Construction of conservation laws for the Gardner equation, Landau&ndash;Ginzburg&ndash;Higgs equation, and Hirota&ndash;Satsuma equation

Cheng Chen; Faiza Afzal; Yufeng Zhang

doi:10.1088/1572-9494/ad19d6

PDF(367 KB)

Communications in Theoretical Physics ›› 2024, Vol. 76 ›› Issue (5) : 55004. DOI: 10.1088/1572-9494/ad19d6

Mathematical Physics

Construction of conservation laws for the Gardner equation, Landau–Ginzburg–Higgs equation, and Hirota–Satsuma equation

Author information +

History +

Abstract

In this paper, two different methods for calculating the conservation laws are used, these are the direct construction of conservation laws and the conservation theorem proposed by Ibragimov. Using these two methods, we obtain the conservation laws of the Gardner equation, Landau–Ginzburg–Higgs equation and Hirota–Satsuma equation, respectively.

Key words

conservation laws / Landau–Ginzburg–Higgs equation / Hirota–Satsuma equation / Gardner equation

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

Cheng Chen, Faiza Afzal, Yufeng Zhang. Construction of conservation laws for the Gardner equation, Landau–Ginzburg–Higgs equation, and Hirota–Satsuma equation[J]. Communications in Theoretical Physics, 2024, 76(5): 55004 https://doi.org/10.1088/1572-9494/ad19d6

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 equation

\begin{array}{rcl} G \equiv u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x} = 0. \end{array}

(2.1)

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{array}{rcl} \begin{array}{l} 0 = D_{t} η + 6 u_{x} η + 6 {u D}_{x} η - 12 {u u}_{x} η \\ - 6 u^{2} D_{x} η + D_{x}^{3} η w h e n G = 0, \end{array} \end{array}

(2.2)

where

D_{t} = \partial_{t} + u_{t} \partial_{u} + u_{t x} \partial_{u_{x}} + u_{t t} \partial_{u_{t}} + \dots

and

D_{x} = \partial_{x} + u_{x} \partial_{u} + u_{x x} \partial_{u_{x}} + u_{t x} \partial_{u_{t}} + \dots

are total derivative operators with respect to t and x. The adjoint of the symmetry equation (2.2) is the determining equation for adjoint symmetries

G^{' *} (ω) |_{ϵ} = 0

, where ε will denote the solution space of the equation (2.1). Therefore, the adjoint of the equation (2.2) is given by

\begin{array}{rcl} \begin{array}{l} 0 = - D_{t} ω - 6 {u D}_{x} ω + 6 u^{2} D_{x} ω - D_{x}^{3} ω \\ w h e n G = 0, \end{array} \end{array}

(2.3)

which is the determining equation for the adjoint symmetries ω of the Gardner equation. Next, we consider local conservation laws D_tΦ^t + D_xΦ^x = 0 on all solutions u(t, x) of the equation (2.1). 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, u_x and u_xx. In particular, by moving off the Gardner equation solution space, we have

\begin{array}{rcl} \begin{array}{l} D_{t} Φ^{t} + D_{x} Φ^{x} = (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) Λ_{0} \\ + D_{x} (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) Λ_{1} + \dots, \end{array} \end{array}

(2.4)

for some expression Λ₀, Λ₁, … with no dependence on u_t or differential consequence. This yields the multiplier

\begin{array}{rcl} \begin{array}{l} D_{t} Φ^{t} + D_{x} (Φ^{x} - Γ) \\ = (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) Λ, \\ Λ = Λ_{0} - D_{x} Λ_{1} + \dots, \end{array} \end{array}

(2.5)

where Γ = 0 when u is restricted to be a Gardner equation solution.

