the Lie symmetries for equation ( $\begin{eqnarray}\begin{array}{l}{V}_{1}=\displaystyle \frac{\partial }{\partial x},\,\,\,\,\,{V}_{2}=\displaystyle \frac{\partial }{\partial t},\,\,\,\,\,\,{V}_{3}=-x\displaystyle \frac{\partial }{\partial x}+t\displaystyle \frac{\partial }{\partial t}\\ \,\,\,\,\,\,-\,2v\displaystyle \frac{\partial }{\partial v},\,\,\,{V}_{4}=u\displaystyle \frac{\partial }{\partial u}-v\displaystyle \frac{\partial }{\partial v}.\end{array}\end{eqnarray}$
The Lie symmetries for equation (4 ) are new, and they can be used to discuss the corresponding symmetry groups, similarity reductions and group invariant solutions of equation (4 ). Similar work has been done in [32], so here we omit it.
By the same way with equation (4 ), we can get the Lie symmetries of equation (5 )
$\begin{eqnarray}\begin{array}{l}{V}_{1}=\displaystyle \frac{\partial }{\partial x},\,{V}_{2}=\displaystyle \frac{\partial }{\partial t},\,{V}_{3}=-x\displaystyle \frac{\partial }{\partial x}+t\displaystyle \frac{\partial }{\partial t}-u\displaystyle \frac{\partial }{\partial u}\\ \,-\,v\displaystyle \frac{\partial }{\partial v}-p\displaystyle \frac{\partial }{\partial p}-q\displaystyle \frac{\partial }{\partial q},\,{V}_{4}=q\displaystyle \frac{\partial }{\partial p}-p\displaystyle \frac{\partial }{\partial q},\,\\ {V}_{5}=-p\displaystyle \frac{\partial }{\partial u}+u\displaystyle \frac{\partial }{\partial p},{V}_{6}=-q\displaystyle \frac{\partial }{\partial u}+u\displaystyle \frac{\partial }{\partial q},\,{V}_{7}=-q\displaystyle \frac{\partial }{\partial v}\\ \,+\,v\displaystyle \frac{\partial }{\partial q},\,\,{V}_{8}=v\displaystyle \frac{\partial }{\partial u}-u\displaystyle \frac{\partial }{\partial v},\,\,\,{V}_{9}=-p\displaystyle \frac{\partial }{\partial v}+v\displaystyle \frac{\partial }{\partial p}.\end{array}\end{eqnarray}$
In [18], equation (2 ) have been turned into the same real systems as equation (5 ). They also studied the Lie symmetries of equation (5 ), but their results are particular cases of (11 ).
3. Conservation laws of equations (4 ) and (5 )
3.1. Conservation laws of equation (4 )
According to [25, 26], one obtains the linearizing operator of equation (4 )
