Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures

1.   Introduction

In the study of differential equations in the past, the conservation law is undoubtedly a very important part, especially in terms of integrability and linearization. How to solve the conservation law of the equation has become a first problem to be faced ^[1,2,3,4,5]. Hence in this paper, we select three different types of differential equations and apply two different methods to solve the conservation laws of the partial differential equations.

The first method of calculating conservation laws does not require the equation to have variational conditions ^[6,7,8,9]. Its principle is to replace symmetry with the adjoint symmetry of a partial differential equation. Then the invariance condition on symmetry is replaced by the invariance condition on adjoint symmetry, and there is a direct explicit formula to calculate the multiplier and obtain the corresponding conservation law. In the calculation process, the adjoint invariance condition is replaced by an additional deterministic equation, and the determinant equation of the adjoint equation is expanded by writing an additional equation, thereby obtaining a linear deterministic equation system. If the adjoint symmetry of the equation is solved by this system, the multiplier that can be used to solve the conservation law is found.

The second part is to use a method to solve the conservation law of partial differential equations, which is actually a derivation of Norther's theorem ^{[10,11,12,13,14]}. In theory, Any Lie point, Lie-Bäcklund, and nonlocal symmetry can derive the corresponding conservation laws. In the last part of this paper, we briefly analyze the Hamiltonian structures and adjoint symmetries ^{[15,16,17,18,19]} of the selected equations, and calculate their linear soliton solutions by traveling wave transformation ^[20,21].

In this paper, we apply these methods to three nonlinear partial differential equations: the combined KdV-MKdV equation ^{[22,23,24,25,26]}, the Klein-Gordon equation ^[27] and the generalized coupled KdV equations ^[28,29].

2.   Brief introduction to the method of directly constructing conservation law

First we consider a system of partial differential equations with $N$ independent variables $\boldsymbol{u} = (u^{1}, \dots, u^{N})$ and $n+1$ independent variables $(t, \boldsymbol{x})$, the forms are as follows:

with $\boldsymbol{x}$ derivatives of $\boldsymbol{u}$ up to some order $m$. And we can use $\partial_{\boldsymbol{x}}\boldsymbol{u}, \partial_{\boldsymbol{x}}^{2}\boldsymbol{u}$, etc. to represent all the derivatives of $u^{j}$ with respect to ${x^{i}}$. We denote partial derivatives $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial x^{i}}$ with $\boldsymbol{x}$ derivatives of $\boldsymbol{u}$ up to some order $m$. We denote partial derivatives $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial x^{i}}$ by subscripts $t$ and $i$ respectively. And likewise, $D_{t}$ and $D_{i}$ represent the total derivatives with respect to $x^{i}$ and $t$. We set

and we can let $(\mathcal{L}^{*}_{g})$ denote the adjoint operator defined by

acting on arbitrary functions $V^{\alpha}, W^{j}$ respectively. It is well known that the determining equation of the system (2.1) with respect to symmetric $X = \eta^{j}$$\frac{\partial}{\partial u^{j}}$ is

for all solutions $\boldsymbol{u}(t, \boldsymbol{x})$ of Eqs. (2.1). The above decision equation can also be used to solve the higher order symmetry of the Eqs. (2.1), such as $\eta^{j}(t, \boldsymbol{x}, \boldsymbol{u}, \partial\boldsymbol{u}, \dots, \partial^{p}\boldsymbol{u})$, where $\partial^{k}\boldsymbol{u}$ denotes all $k$th order derivatives of $\boldsymbol{u}$ with respect to all independent variables $t, \boldsymbol{x}$. And the adjoint of Eq. (2.4) is given by

which is the determining equation for the adjoint symmetries $\omega_{j}(t, \boldsymbol{x}, \boldsymbol{u}, \partial\boldsymbol{u}, \dots, \partial^{p}\boldsymbol{u})$ of the Eqs. (2.1). In general, solutions of the adjoint symmetry Eq. (2.5) are not solutions of the symmetry Eq. (2.4), and there is no interpretation of adjoint symmetries in terms of an infinitesimal generator leaving anything invariant. Then Let