The definition for multipliers Λ(t, x, u, u_x, u_xx) is that (u_t + 6uu_x − 6u²u_x + u_xxx)Λ must be a divergence expression for all functions u(t, x). The determining condition is expressed by

\begin{array}{rcl} \begin{array}{rcl} 0 & = & E_{u} (G Λ) = E_{u} ((u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) Λ) \\ = & - D_{t} Λ - 6 {u D}_{x} Λ + 6 u^{2} D_{x} Λ - D_{x}^{3} Λ \\ + G Λ_{u} - D_{x} (G Λ_{u_{x}}) + D_{x}^{2} (G Λ_{u_{x x}}), \end{array} \end{array}

(2.6)

where

E_{u} = \partial_{u} - D_{t} \partial_{u_{t}} - D_{x} \partial_{u_{x}} + D_{t} D_{x} \partial_{u_{t x}} + D_{x}^{2} \partial_{u_{x x}} + \dots

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{array}{rcl} \begin{array}{rcl} Λ & = & c_{1} [(2 u^{3} - 3 u^{2} + u - u_{x x}) t - \frac{x}{6} + \frac{1}{3} x u] \\ + c_{2} [- 2 u^{3} + 3 u^{2} + u_{x x}] + c_{3} u + c_{4}, \end{array} \end{array}

(2.7)

where c₁, c₂, c₃ and c₄ are arbitrary constants. So for the Gardner equation, here are some multipliers

\begin{array}{rcl} \begin{array}{rcl} Λ_{1} & = & (2 u^{3} - 3 u^{2} + u - u_{x x}) t - \frac{x}{6} + \frac{1}{3} x u, \\ Λ_{2} & = & - 2 u^{3} + 3 u^{2} + u_{x x}, \\ Λ_{3} & = & u, Λ_{4} = 1. \end{array} \end{array}

(2.8)

For the multiplier

Λ_{1} = (2 u^{3} - 3 u^{2} + u - u_{x x}) t - \frac{x}{6} + \frac{1}{3} x u

, according to the definition of multiplier

Λ_{1} G = D_{t} (Φ_{1}^{t}) + D_{x} (Φ_{1}^{x})

, we have the following conserved vector:

\begin{array}{rcl} \begin{array}{rcl} Φ_{1}^{t} & = & t (\frac{1}{2} u^{4} - u^{3} + \frac{1}{2} u^{2} - u_{t} u_{x} + \frac{1}{2} u_{x}^{2}) \\ - \frac{x}{6} u + \frac{x}{6} u^{2}, \\ Φ_{1}^{x} & = & t (- 2 u^{6} + 6 u^{5} - 6 u^{4} + 2 u^{3} + 2 u^{3} u_{x x} \\ - 3 u^{2} u_{x x} + {u u}_{x x} - \frac{1}{2} u_{x}^{2}) \\ - \frac{x}{2} u^{4} + {x u}^{3} - \frac{x}{2} u^{2} - \frac{x}{6} u_{x x} \\ + \frac{1}{6} u_{x} + \frac{x}{3} {x u u}_{x x} - \frac{x}{2} u_{x}^{2} - {u u}_{x} . \end{array} \end{array}

(2.9)

Similarly, for the multiplier Λ₂ = − 2u³ + 3u² + u_xx, Λ₃ = u and Λ₄ = 1, we have

\begin{array}{rcl} \begin{array}{rcl} Φ_{2}^{t} & = & - \frac{1}{2} u^{4} + u^{3} - \frac{1}{2} u_{x}^{2}, \\ Φ_{2}^{x} & = & 2 u^{6} - 6 u^{5} + \frac{9}{2} u^{4} + \frac{1}{2} u_{x x}^{2} \\ + 3 u^{2} u_{x x} - 2 u^{3} u_{x x} + u_{x} u_{t}, \end{array} \end{array}

(2.10)

\begin{array}{rcl} \begin{array}{rcl} Φ_{3}^{t} & = & \frac{1}{2} u^{2}, \\ Φ_{3}^{x} & = & - \frac{3}{2} u^{4} + 2 u^{3} - \frac{1}{2} u_{x}^{2} + {u u}_{x x}, \end{array} \end{array}