$\begin{eqnarray}\begin{array}{l}L\,=\,\left[\begin{array}{lr}\left(1+\displaystyle \frac{1}{2}{u}_{x}{v}_{x}+\displaystyle \frac{1}{2}v{u}_{xx}\right)+\left({u}_{x}v+\displaystyle \frac{1}{2}u{v}_{x}\right){D}_{x}+\displaystyle \frac{1}{2}uv{D}_{xx}+{D}_{xt} & \left(\displaystyle \frac{1}{2}{{u}_{x}}^{2}+\displaystyle \frac{1}{2}u{u}_{xx}\right)+\displaystyle \frac{1}{2}u{u}_{x}{D}_{x}\\ \left(\displaystyle \frac{1}{2}{{v}_{x}}^{2}+\displaystyle \frac{1}{2}v{v}_{xx}\right)+\displaystyle \frac{1}{2}{v}_{x}v{D}_{x} & \left(1+\displaystyle \frac{1}{2}{u}_{x}{v}_{x}+\displaystyle \frac{1}{2}u{v}_{xx}\right)+\displaystyle \frac{1}{2}uv{D}_{xx}+\left(u{v}_{x}+\displaystyle \frac{1}{2}{u}_{x}v\right){D}_{x}+{D}_{xt}\end{array}\right]\,,\end{array}\end{eqnarray}$
and the adjoint linearizing operator $\begin{eqnarray}{L}^{* }\,=\,\left[\begin{array}{cc}1+\displaystyle \frac{1}{2}u{v}_{x}{D}_{x}+\displaystyle \frac{1}{2}uv{D}_{xx}+{D}_{xt} & -\displaystyle \frac{1}{2}v{v}_{x}{D}_{x}\\ -\displaystyle \frac{1}{2}u{u}_{x}{D}_{x} & 1+\displaystyle \frac{1}{2}{u}_{x}v{D}_{x}+\displaystyle \frac{1}{2}uv{D}_{xx}+{D}_{xt}\end{array}\right]\,.\end{eqnarray}$
Based on the symmetries (10 ) of equation (4 ), the symmetry components ${\hat{\eta }}^{\rho }$ of equation (4 ) are given by
$\begin{eqnarray}\begin{array}{l}{\hat{\eta }}^{1}=({\hat{\eta }}_{1}^{1},{\hat{\eta }}_{2}^{1})=(-{u}_{x},-{v}_{x}),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\hat{\eta }}^{2}=({\hat{\eta }}_{1}^{2},{\hat{\eta }}_{2}^{2})=(-{u}_{t},-{v}_{t}),\\ {\hat{\eta }}^{3}=({\hat{\eta }}_{1}^{3},{\hat{\eta }}_{2}^{3})=(x{u}_{x}-t{u}_{t},-2v+x{v}_{x}-t{v}_{t}),\,\,{\hat{\eta }}^{4}=({\hat{\eta }}_{1}^{4},{\hat{\eta }}_{2}^{4})=(u,-v).\end{array}\end{eqnarray}$
From (13 ), one obtains the following adjoint linearizing system15 ) is4 ) and we take $C=1$ in the calculation of conservation laws.
$\begin{eqnarray}\begin{array}{l}{L}^{* }{\omega }_{\sigma }=\,\left[\begin{array}{cc}1+\displaystyle \frac{1}{2}u{v}_{x}{D}_{x}+\displaystyle \frac{1}{2}uv{D}_{xx}+{D}_{xt} & -\displaystyle \frac{1}{2}v{v}_{x}{D}_{x}\\ -\displaystyle \frac{1}{2}u{u}_{x}{D}_{x} & 1+\displaystyle \frac{1}{2}{u}_{x}v{D}_{x}+\displaystyle \frac{1}{2}uv{D}_{xx}+{D}_{xt}\end{array}\right]\left[\begin{array}{c}{\omega }^{1}\\ {\omega }^{2}\end{array}\right]\,=\,\left[\begin{array}{c}0\\ 0\end{array}\right],\end{array}\end{eqnarray}$
with ${\omega }^{1}$ and ${\omega }^{2}$ are functions of $x,t,u$ and $v.$ After complicated calculations, we find that the solution to ( $\begin{eqnarray}({\omega }^{1},{\omega }^{2})=(Cv,-Cu),\end{eqnarray}$