which is the total derivative with respect to $t$ on the solution space of Eqs. (2.1) (In particular, $D_{t} = \mathcal{D}_{t}$ when acting on all solutions $\boldsymbol{u}(t, \boldsymbol{x}).$) Then the determining equations explicitly become

for $\eta^{j}(t, \boldsymbol{x}, \boldsymbol{u}, \partial_{\boldsymbol{x}}\boldsymbol{u}, \dots, \partial^{p}_{\boldsymbol{x}}\boldsymbol{u})$ and

for $\omega_{j}(t, \boldsymbol{x}, \boldsymbol{u}, \partial_{\boldsymbol{x}}\boldsymbol{u}, \dots, \partial^{p}_{\boldsymbol{x}}\boldsymbol{u}).$ The solutions of Eqs. (2.7) and (2.8) yield all symmetries and adjoint symmetries up to any given order $p$.

Definition 1 A local conservation law of PDE system (2.1) is a divergence expression

for all solutions $u(t, \boldsymbol{x})$ of Eqs. (2.1); $\phi^{t}$ and $\phi^{i}$ are called the conserved densities.

Definition 2 Multipliers for PDE system (2.1) are a set of expressions

satisfying

for some expression $\phi^{t}(t, \boldsymbol{x}, \boldsymbol{u}, \partial_{\boldsymbol{x}}\boldsymbol{u}, \dots, \partial^{k}_{\boldsymbol{x}}\boldsymbol{u})$ and $\phi^{i}(t, \boldsymbol{x}, \boldsymbol{u}, \partial_{\boldsymbol{x}}\boldsymbol{u}, \dots, \partial^{k}_{\boldsymbol{x}}\boldsymbol{u})$ for all functions $\boldsymbol{u}(t, \boldsymbol{x}).$

Obviously $\phi^{t} = D_{i}\theta^{i}$ and $\phi^{i} = -D_{t}\theta^{i}+D_{j}\psi^{ij}$ are trivial conservation laws of the system, for some erpressions $\theta^{i}(t, \boldsymbol{x}, \boldsymbol{u}, \partial_{\boldsymbol{x}}\boldsymbol{u}, \dots, \partial^{k-1}_{\boldsymbol{x}}\boldsymbol{u})$ and $\psi^{ij}(t, \boldsymbol{x}, \boldsymbol{u}, \partial_{\boldsymbol{x}}\boldsymbol{u}, \dots, \partial^{q}_{\boldsymbol{x}}\boldsymbol{u})$ with $\psi^{ij} = -\psi^{ji}$, i.e. satisfying

The purpose of the following article is to calculate the corresponding conservation law multipliers based on differential equations. Next we consider $D_{t}\phi^{t}+D_{i}\phi^{i}$ and it is well konwn that

where $(\mathcal{L}_{\phi^{t}})_{j} = (\frac{\partial \phi^{t}}{\partial u^{j}})+(\frac{\partial \phi^{t}}{\partial u^{j}_{t}})D_{i}+\cdots+(\frac{\partial \phi^{t}}{\partial u^{j}_{i_{1}\cdots i_{k}}}D_{i_{1}\cdots D_{i_{k}}})$ denotes the linearization operator of $\phi^{t}$. Then we can obtain

where $\Gamma^{i}$ is given by an expression proportional to $u_{t}^{j}+g^{j}$ and where

is a restricted Euler operator. So we have

Then in order to ensure that the Eq. (2.10) holds, $\partial_{t}\phi^{t}-(\mathcal{L}_{\phi^{t}})_{j}g^{j}$ which involes $u_{t}^{j}+g^{j}$ must cancel $D_{i}\phi^{i}$, so we have