(2.11)

and

\begin{array}{rcl} \begin{array}{rcl} Φ_{4}^{t} & = & u, \\ Φ_{4}^{x} & = & 3 u^{2} - 2 u^{3} + u_{x x} . \end{array} \end{array}

(2.12)

2.2. Landau–Ginzburg–Higgs equation

Consider the LGH equation

\begin{array}{rcl} G = u_{t t} - u_{x x} + g^{2} u^{N} = 0, \end{array}

(2.13)

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 equation

\begin{array}{rcl} \begin{array}{c} D_{t}^{2} η - D_{x}^{2} η + {N g}^{2} u^{N - 1} η = 0 \\ w h e n G = 0. \end{array} \end{array}

(2.14)

Obviously, equation (2.14) is self-adjointed. For any local conservation laws

\begin{array}{rcl} D_{t} Φ^{t} + D_{x} Φ^{x} = 0, \end{array}

(2.15)

by moving off the LGH equation solution space, we have

\begin{array}{rcl} \begin{array}{l} D_{t} Φ^{t} + D_{x} Φ^{x} = (u_{t t} - u_{x x} + g^{2} u^{N}) Λ_{0} \\ + D_{x} (u_{t t} - u_{x x} + g^{2} u^{N}) Λ_{1} + \dots, \end{array} \end{array}

(2.16)

for some expressions Λ₀, Λ₁, … with no dependence on u_tt and differential consequences. This yields the multiplier

\begin{array}{rcl} \begin{array}{l} D_{t} Φ^{t} + D_{x} (Φ^{x} - Γ) = (u_{t t} - u_{x x} + g^{2} u^{N}), \\ Λ = Λ_{0} - D_{x} Λ_{1} + \dots, \end{array} \end{array}

(2.17)

where Γ = 0 when u is restricted to be a LGH equation solution. Multipliers Λ are defined by the condition that (u_tt − u_xx + g²u^N)Λ is a divergence expression for all functions u(t, x). We restrict attention to Λ of first-order, depending on t, x, u, u_t, u_x. This leads to the necessary and sufficient determining condition

\begin{array}{rcl} \begin{array}{rcl} 0 & = & E_{u} ((u_{t t} - u_{x x} - m^{2} u + g^{2} u^{3}) Λ) \\ = & D_{t}^{2} Λ - D_{x}^{2} Λ + {N g}^{2} u^{N - 1} Λ \\ + (u_{t t} - u_{x x} + g^{2} u^{N}) Λ_{u} \\ - D_{x} (Λ_{u_{x}} (u_{t t} - u_{x x} + g^{2} u^{N})) \\ - D_{t} (Λ_{u_{t}} (u_{t t} - u_{x x} + g^{2} u^{N})) . \end{array} \end{array}

(2.18)

By solving equations (2.18), we can obtain the general form of the multiplier of the Hirota–Satsuma equation as follows (see appendix B) :

\begin{array}{rcl} Λ = a_{1} (u_{x} - u_{t}) + a_{2} (u_{t} + u_{x}) + c_{1} u_{t}, \end{array}

(2.19)

where a₁, a₂ and c₁ are arbitrary constants. For the LGH equation, here are some multipliers

\begin{array}{rcl} Λ_{1} = u_{x} - u_{t}, Λ_{2} = u_{x} + u_{t}, Λ_{3} = u_{t} . \end{array}

(2.20)

For the multiplier Λ₁ = u_t + u_x, according to the definition of multiplier

Λ_{1} G = D_{t} (Φ_{1}^{t}) + D_{x} (Φ_{1}^{x})

, we have the following conserved vector:

\begin{array}{rcl} \begin{array}{rcl} Φ_{1}^{t} & = & u_{x} u_{t} - \frac{1}{2} u_{t}^{2} - \frac{1}{2} u_{x}^{2} - \frac{g^{2}}{N + 1} u^{N + 1}, \\ Φ_{1}^{x} & = & u_{x} u_{t} - \frac{1}{2} u_{t}^{2} - \frac{1}{2} u_{x}^{2} + \frac{g^{2}}{N + 1} u^{N + 1} . \end{array} \end{array}