where $C$ is a constant. According to [25, 26], this is the adjoint symmetry of equation (Substituting ${\hat{\eta }}^{1}=({\hat{\eta }}_{1}^{1},{\hat{\eta }}_{2}^{1})$ in (14 ) and $({\omega }^{1},{\omega }^{2})=(v,-u)$ into the conservation laws identity of Theorem 1 in [25], one can obtain the following conservation laws of equation (4 ) with respect to ${V}_{1}$
$\begin{eqnarray}\begin{array}{l}{X}_{1}=-\displaystyle \frac{1}{2}{{u}_{x}}^{2}{v}^{2}-\displaystyle \frac{1}{2}u{v}^{2}{u}_{xx}+\displaystyle \frac{1}{2}{{v}_{x}}^{2}{u}^{2}+\displaystyle \frac{1}{2}{u}^{2}v{v}_{xx}\\ \,\,\,+\,{u}_{x}{v}_{t}-{v}_{x}{u}_{t},\,{T}_{1}=-{u}_{xx}v+{v}_{xx}u.\end{array}\end{eqnarray}$
Using other pairs ${\hat{\eta }}^{i}\,(i=2,3,4)$ in (14 ) and $({\omega }^{1},{\omega }^{2})=(v,-u),$ one can obtain other three conservation laws of equation (4 )
$\begin{eqnarray}\begin{array}{l}{X}_{2}=-\displaystyle \frac{1}{2}{u}_{x}{u}_{t}{v}^{2}+\displaystyle \frac{1}{2}{v}_{x}{v}_{t}{u}^{2}-\displaystyle \frac{1}{2}u{v}^{2}{u}_{xt}+\displaystyle \frac{1}{2}{u}^{2}v{v}_{xt}\\ \,\,\,\,+\,uv{u}_{t}{v}_{x}-uv{u}_{x}{v}_{t},\,{T}_{2}=-{u}_{xt}v+{v}_{xt}u,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{X}_{3}=\displaystyle \frac{1}{2}x{v}^{2}{{u}_{x}}^{2}+\displaystyle \frac{3}{2}{u}^{2}v{v}_{x}-\displaystyle \frac{1}{2}x{u}^{2}{{v}_{x}}^{2}-\displaystyle \frac{3}{2}u{v}^{2}{u}_{x}\\ \,\,-\,2v{u}_{t}+x{v}_{x}{u}_{t}-x{u}_{x}{v}_{t}-\displaystyle \frac{1}{2}t{v}^{2}{u}_{x}{u}_{t}+\displaystyle \frac{1}{2}t{u}^{2}{v}_{x}{v}_{t}\\ \,\,+\,\displaystyle \frac{1}{2}xu{v}^{2}{u}_{xx}-\displaystyle \frac{1}{2}tu{v}^{2}{u}_{xt}-\displaystyle \frac{1}{2}x{u}^{2}v{v}_{xx}\\ \,\,+\,\displaystyle \frac{1}{2}t{u}^{2}v{v}_{xt}+tuv{u}_{t}{v}_{x}-tuv{u}_{x}{v}_{t},\,{T}_{3}={u}_{x}v\\ \,\,+\,xv{u}_{xx}-vt{u}_{xt}+u{v}_{x}-xu{v}_{xx}+tu{v}_{xt},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{X}_{4}=-u{v}_{t}-v{u}_{t},\\ {T}_{4}={u}_{x}v+{v}_{x}u.\end{array}\end{eqnarray}$
The validity of the above conservation laws has been checked by Maple. According to [33], there are two different types of trivial conservation laws. In the first type the divergence of the corresponding conserved vector vanishes identically. The conservation laws $({X}_{4},{T}_{4})$ belongs to the first type and is trivial, since ${D}_{x}{X}_{4}+{D}_{t}{T}_{4}\equiv 0.$ The second type of trivial conservation laws are provided by those conserved vectors whose components vanish on the solutions of the considered systems of differential equations. The conservation laws $({X}_{1},{T}_{1})$ and $({X}_{2},{T}_{2})$ belongs to the second type and is also trivial. For example,