Then after combining the expressions (2.12) and (2.13) we obtain

with

Next, by solving the determination equation

where

Then this yeilds

where

and

acting on arbitrary functions $V^{\alpha}, W^{j}$. And we know

which yeilds $D_{t}\Lambda_{j} = \mathcal{D}_{t}\Lambda_{j}+(\mathcal{L}_{\Lambda})_{j\alpha}G^{\alpha}$. So by expansion (2.16), we can get

Then by comparing the coefficients of $G^{j}, D_{i}G^{j}, \dots, D_{i_{1}}\cdots D_{i_{p}}G^{j}$, we can get the corresponding determining equations:

where $C_{r}^{q} = \frac{r!}{q!(r-q)!}$. Then the expression of the corresponding multiplier $\Lambda_{j}$ is obtained by solving the above decision equations. After obtaining the multiplier, we can solve the corresponding conservation law. Next, we introduce the theorem:

Theorem 2: For the differential equation system (2.1), the conserved densities of any nontrivial conservation law in normal form are given in terms of the multipliers by

where

3.   Application of directly constructing conservation laws

3.1.   Combined KdV-MKdV equation

Firstly, we choose an equation named the combined KdV-MKdV equation and its form is as follows:

where $\alpha$ and $\beta$ are arbitrary constants. And its symmetries with infinitesimal genertor $Xu = \eta$ satisfies the determining equation

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 $x$. And the adjoint of Eq. (3.1) is given by

when $G = 0,$ which is the determining equation for the adjoint symmetries $\omega$ of the cKMK(the combined KdV-MKdV equation) equation. Next, we calculate the conservation law multiplier of the equation, namely finding $\Lambda$, which satisfies

with no dependence on $u_{t}$ and its differential consequences. This yields the multiplier

where $\Gamma = 0$ when $u$ is restricted to be a cKMK solution. Next we set $\Lambda$ to be related to $x, t, u, u_{x}, u_{xx}$, the determing equation becomes

Then by comparing the coefficients of $G$ and $D_{i}G$, we can get the equation

and we can notice that (3.7) is a differential consequence of (3.8). The highest coefficient in formula (3.8) is $\Lambda_{u_{xx}u_{xx}}u_{xxx}$, and we know that $\Lambda$ is not related to $u_{xx}$. It yeilds $\Lambda_{u_{xx}u_{xx}} = 0$, then $\Lambda$ has the following form:

Then the remaining terms in formular (3.8), after some cancellations, are of first order

Next we extract the coefficient of $u_{xxxx}$ in (3.6) and these yeild

and $a$ is not related with $u_{xx}$, so we can obtain $a_{u_{x}} = 0$, namely $a(t, x, u)$. Similarly, we can easily get

According to (3.10), we can deduce that $b_{u_{x}} = 0$. We replace (3.9) into (3.6), then we can find the coefficient of $u_{xxx}$, i.e.

and the coefficient of $u_{xx}$ is

It yeilds that $b_{uu} = \alpha a+2\beta ua$, $a_{t} = -3b_{xu}.$ Hence we have

from (3.14) and

We can obtain the form of $b(x, t, u)$:

Then $\Lambda$ has the form:

Then after taking (3.18) back into the decision Eq. (3.6), we can get

Then by comparing the coefficients of $u$ and $u^{2}$, we can obtain

then solve them. We get the general forms of $a(t), b_{1}(t)$ and $b_{2}(x, t)$

So the general form of $\Lambda$ is

where $c_{i}(i = 1, 2, 3, 4)$ are arbitrary constants. It yields that

Next, we find the conservation law $\phi^{x}_{i}, \phi^{t}_{i}$ according to $\Lambda_{i}$.

According to the Theorem.1, we can take $\widetilde{u} = 0$ so that $K = 0$. So we have

Firstly, for $\Lambda_{1}$,

and similarly, the $\phi^{x}_{1}$ has the following form:

We can simply verify that

when $u$ is a solution of $G$.