(2.21)

Similarly, for the multiplier Λ₂ = u_x and Λ₃ = u_t, we have

\begin{array}{rcl} \begin{array}{rcl} Φ_{2}^{t} & = & u_{x} u_{t} + \frac{1}{2} u_{t}^{2} + \frac{1}{2} u_{x}^{2} + \frac{g^{2}}{N + 1} u^{N + 1}, \\ Φ_{2}^{x} & = & - u_{x} u_{t} - \frac{1}{2} u_{t}^{2} - \frac{1}{2} u_{x}^{2} + \frac{g^{2}}{N + 1} u^{N + 1}, \end{array} \end{array}

(2.22)

and

\begin{array}{rcl} \begin{array}{rcl} Φ_{3}^{t} & = & \frac{1}{2} u_{t}^{2} + \frac{1}{2} u_{x}^{2} + \frac{g^{2}}{N + 1} u^{N + 1}, \\ Φ_{3}^{x} & = & - u_{t} u_{x} . \end{array} \end{array}

(2.23)

2.3. Hirota–Satsuma equation

Consider the Hirota–Satsuma (HS) equation

\begin{array}{rcl} \begin{array}{c} {\begin{matrix} G^{1} = u_{t} - 6 {p u u}_{x} + {r v v}_{x} - {p u}_{x x x} = 0, \\ G^{2} = v_{t} + 3 {p u v}_{x} + {p v}_{x x x} = 0, \end{matrix} \end{array} \end{array}

(2.24)

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)

\begin{array}{rcl} \begin{array}{rcl} {(L_{g})}_{1}^{1} V^{1} & = & - 6 {p u}_{x} V^{1} - 6 {p u D}_{x} V^{1} - {p D}_{x}^{3} V^{1}, \\ {(L_{g})}_{1}^{2} V^{1} & = & 3 {p v}_{x} V^{1}, \\ {(L_{g})}_{2}^{1} V^{2} & = & 2 {r v}_{x} V^{2} + {r v D}_{x} V^{2}, \\ {(L_{g})}_{2}^{2} V^{2} & = & 3 {p u D}_{x} V^{2} + {p D}_{x}^{3} V^{2} . \end{array} \end{array}

(2.25)

And the adjoint system of system (2.25) is given by

\begin{array}{rcl} \begin{array}{rcl} {(L_{g}^{*})}_{1}^{1} W_{1} & = & 6 {p u D}_{x} W_{1} + {p D}_{x}^{3} W_{1}, \\ {(L_{g}^{*})}_{1}^{2} W_{2} & = & 3 {p v}_{x} W_{2}, \\ {(L_{g}^{*})}_{2}^{1} W_{1} & = & - {r v D}_{x} W_{1}, \\ {(L_{g}^{*})}_{2}^{2} W_{2} & = & - 3 {p u}_{x} W_{2} - 3 {p u D}_{x} W_{2} - {p D}_{x}^{3} W_{2}, \end{array} \end{array}

(2.26)

here

L_{g}

and

L_{g}^{*}

denote a linearization operator defined as follows:

\begin{array}{rcl} \begin{array}{l} {(L_{Λ})}_{σ ρ} V^{ρ} = \frac{\partial Λ_{σ}}{\partial u^{ρ}} V^{ρ} + \frac{\partial Λ_{σ}}{\partial u_{i}^{ρ}} D_{i} V^{ρ} + \dots \\ + \frac{\partial Λ_{σ}}{\partial u_{i_{1} \dots i_{p}}^{ρ}} D_{i_{1}} \dots D_{i_{p}} V^{ρ}, \end{array} \end{array}