$\begin{eqnarray*}\begin{array}{l}{D}_{x}{X}_{1}+{D}_{t}{T}_{1}={D}_{x}{X}_{1}+{D}_{t}{D}_{x}({v}_{x}u-{u}_{x}v)\\ \,=\,{D}_{x}\left({X}_{1}+{D}_{t}({v}_{x}u-{u}_{x}v)\right)+{D}_{t}0,\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{l}{\left.\left({X}_{1}+{D}_{t}({v}_{x}u-{u}_{x}v)\right)\right|}_{{{\rm{\Delta }}}_{1}=0,{{\rm{\Delta }}}_{2}=0}=\\ {\left.\left(-\displaystyle \frac{1}{2}{{u}_{x}}^{2}{v}^{2}-\displaystyle \frac{1}{2}u{v}^{2}{u}_{xx}-{u}_{xt}v+\displaystyle \frac{1}{2}{{v}_{x}}^{2}{u}^{2}+\displaystyle \frac{1}{2}{u}^{2}v{v}_{xx}+{v}_{xt}u\right)\right|}_{{{\rm{\Delta }}}_{1}=0,{{\rm{\Delta }}}_{2}=0}\\ \,=\,0.\end{array}\end{eqnarray*}$
Therefore, $({X}_{1},{T}_{1})$ belongs to the second type of trivialness, so is $({X}_{2},{T}_{2}).$ The conservation laws $({X}_{3},{T}_{3})$ is nontrivial and can be simplified to $\begin{eqnarray*}\begin{array}{l}{X}_{3}={u}^{2}v{v}_{x}-u{v}^{2}{u}_{x}-2v{u}_{t},\\ {T}_{3}=2u{v}_{x}.\end{array}\end{eqnarray*}$
By means of the above nontrivial conservation laws, equation ( $\begin{eqnarray*}\begin{array}{l}{D}_{x}({u}^{2}v{v}_{x}-u{v}^{2}{u}_{x}-2v{u}_{t})\\ \,+\,{D}_{t}(2u{v}_{x})=-2v{{\rm{\Delta }}}_{1}+2u{{\rm{\Delta }}}_{2}=0.\end{array}\end{eqnarray*}$
The conservation form of (From (
$\begin{eqnarray}\begin{array}{l}{{{\rm{\Delta }}}_{1}}^{{\prime} }={\omega }^{1}+{\omega }_{xt}^{1}+\displaystyle \frac{1}{2}u{v}_{x}{\omega }_{x}^{1}-\displaystyle \frac{1}{2}v{v}_{x}{\omega }_{x}^{2}+\displaystyle \frac{1}{2}uv{\omega }_{xx}^{1}=0,\\ {{{\rm{\Delta }}}_{2}}^{{\prime} }={\omega }^{2}+{\omega }_{xt}^{2}-\displaystyle \frac{1}{2}u{u}_{x}{\omega }_{x}^{1}+\displaystyle \frac{1}{2}{u}_{x}v{\omega }_{x}^{2}+\displaystyle \frac{1}{2}uv{\omega }_{xx}^{2}=0.\end{array}\end{eqnarray}$
Taking the solution $({\omega }^{1},{\omega }^{2})=(Cv,-Cu)$ into equation ( $\begin{eqnarray*}\begin{array}{l}{\left.{{{\rm{\Delta }}}_{1}}^{^{\prime} }\right|}_{{\omega }^{1}=Cv,{\omega }^{2}=-Cu}=C{{\rm{\Delta }}}_{{\rm{2}}},\\ {\left.{{{\rm{\Delta }}}_{2}}^{^{\prime} }\right|}_{{\omega }^{1}=Cv,{\omega }^{2}=-Cu}=-C{{\rm{\Delta }}}_{{\rm{1}}}.\end{array}\end{eqnarray*}$