And for $\Lambda_{2} = 1$, the we can easily obtain

We can also verify that

when $u$ is a solution of $G$. Identically

for $\Lambda_{3} = u_{xx}+\frac{\alpha}{2}u^{2}+\frac{\beta}{3}u^{3}$. We can verify it by

when $u$ is a solution of $G$.

The last one is $\Lambda_{4} = u$,

Then we can easily obtain

when $u$ is a solution of $G$.

3.2.   Klein-Gordon equation

We choose the Klein-Gordon equation as the second equation to study and it has the following form:

It is obvious that its self-adjoint and the determing equation for its symmetries with infinitesimal generator $Xu = \eta$ and the adjoint of it is all

We set the $\Lambda$ has the expression $\Lambda(t, x, u, u_{x}, u_{t}).$ Then the determining equation for the conservation law multiplier is

By comparing the coefficient of $G$, we can sort out two determining equations:

where $\mathcal{D}_{t} = \Lambda_{t}+u_{t}\Lambda_{u}+u_{xt}\Lambda_{u_{x}}+(u_{xx}-\alpha u-\beta u^{3})\Lambda_{u_{t}}.$ We start from the Eq. (3.37) and it yeilds

and since $\Lambda$ does not contain $u_{xx}$, we can seperat it from Eq.(3.38)

Then we deal with (3.36). It is easy to verify that the coefficients of $u_{xxx}$ and $u_{xxt}$ are 0. So we consider the coefficient of the second order $u_{xx}$ and $u_{xt}$, i.e.

and from (3.36) we can get

and the coefficient of $u_{xt}$ is

Then by solving the four equations (3.39), (3.40), (3.42) and (3.43) simultaneously, we can obtain the general form of the multiplier

where $f_{1}(t-x)$ is any function related to $t-x$, $f_{2}(t+x)$ is any function related to $t+x$, $c_{1}$ is an arbitrary constant and $d(x, t)$ is any function related to $x, t$. Next, we substitute the multiplier into (3.38), and compare the coefficients of $u_{xx}, u_{xt}$. We can get

We can obtain

We can take their simplest form for $f_{1}$ and $f_{2},$

where $c_{i}(i = 1, 2, 3, 4)$ are arbitary constants. Therefore, the general form of multiplier is

It yeilds that

Next we solve the conservation law with the formula

and we choose $\tilde{u} = u_{0}$ which is a constant and $K = 0$. Then we can obtain that

for multiplier $\Lambda_{1}$.

for multiplier $\Lambda_{2}$.

for multiplier $\Lambda_{3}$.

for multiplier $\Lambda_{4}$.

3.3.   The generalized coupled KdV equation

In the third example we try to apply to a multi-potential differential equation system and we choose the generalized coupled KdV equation. It has the form as follows:

The famous KdV equation is considered to be one of the most important equations in the theory of integrable systems. It gives multiple soliton solutions with infinite number of conservation laws, double Hamiltonian structures, Lax pairs and many other physical properties. The coupled KdV equation has attracted a lot of research due to its importance in theoretical physics and many scientific applications. According to the formula (2.2) and (2.3), we can obtain by calculation

and we can also obtain the adjoint form of them

We set $\mathcal{D}_{t} = \partial_{t}-(g^{1}\partial_{u}+g^{2}\partial_{v}+g^{3}\partial_{\omega})+\cdots,$ where $g^{1} = -\frac{1}{4}u_{xxx}-3uu_{x}+6vv_{x}-3\omega_{x}, g^{2} = 3uv_{x}+\frac{1}{2}v_{xxx}, g^{3} = 3u\omega_{x}+\frac{1}{2}\omega_{xxx}.$ Then we will try to calculate the form $\Lambda(t, x, \boldsymbol{u}, \boldsymbol{u}_{x}, \boldsymbol{u}_{xx}),$ and $\boldsymbol{u}$ means $u, v, \omega$. The determing equations according to (2.16) become

The specific forms are

Then by comparing the coefficients of the derivative of $G$, we can separate the following determining equations:

By the formula (3.65) we can know that the hignest coefficient of it is $\Lambda_{1u_{xx}u_{xx}}u_{xxx},$ and $\Lambda_{1}$ is not related with $u_{xxx},$ so we obtain $\Lambda_{1u_{xx}u_{xx}} = 0$,

Similarly, we can get

and accoring to the formula (3.66), we can obtain

and since the formula (3.64), we have $\frac{\partial \Lambda_{1}}{\partial v_{xx}} = \frac{\partial \Lambda_{2}}{\partial u_{xx}}.$

Then the genreal forms of the multiplicators become

Then we start from formula (3.65), and according to the remaining items we can get

According to its $u_{xxxx}$ coefficient of the formula (3.61), we have

then it yeilds $3D_{x}a_{1}-a_{2u_{x}}v_{xx}-a_{3u_{x}}\omega_{xx} = 0,$ so we get $a_{2u_{x}} = a_{3u_{x}} = a_{1u_{x}} = a_{1u} = a_{1x} = 0.$ Similiarly according to the forlumas (3.62) and (3.63), finally we obtain the expressions $a_{i}(t), i = 1, \cdots, 6.$ Then we calculate from formula (3.75), we have $b_{1u_{x}} = 0$, similarly, $b_{2v_{x}} = b_{3\omega_{x}} = 0$. Accoring to the formula (3.66), we get

We can take the simplest forms, i.e. $b_{1}, \ b_{2}, \ b_{3}$ are not related with $\boldsymbol{u}_{x}$. Then let us start with the formula (3.58). The coefficient of $u_{xxx}$ is $3ua_{1}-3ua_{1}+\frac{1}{4}b_{u}-\frac{1}{4}b_{u} = 0,$ and the coefficient of $u_{xx}$ is

Since $a_{1}, a_{2}, a_{3}$ don't contain $u_{x}, v_{x}, \omega_{x}$, we obtain

So we have

and since $b_{1uv} = -8a_{2}$, we get $c_{v} = -8a_{x}$. It yeilds $c = -8a_{2}v+d^{'}$, since $\frac{\partial d}{\partial \omega} = -8a_{3}$.

We can denote it by

And according to the formula (3.62), the coefficient of $v_{xx}$ is

It yeilds that

We can calculate the expression

Similarly, in the formula (3.63), the coffecient of $\omega_{xx}$ is

It yeilds that

Then we can calculate the expression

Then the general forms become

And according to the formula (3.61), the coefficient of $v_{xxxxx}$ is $(\frac{1}{4}-\frac{1}{2})a_{2} = 0$, so we obtain $a_{2} = 0$. Similarly, we also get $a_{3} = 0$. Then we separated the coefficient of $u_{xx}$ of the formula (3.61):

and it yeilds that

According to the coefficients of $\omega_{xxx}$:$\frac{3}{4}p_{2\omega} = 3a_{1}(t) = 0, \ v_{xxx}$:${3}{4}p_{2}v = -6a_{1}(t)v = 0, \ v_{x}v_{xx}$:$3a_{4}+12a_{1} = 0, \ v_{x}\omega_{xx}$:$3a_{5} = 0, \ \omega_{x}v_{xx}$:$3a_{5} = 0, \ \omega_{x}\omega_{xx}$:$3a_{6} = 0.$ We obtain that $p_{2} = 4a_{1}(t)\omega, \ p_{2} = -4a_{1}(t)v^{2}, \ a_{5} = a_{6} = 0, \ a_{4} = -4a_{1}(t).$ And the coefficient of $vv_{x}$ is

It yeilds that $a_{1}(t) = 0.$ The remaining coefficients of the formula (3.61) can be separated into three equations for the coefficients of $v_{x}$ and $\omega_{x}$:

They yeild that

Let us bring back them into the formulas (3.61), (3.62), (3.63), the remaining items of them can be separated into four equations

By solving the equations, we can get

Then we get the genreal form of $\Lambda_{i}, i = 1, 2, 3.$

They yeild that

Then we can solve the conservation laws $\phi^{t}, \phi^{x}$ by multipliers according to (2.23), (2.24)

for $\Lambda_{11}, \Lambda_{12}, \Lambda_{13}$.

for $\Lambda_{21}, \Lambda_{22}, \Lambda_{23}$.

for $\Lambda_{31}, \Lambda_{32}, \Lambda_{33}$.

for $\Lambda_{41}, \Lambda_{42}, \Lambda_{43}$.