(2.27)

and

\begin{array}{rcl} \begin{array}{l} {(L_{Λ}^{*})}_{σ ρ} W^{ρ} = \frac{\partial Λ_{ρ}}{\partial u^{σ}} W^{ρ} - D_{i} (\frac{\partial Λ_{ρ}}{\partial u_{i}^{σ}} W^{ρ}) + \dots \\ + {(- 1)}^{p} D_{i_{1}} \dots D_{i_{p}} (\frac{\partial Λ_{ρ}}{\partial u_{i_{1} \dots i_{p}}^{σ}} W^{ρ}) . \end{array} \end{array}

(2.28)

Let

D_{t} = \partial_{t} - (g^{1} \partial_{u} + g^{2} \partial_{v} + \dots)

, where g¹ = − 6puu_x + rvv_x − pu_xxx, and g² = 3puv_x + pv_xxx. For multipliers of the form Λ(x, t, u, u_x, u_xx) and u means u, v we have

\begin{array}{rcl} \begin{array}{rcl} 0 & = & E_{u^{σ}} (u_{t} Λ_{1} + g^{1} Λ_{1} + v_{t} Λ_{2} + g^{2} Λ_{2}) \\ = & - D_{t} Λ_{σ} + {(L_{g}^{*})}_{σ}^{ρ} Λ_{ρ} + {(L_{g}^{*})}_{σ ρ} (u_{t}^{ρ} + g^{ρ}), \\ σ & = & 1, 2. \end{array} \end{array}

(2.29)

By solving equations (2.29), we can obtain the general form of the multiplier of the HS equation as follows (see appendix C):

\begin{array}{rcl} \begin{array}{rcl} Λ_{1} & = & a_{1} u_{x x} - \frac{r}{4 p} a_{1} v^{2} + 3 a_{1} u^{2} + k_{1}, \\ Λ_{2} & = & - \frac{r}{2 p} a_{1} v_{x x} - \frac{r}{2 p} a_{1} u v, \end{array} \end{array}

(2.30)

where a₁ and k₁ are arbitrary constants. So for the HS equation, here are some multipliers

\begin{array}{rcl} \begin{array}{rcl} Λ_{11} & = & 1, Λ_{12} = 0, \\ Λ_{21} & = & u_{x x} - \frac{r}{4 p} v^{2} + 3 u^{2}, \\ Λ_{22} & = & \frac{r}{2 p} v_{x x} - \frac{r}{2 p} u v . \end{array} \end{array}

(2.31)

For the multiplier Λ₁ = (1, 0), according to the definition of multiplier

Λ_{i} G^{i} = D_{t} (Φ_{1}^{t}) + D_{x} (Φ_{1}^{x})

, we have the following conserved vector:

\begin{array}{rcl} \begin{array}{rcl} Φ_{1}^{t} & = & u, \\ Φ_{1}^{x} & = & - 3 {p u}^{2} + \frac{r}{2} v^{2} - {p u}_{x x} . \end{array} \end{array}

(2.32)

Similarly, for the multiplier

Λ_{2} = (u_{x x} - \frac{r}{4 p} v^{2} + 3 u^{2}, \frac{r}{2 p} v_{x x} - \frac{r}{2 p} u v)

, we also have the following conserved vector:

\begin{array}{rcl} \begin{array}{rcl} Φ_{2}^{t} & = & \frac{1}{2} {u u}_{x x} - \frac{r}{12 p} {u v}^{2} + u^{3} - \frac{r}{4 p} {v v}_{x x} - \frac{r}{6 p} {u v}^{2}, \\ Φ_{2}^{x} & = & \frac{1}{24} (- 216 p^{2} u^{4} + 36 {p r u}^{2} v^{2} - 3 r^{2} v^{4} \\ - 312 p^{2} u^{2} u_{x x} - 32 {p r u v v}_{x x} \\ + 34 {p r v}_{x}^{2} u + 6 {p r u}_{x x} v^{2} - 34 {p r v v}_{x} u_{x} \\ - 24 p^{2} {u u}_{x x x x} - 24 p^{2} u_{x x}^{2} \\ - 12 {p r v v}_{x x x x} + 12 {p r v}_{x} v_{x x x} - 12 {p r v}_{x x}^{2} + p^{2} u_{x} u_{x x x}) . \end{array} \end{array}

(2.33)

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 (2.24).