According to the definition of quasi-self adjointness [29], we can conclude that equation (3.2. Conservation laws of equation (5 )
According to [25, 26], one obtains the linearizing operator of equation (5 )
$L=\left[\begin{array}{cccc}{(u{v}_{x})}_{x}+u{v}_{x}{D}_{x} & \begin{array}{l}({{v}_{x}}^{2}+v{v}_{xx}+1)+\\ (R+v{v}_{x}){D}_{x}+H\end{array} & {(p{v}_{x})}_{x}+p{v}_{x}{D}_{x} & {(q{v}_{x})}_{x}+q{v}_{x}{D}_{x}\\ \begin{array}{c}({{u}_{x}}^{2}+u{u}_{xx}+1)+\\ (R+u{u}_{x}){D}_{x}+H\end{array} & {({u}_{x}v)}_{x}+{u}_{x}v{D}_{x} & {({u}_{x}p)}_{x}+{u}_{x}p{D}_{x} & {({u}_{x}q)}_{x}+{u}_{x}q{D}_{x}\\ {({q}_{x}u)}_{x}+{q}_{x}u{D}_{x} & {({q}_{x}v)}_{x}+{q}_{x}v{D}_{x} & {({q}_{x}p)}_{x}+{q}_{x}p{D}_{x} & \begin{array}{l}({{p}_{x}}^{2}+p{p}_{xx}+1)+\\ (R+p{p}_{x}){D}_{x}+H\end{array}\\ {({p}_{x}u)}_{x}+{p}_{x}u{D}_{x} & {({p}_{x}v)}_{x}+{p}_{x}v{D}_{x} & \begin{array}{l}({{p}_{x}}^{2}+p{p}_{xx}+1)+\\ (R+p{p}_{x}){D}_{x}+H\end{array} & {({p}_{x}q)}_{x}+{p}_{x}q{D}_{x}\end{array}\right],$
and the adjoint linearizing operator $\begin{eqnarray}{L}^{* }=\left[\begin{array}{cccc}-u{v}_{x}{D}_{x} & 1+(R-u{u}_{x}){D}_{x}+H & -u{q}_{x}{D}_{x} & -u{p}_{x}{D}_{x}\\ 1+(R-v{v}_{x}){D}_{x}+H & -v{u}_{x}{D}_{x} & -v{q}_{x}{D}_{x} & -v{p}_{x}{D}_{x}\\ -p{v}_{x}{D}_{x} & -p{u}_{x}{D}_{x} & -p{q}_{x}{D}_{x} & 1+(R-p{p}_{x}){D}_{x}+H\\ -q{v}_{x}{D}_{x} & -q{u}_{x}{D}_{x} & 1+(R-q{q}_{x}){D}_{x}+H & -q{p}_{x}{D}_{x}\end{array}\right],\end{eqnarray}$
with $R=u{u}_{x}+v{v}_{x}+p{p}_{x}+q{q}_{x},S= \frac{1}{2}\left({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}\right), \\ \,H=S{D}_{xx}+{D}_{xt}.$
From (11 ), there are nine symmetry components of equation (5 ), they are given by
$\begin{eqnarray}\begin{array}{l}{\hat{\eta }}^{1}=({\hat{\eta }}_{1}^{1},{\hat{\eta }}_{2}^{1},{\hat{\eta }}_{3}^{1},{\hat{\eta }}_{4}^{1})=(-{u}_{x},-{v}_{x},-{p}_{x},-{q}_{x}),\,\,\\ {\hat{\eta }}^{2}=(-{u}_{t},-{v}_{t},-{p}_{t},-{q}_{t}),\\ {\hat{\eta }}^{3}=\left(-u+x{u}_{x}-t{u}_{t},-v+x{v}_{x}-t{v}_{t},-p+x{p}_{x}\right.\\ \,\,\,\,\,\,\left.-\,t{p}_{t},-q+x{q}_{x}-t{q}_{t}\right),\,\,\\ {\hat{\eta }}^{4}=(0,0,q,-p),\,\,\,{\hat{\eta }}^{5}=(-p,0,u,0),\\ {\hat{\eta }}^{6}=(-q,0,0,u),\,\,\,{\hat{\eta }}^{7}=(0,-q,0,v),\\ {\hat{\eta }}^{8}=(v,-u,0,0),\,\,\,\,{\hat{\eta }}^{9}=(0,-p,v,0).\end{array}\end{eqnarray}$
From (23 ), one obtains the following adjoint linearizing system25 ) is5 ) and we take $C=1$ in the following for simplicity.