4.   Solving conservation law by Ibragimov method

In this part we try to solve the equation by Ibragimov method to obtain the conservation laws. Firstly, we introduce a theorem:

Theorem 2. Any Lie point, Lie-Bäcklund, and nonlocal symmetry

leads to the conservation law $D_{i}(C^{i}) = 0$, i.e.

where

By using maple, we can obtain some symmetries of the target equations. Firstly, for the combined KdV-MKdV equation (3.1), it has

We choose $X = (\frac{x}{3}-\frac{\alpha^{2}t}{6\beta})\frac{\partial}{\partial x}+t\frac{\partial}{\partial t}+(\frac{-2\beta u-\alpha}{6\beta})\frac{\partial}{\partial u}$ as the symmetry used to calculate the conservation laws. According to the formula (4.1), we have

And we note $\mathcal{L} = v(u_{t}+\alpha uu_{x}+\beta u^{2}u_{x}+u_{xxx}).$ Then the conservation laws calculated directly become

We can simply verify it:

when $u$ is a solution of Eq. (3.1) and $v = -u$.

Similarly, for the Klein-Gordon equation (3.33), it has symmetries which are as follows:

and we choose $X = t\frac{\partial}{\partial x}+x\frac{\partial}{\partial t}$ as the symmetry to compute conservation law.

where $\mathcal{L} = v(u_{tt}-u_{xx}+\alpha u+\beta u^{3}), W = -tu_{x}-xu_{t}$.

At last, we consider the generalized coupled KdV equation (3.54), it has the symmetries which are as follows:

and we choose $X = \frac{x}{3}\frac{\partial}{\partial x}+t\frac{\partial}{\partial t}-\frac{4\omega}{3}\frac{\partial}{\partial \omega}-\frac{2u}{3}\frac{\partial}{\partial u}-\frac{2v}{3}\frac{\partial}{\partial v}$ as the symmetry to compute conservation law. We can calculate that

Then the conservation law is

5.   Hamiltonian Structure and Line Soliton Solution

5.1.   Hamiltonian structure

For the combined KdV and MKdV equation, We notice that it has a Hamiltonian formulation $u_{t} = -\mathcal{D}(\frac{\delta H}{\delta u})$, where $H = \int \frac{\alpha u^3}{6}+\frac{\beta}{12 u^4}-\frac{1}{2}u_{x}^{2}dx$ is the Hamiltonian functional, and $\mathcal{D} = D_{x}$ is a Hamiltonian operator. Then since $D_{x}$ is a Hamiltonian operator, it can map adjoint-symmetries into symmetries, so $D_{x}^{-1}$ can map symmetries into adjoint-symmetries. And we can use the above symmetry to get the adjoint symmetry of some objective equations. Applying this latter operator to the scaling symmetries, we obtain the adjoint-symmetries:

where $u = v_{x}.$ In fact, the multiplier $\Lambda$ calculated earlier in this paper is the adjoint symmetry of the variation of the objective equation.

Then for the Klein-Gordon equation, to obtain the Hamiltonian formulation, we transform the Eq. (3.33) into an equation system:

The associated Hamiltonian formulation for this system is then given by

where $H = \int \frac{v^{2}}{2}+\frac{u_{x}^{2}}{2}+\frac{\alpha}{2}u^{2}+\frac{\beta}{4}u^{4}dx$. We note that the determining equation of the objective equation and its self-adjoint determining equation is consistent:

Hence, the symmetry of the equation is consistent with the adjoint symmetry.