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 systems

\begin{array}{rcl} F_{α} (x, u, u_{(1)}, \dots, u_{(r)}) = 0, α = 1, 2, \dots, m, \end{array}

(3.1)

where x = (x¹, x², …,xⁿ) denotes n independent variables, u = (u¹, u²,…,u^m) 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 (3.1) is given by

\begin{array}{rcl} F_{α}^{*} (x, u, v, u_{(1)}, v_{(1)}, \dots, u_{(r)}, v_{(r)}) = 0, α = 1, 2, \dots, m, \end{array}

(3.2)

with

\begin{array}{rcl} \begin{array}{c} F_{α}^{*} (x, u, v, u_{(1)}, v_{(1)}, \dots, u_{(r)}, v_{(r)}) \equiv \frac{δ L}{δ u^{α}} = 0, \\ α = 1, 2, \dots, m . \end{array} \end{array}

(3.3)

Here

L = \sum_{β = 1}^{m} v^{β} F_{β}

is a Lagrangian form of (3.1), with v = (v¹,…,v^m) being new dependent variables. We use

\frac{δ}{δ u^{α}}

for the Euler–Lagrange operator

\begin{array}{rcl} \begin{array}{l} \frac{δ}{δ u^{α}} = \frac{\partial}{\partial u^{α}} - D_{i_{1}} \frac{\partial}{\partial u_{i_{1}}^{α}} + D_{i_{1}} D_{i_{2}} \frac{\partial}{\partial u_{i_{1} i_{2}}^{α}} + \dots \\ + {(- 1)}^{s} D_{i_{1}} D_{i_{2}} \dots D_{i_{s}} \frac{\partial}{\partial u_{i_{1} i_{2} \dots i_{s}}^{α}} . \end{array} \end{array}

(3.4)

Next, the definition of nonlinear self-adjointness and the theorem of conservation laws are given in [15].

Definition 1. The systems of m differential equations (3.1) are said to be nonlinearly self-adjointed if the adjoint equations (3.3) are satisfied for all solutions $u$ of the original system (3.1) upon a substitution

\begin{array}{rcl} v^{α} = ψ^{α} (x, u), α = 1, \dots, m, \end{array}

(3.5)

such that

ψ (x, u) \neq 0

Theorem 1. Let the system of differential equations (3.1) be nonlinearly self-adjoint. Specifically, let the adjoint system (3.3) to (3.1) be satisfied for all solutions of equations (3.1) upon a substitution (3.5). Then any Lie point, contact or Lie-Bäcklund symmetry

\begin{array}{rcl} X = ξ^{i} (x, u, u_{(1)}, \dots) \frac{\partial}{\partial x^{i}} + η^{j} (x, u, u_{(1)}, \dots) \frac{\partial}{\partial u^{j}}, \end{array}

(3.6)

as well as a nonlocal symmetry of equations (3.1) leads to a conservation law

D_{i} C^{i} |_{(4.1)} = 0

constructed by the following formulas:

\begin{array}{rcl} \begin{array}{rcl} C^{i} & = & ξ^{i} L + W^{α} [\frac{\partial L}{\partial u_{i}^{α}} - D_{j} (\frac{\partial L}{\partial u_{i j}^{α}}) + D_{j} D_{k} (\frac{\partial L}{\partial u_{i j k}^{α}}) - \dots] \\ + D_{j} (W^{α}) [\frac{\partial L}{\partial u_{i j}^{α}} - D_{k} (\frac{\partial L}{\partial u_{i j k}^{α}}) + \dots] + D_{j} D_{k} (W^{α}) \\ \times [\frac{\partial L}{\partial u_{i j k}^{α}} - \dots], \end{array} \end{array}

(3.7)

where

\begin{array}{rcl} W^{α} = η^{α} - ξ^{j} u_{j}^{α}, \end{array}

(3.8)

and

L

is the formal Lagrangian for the system (3.1).