$\begin{eqnarray}{L}^{* }{\omega }_{\sigma }=\left[\begin{array}{cccc}-u{v}_{x}{D}_{x} & 1+(R-u{u}_{x}){D}_{x}+H & -u{q}_{x}{D}_{x} & -u{p}_{x}{D}_{x}\\ 1+(R-v{v}_{x}){D}_{x}+H & -v{u}_{x}{D}_{x} & -v{q}_{x}{D}_{x} & -v{p}_{x}{D}_{x}\\ -p{v}_{x}{D}_{x} & -p{u}_{x}{D}_{x} & -p{q}_{x}{D}_{x} & 1+(R-p{p}_{x}){D}_{x}+H\\ -q{v}_{x}{D}_{x} & -q{u}_{x}{D}_{x} & 1+(R-q{q}_{x}){D}_{x}+H & -q{p}_{x}{D}_{x}\end{array}\right]\left[\begin{array}{c}{\omega }^{1}\\ {\omega }^{2}\\ {\omega }^{3}\\ {\omega }^{4}\end{array}\right]\,=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],\end{eqnarray}$
with ${\omega }^{i}\,(i=1,2,3,4)$ being functions of $x,t,u,v,p$ and $q.$ The solution to ( $\begin{eqnarray}({\omega }^{1},{\omega }^{2},{\omega }^{3},{\omega }^{4})=(-Cu,Cv,0,0),\end{eqnarray}$
where $C$ is a constant. This is the adjoint symmetry of equation (Substituting ${\hat{\eta }}^{1}=({\hat{\eta }}_{1}^{1},{\hat{\eta }}_{2}^{1},{\hat{\eta }}_{3}^{1},{\hat{\eta }}_{4}^{1})$ in (24 ) and $({\omega }^{1},{\omega }^{2},{\omega }^{3},{\omega }^{4})$ in (26 ) into the conservation laws identity of Theorem 1 in [26], one can obtain the following conservation laws of equation (5 ) with respect to ${V}_{1}$ 24 ) and $({\omega }^{1},{\omega }^{2},{\omega }^{3},{\omega }^{4})$ in (26 ), one can obtain other conservation laws of equation (5 ) with respect to ${V}_{i}\,\,(i=2,3,\,{K}\ldots ,9)$
$\begin{eqnarray*}\begin{array}{l}{X}_{1}=-{u}_{t}{v}_{x}-{u}_{x}vp{p}_{x}-{u}_{x}vq{q}_{x}+u{v}_{x}p{p}_{x}\\ \,\,\,\,+u{v}_{x}q{q}_{x}+uv{{v}_{x}}^{2}+{u}^{2}{u}_{x}{v}_{x}-u{{u}_{x}}^{2}v-{u}_{x}{v}^{2}{v}_{x}\\ \,\,\,\,-\,\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){u}_{xx}v\\ \,\,\,\,+\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){v}_{xx}u+{u}_{x}{v}_{t},\\ {T}_{1}=-{u}_{xx}v+{v}_{xx}u.\end{array}\end{eqnarray*}$
Substituting other ${\hat{\eta }}^{i}\,(i=2,3,\,{K}\ldots ,9)$ in ( $\begin{eqnarray*}\begin{array}{l}{X}_{2}=-u{u}_{x}{u}_{t}v+u{v}_{x}{v}_{t}v+u{v}_{x}p{p}_{t}-{u}_{x}vp{p}_{t}+u{v}_{x}q{q}_{t}\\ \,\,\,\,-\,{u}_{x}vq{q}_{t}-\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){u}_{xt}v\\ \,\,+\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2})u{v}_{xt}-\displaystyle \frac{1}{2}({p}^{2}+{q}^{2}+{u}^{2}+3{v}^{2})\\ \,\,\times \,{u}_{x}{v}_{t}+\displaystyle \frac{1}{2}({p}^{2}+{q}^{2}+{v}^{2}+3{u}^{2}){u}_{t}{v}_{x},\,\\ {T}_{2}=-{u}_{xt}v+{v}_{xt}u,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{X}_{3}=\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){\left(tu{v}_{xt}-tv{u}_{xt}+xvu\right.}_{xx}\\ \left.