5.2.   Line soliton solutions

A line soliton is a solitary travelling wave $u = U(x-\mu t)$ in one dimension where the parameter $\mu$ means the speed of the wave. Then we study the conservation laws of the combined KdV and MKdV equation and the Klein-Gordon equation $\phi^{t}, \ \phi^{x}$ which doesn't contain the variables $t, x$. Then the conservation law is obtained by reduction

yielding

So $(\phi^{x}-\mu \phi^{t}) = C.$ Then we begin with the combined KdV and MKdV equation. Using the transformation $u(x, t) = U(\xi),$ we can obtain the nonlinear ordinary differential equation:

for $U(\xi)$. Conservation laws (3.28), (3, 30), (3.32) do not contain the variables $t, x$. When the first integral formula $(\phi^{x}-\mu \phi^{t}) = C,$ is applied to these three conservation laws, we obtain

We impose the asymptotic conditions $U, U^{'}, U^{''}, U^{'''}\rightarrow 0$ as $|\xi|\rightarrow \infty.$ Then we combine the formulas (5.8), (5.9) and (5.10), then we can calculate its general line soliton solutions:

where $k_{1}$ is an arbitrary constant.

Figure 1 and Figure 2 display the kinds of 3D plots of $U_{1}(\xi)$ and $U_{2}(\xi)$ determined by (5.11) and (5.12), and Figure 3 and Figure 4 display the kinds of density plots of them.

Then for the Klein Gordon equation, we make the transformation $u(x, t) = U(\xi)$, we obtain an ODE:

And we study the related conservation laws (3.50)-(3.53), only the formula (3.50) does not contain variables $x, t$. We can obtain

By calculating (5.14), we can get its soliton solution

and the roots of $\xi-(\int^{U_{3}}\pm\frac{(\mu-1)(\mu+1)}{\sqrt{-(\mu-1)(\mu+1)a(a-u_{0})(a^{2}\beta+\alpha)}}da)-k_{2} = 0,$ where $k_{2}$ is arbitrary constant.

6.   Conclusion

It is well known that the study of conservation laws is very important for studying the integrability of optimal systems. In this paper, two methods are used to solve three different types of partial differential equations and systems, namely the conservation laws of the combined KdV-MKdV equation, the Klein-Gordon equation and the generalized coupled KdV equation. And these two methods are widely applicable. It can be applied not only to the case of multiple independent variables, but also to the case of multiple dependent variables and differential equation systems. In fact, the multipliers obtained in this part of the direct construction of conservation laws are actually some adjoint symmetries of the equation with variational properties. And the linear soliton solutions of the equations can be analyzed by the obtained conservation law.

In fact, the two methods used in this paper to calculate the conservation law of equations have different advantages. The adjoint equation method proposed by Ibragimov can use the symmetry of the equation to calculate the conservation law through the explicit formula. It is convenient to calculate and does not require complex analysis. It has a wide range of applications, but the results are directly affected by the symmetry of the equation. The advantage of constructing conservation laws directly is that it is not necessary to use the variational symmetry of the equation. For a partial differential equation without variational symmetry, the adjoint symmetry of the equation is used to replace the symmetry. At this time, the adjoint symmetry satisfies the linear adjoint symmetry determining equations. The symmetry invariant condition is replaced by the adjoint symmetry invariant condition, and a formula using adjoint symmetry is given. However, this method is computationally complex and does not apply to any type of equation and equation system. Both methods can be naturally applied to higher dimensional differential equations and differential equation systems. This paper mainly integrates the two methods and applies them to different types of equations and equation systems. The examples in Anco's paper ^[6,7] basically apply the direct construction method to the (1+1) dimensional differential equations, and this paper attempts to apply the method to the equation system.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11971475).

Conflict of interest

The authors declare that they have no known competing financial interests.