Proposition 1. The Gardner equation is nonlinearly self-adjoint.

Proof. Taking the multiplier $μ = \frac{1}{u}$ so that for

\begin{array}{rcl} \frac{1}{u} (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) = 0. \end{array}

(3.9)

Then, the adjoint system of equation (3.9) is obtained as follows:

\begin{array}{rcl} \begin{array}{rcl} \frac{δ L}{δ u} & = & - \frac{ω}{u^{2}} (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) - D_{t} (\frac{ω}{u}) \\ - D_{x} (6 ω) + 6 u_{x} \frac{ω}{u} - 12 u_{x} ω + D_{x} (6 u ω) - D_{x}^{3} (\frac{ω}{u}), \end{array} \end{array}

(3.10)

with the formal Lagrangian

\begin{array}{rcl} L = \frac{ω}{u} (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) . \end{array}

(3.11)

Substituting

ω = u

into (3.4), one has

\begin{array}{rcl} \frac{δ L}{δ u} = - \frac{1}{u} (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) . \end{array}

(3.12)

By comparing (3.12) and (3.9), we find that equation (2.1) is strictly self-adjoint. Thus, the Gardner equation is nonlinearly self-adjoint.

The Gardner equation admits three infinitesimal generators as follows:

\begin{array}{rcl} \begin{array}{l} X_{1} = \frac{\partial}{\partial t}, X_{2} = \frac{\partial}{\partial x}, \\ X_{3} = (t + \frac{x}{3}) \frac{\partial}{\partial x} + t \frac{\partial}{\partial t} + (- \frac{u}{3} + \frac{1}{6}) \frac{\partial}{\partial u} . \end{array} \end{array}

(3.13)

The conservation laws of symmetry (3.13) have forms as follows:

\begin{array}{rcl} [D_{t} C^{t} + D_{x} C^{x}] |_{(3.1)} = 0, \end{array}

(3.14)

where the conserved vector C = (C^t, C^x) is given by (3.7).

For the generator X₁, the corresponding Lie characteristic function is given by W = − u_t. Thus, formulas (3.7) yield the following conserved vector:

\begin{array}{rcl} \begin{array}{rcl} C^{t} & = & ξ^{t} L + W [\frac{\partial L}{\partial u_{t}}] \\ = & v (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) - u_{t} v, \\ C^{x} & = & ξ^{x} L + W [\frac{\partial L}{\partial u_{x}} + D_{x}^{2} \frac{\partial L}{\partial U_{x x x}}] \\ + D_{x} W [- D_{x} \frac{\partial L}{\partial u_{x x x}}] + D_{x}^{2} W [\frac{\partial L}{\partial u_{x x x}}] \\ = & - u_{t} (6 u v - 6 u^{2} v + v_{x x}) + u_{t x} v_{x} - u_{t x x} v . \end{array} \end{array}

(3.15)

Similarly, for the generator X₂, we have W = − u_x. Thus, formulas (3.7) yield the following conserved vector

\begin{array}{rcl} \begin{array}{rcl} C^{t} & = & - u_{x} v, \\ C^{x} & = & v (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) \\ - u_{x} (6 u v - 6 u^{2} v + v_{x x}) + u_{x x} v_{x} - u_{x x x} v . \end{array} \end{array}

(3.16)