\,\,\,\,-\,xu{v}_{xx}+3u{v}_{x}-3v{u}_{x}\right)-x{u}^{2}{u}_{x}{v}_{x}+xu{{u}_{x}}^{2}v\\ \,\,\,\,+\,\displaystyle \frac{t}{2}{u}_{t}{v}_{x}(3{u}^{2}+{v}^{2}+{p}^{2}+{q}^{2})-\displaystyle \frac{t}{2}{v}_{t}{u}_{x}\\ \,\,\,\,\times \,({u}^{2}+3{v}^{2}+{p}^{2}+{q}^{2})+x{u}_{x}{v}^{2}{v}_{x}-xuv{{v}_{x}}^{2}\\ \,\,\,\,-x{u}_{x}{v}_{t}+\,x{u}_{t}{v}_{x}+x{u}_{x}vq{q}_{x}-tu{u}_{x}{u}_{t}v-xu{v}_{x}p{p}_{x}\\ \,\,\,\,-\,xu{v}_{x}q{q}_{x}+tuv{v}_{x}{v}_{t}+tu{v}_{x}p{p}_{t}-t{u}_{x}vp{p}_{t}\\ \,\,\,\,+\,tu{v}_{x}q{q}_{t}-t{u}_{x}vq{q}_{t}+{v}_{t}u-{u}_{t}v+x{u}_{x}vp{p}_{x},\\ {T}_{3}=xv{u}_{xx}-tv{u}_{xt}-xu{v}_{xx}+tu{v}_{xt},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{X}_{4}=0,\\ {T}_{4}=0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{X}_{5}=\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2})({v}_{x}p-{p}_{x}v)+p{v}_{t},\\ {T}_{5}=-v{p}_{x},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{X}_{6}=\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2})({v}_{x}q-{q}_{x}v)+q{v}_{t},\\ {T}_{6}=-v{q}_{x},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{X}_{7}=\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2})({q}_{x}u-{u}_{x}q)-q{u}_{t},\\ {T}_{7}=u{q}_{x},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{X}_{8}=u{u}_{t}-v{v}_{t},\\ {T}_{8}=v{u}_{x}+u{u}_{x},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{X}_{9}=\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2})(u{p}_{x}-{u}_{x}p)-p{u}_{t},\\ {T}_{9}=u{p}_{x}.\end{array}\end{eqnarray*}$
According to [33], $({X}_{4},{T}_{4})$ and $({X}_{8},{T}_{8})$ is trivial and belongs to the first type, since
$\begin{eqnarray*}{D}_{x}{X}_{4}+{D}_{t}{T}_{4}\equiv 0,\,{D}_{x}{X}_{8}+{D}_{t}{T}_{8}\equiv 0.\end{eqnarray*}$
$({X}_{1},{T}_{1})$ and $({X}_{2},{T}_{2})$ is trivial and belongs to the second type, for example $\begin{eqnarray*}\begin{array}{l}{D}_{x}{X}_{1}+{D}_{t}{T}_{1}={D}_{x}{X}_{1}+{D}_{t}{D}_{x}({v}_{x}u-{u}_{x}v)\\ \,=\,{D}_{x}\left({X}_{1}+{D}_{t}({v}_{x}u-{u}_{x}v)\right)+{D}_{t}0\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}{\left.\left({X}_{1}+{D}_{t}({v}_{x}u-{u}_{x}v)\right)\right|}_{{G}_{1}=0,{G}_{2}=0,{G}_{3}=0,{G}_{4}=0}=0.\end{eqnarray*}$
Other conservation laws for equation $\begin{eqnarray*}\begin{array}{l}{X}_{3}=({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2})(u{v}_{x}-v{u}_{x})+u{v}_{t}-v{u}_{t},\\ {T}_{3}=u{v}_{x}-v{u}_{x}.\end{array}\end{eqnarray*}$
Using the nontrivial conservation laws, equation (Similar to the case in Remark 2, we can directly obtain the adjoint equations of equation (
$\begin{eqnarray}\begin{array}{l}{{G}_{1}}^{{\prime} }=-u{v}_{x}{\omega }_{x}^{1}+{\omega }^{2}+(v{v}_{x}+p{p}_{x}+q{q}_{x}){\omega }_{x}^{2}\\ \,+\,\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){\omega }_{xx}^{2}+{\omega }_{xt}^{2}-u{q}_{x}{\omega }_{x}^{3}\\ -u{p}_{x}{\omega }_{x}^{4}=0,\,\\ {{G}_{2}}^{{\prime} }={\omega }^{1}+(u{u}_{x}+p{p}_{x}+q{q}_{x}){\omega }_{x}^{1}\\ +\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2})\,{\omega }_{xx}^{1}+{\omega }_{xt}^{1}-v{u}_{x}{\omega }_{x}^{2}\\ -v{q}_{x}{\omega }_{x}^{3}-v{p}_{x}{\omega }_{x}^{4}=0,\,\\ {{G}_{3}}^{{\prime} }=-p{v}_{x}{\omega }_{x}^{1}-p{u}_{x}{\omega }_{x}^{2}\\ -p{q}_{x}{\omega }_{x}^{3}+{\omega }^{4}+(u{u}_{x}+v{v}_{x}+q{q}_{x}){\omega }_{x}^{4}\\ +\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){\omega }_{xx}^{4}+{\omega }_{xt}^{4}=0,\\ {{G}_{4}}^{{\prime} }=-q{v}_{x}{\omega }_{x}^{1}-q{u}_{x}{\omega }_{x}^{2}+{\omega }^{3}+(u{u}_{x}+v{v}_{x}+p{p}_{x}){\omega }_{x}^{3}\\ \,+\,\displaystyle \frac{1}{2}({u}^{2}+{v}^{2}+{p}^{2}+{q}^{2}){\omega }_{xx}^{3}+{\omega }_{xt}^{3}-q{p}_{x}{\omega }_{x}^{4}=0.\end{array}\end{eqnarray}$
Taking the solution $({\omega }^{1},{\omega }^{2},{\omega }^{3},{\omega }^{4})=(-Cu,Cv,0,0)$ into equation ( $\begin{eqnarray*}\begin{array}{l}{\left.{{G}_{1}}^{^{\prime} }\right|}_{{\omega }^{1}=-Cu,{\omega }^{2}=Cv,{\omega }^{3}=0,{\omega }^{4}=0}=C{G}_{{\rm{1}}},\\ {\left.{{G}_{2}}^{^{\prime} }\right|}_{{\omega }^{1}=-Cu,{\omega }^{2}=Cv,{\omega }^{3}=0,{\omega }^{4}=0}=-C{G}_{2},\\ {\left.{{G}_{3}}^{^{\prime} }\right|}_{{\omega }^{1}=-Cu,{\omega }^{2}=Cv,{\omega }^{3}=0,{\omega }^{4}=0}=0,\\ {\left.{{G}_{4}}^{^{\prime} }\right|}_{{\omega }^{1}=-Cu,{\omega }^{2}=Cv,{\omega }^{3}=0,{\omega }^{4}=0}=0.\end{array}\end{eqnarray*}$
According to the definition of quasi-self adjointness [29], we know equation (From the definition of adjoint symmetry, we can conclude that the determination of the self-adjointness and nonlinear self-adjointness (including quasi self-adjointness, weak self-adjointness, see [27, 30, 31]) in the nonlinear self-adjointness method can be determined by the adjoint symmetry in the SA method.
4. Conclusions and discussions
Recently, the CSP equation and CCSP equations have been proposed and attracted the interest of many researchers. The two equations can be regarded as the NLS equation and coupled NLS equations in the scale of attoseconds. In this paper, explicit conservation laws for equations (4 ) and (5 ) is obtained by the SA method. The relationships between nonlinear self-adjointness method and the SA method are investigated. The determination of the self-adjointness and nonlinear self-adjointness is very essential in the nonlinear self-adjointness method, and we find that they can be determined by the adjoint symmetry in the SA method. Also, we find that the conservation laws of equations (4 ) and (5 ) derived from SA method is all local. Using the derived nontrivial conservation laws, equations (4 ) and (5 ) can be written in conservation form. The conservation form can be employed to derive exact solutions and develop numerical and analytical methods [34].