For the generator X₃, we have

W = - \frac{u}{3} + \frac{1}{6} - (t + \frac{x}{3}) u_{x} - {t u}_{t}

. According to formula (3.7), we have the conservation vector of the Gardner equation as follows:

\begin{array}{rcl} \begin{array}{rcl} C^{t} & = & t (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) v \\ - (\frac{u}{3} - \frac{1}{6} + (t + \frac{x}{3}) u_{x} + {t u}_{t}) v, \\ C^{x} & = & (t + \frac{x}{3}) v (u_{t} + 6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) \\ + (- \frac{u}{3} + \frac{1}{6} - (t + \frac{x}{3}) u_{x} - {t u}_{t}) (6 u v \\ - 6 u^{2} v + v_{x x}) - (- \frac{2}{3} u_{x} - (t + \frac{x}{3}) u_{x x} - {t u}_{t x}) v_{x} \\ + (- u_{x x} - (t + \frac{x}{3}) u_{x x x} - {t u}_{t x x}) v . \end{array} \end{array}

(3.17)

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{array}{rcl} {\begin{cases} G^{1} = u_{t} - 6 {p u u}_{x} + 3 {p v v}_{x} - {p u}_{x x x} = 0, \\ G^{2} = v_{t} + 3 {p u v}_{x} + {p v}_{x x x} = 0. \end{cases} \end{array}

(3.18)

Proposition 2. The HS equation is nonlinearly self-adjoint.

Proof. The formal Lagrangian of the HS equation (3.18) can be written as

\begin{array}{rcl} \begin{array}{l} L = ω^{1} (u_{t} - 6 {p u u}_{x} + 3 {p v v}_{x} - {p u}_{x x x}) \\ + ω^{2} (v_{t} + 3 {p u v}_{x} + {p v}_{x x x}) . \end{array} \end{array}

(3.19)

Then, one has

\begin{array}{rcl} \begin{array}{rcl} \frac{δ L}{δ u} & = & - ω_{t}^{1} + 6 p u ω_{x}^{1} + p ω_{x x x}^{1} - 3 {p v}_{x} ω^{2}, \\ \frac{δ L}{δ v} & = & - 3 p v ω_{x}^{1} - ω_{t}^{2} + 3 {p u}_{x} ω^{2} + 3 p u ω_{x}^{2} - p ω_{x x x}^{2} . \end{array} \end{array}

(3.20)

Substituting

ω^{1} = u

and

ω^{2} = v

into equation (3.20), we have

\begin{array}{rcl} \begin{array}{rcl} \frac{δ L}{δ u} & = & - (u_{t} - 6 {p u u}_{x} + 3 {p v v}_{x} - {p u}_{x x x}), \\ \frac{δ L}{δ v} & = & - (v_{t} + 3 {p u v}_{x} + {p v}_{x x x}) . \end{array} \end{array}

(3.21)

Hence, the special HS equation is nonlinearly self-adjoint.

The special HS equation admits three infinitesimal generators as follows:

\begin{array}{rcl} \begin{array}{l} X_{1} = \frac{\partial}{\partial t}, X_{2} = \frac{\partial}{\partial x}, X_{3} = \frac{x}{3} \frac{\partial}{\partial x} \\ + t \frac{\partial}{\partial t} - \frac{2}{3} u \frac{\partial}{\partial u} - \frac{2}{3} v \frac{\partial}{\partial v} . \end{array} \end{array}

(3.22)

The corresponding conservation law is limited by

\begin{array}{rcl} [D_{t} C^{t} + D_{x} C^{x}] |_{(4.21)} = 0, \end{array}

(3.23)

where C = (C^t, C^x) denotes the conserved vector of (3.7). For the generator, the corresponding Lie characteristic function is given by W¹ = − u_t, W² = − v_t. One can find the conservation vector of equation (3.18) given by

\begin{array}{rcl} \begin{array}{rcl} C^{t} & = & ξ^{t} L + W^{1} [\frac{\partial L}{\partial u_{t}}] + W^{2} [\frac{\partial L}{\partial v_{t}}] \\ = & ω^{1} (u_{t} - 6 {p u u}_{x} + 3 {p v v}_{x} - {p u}_{x x x}) \\ + ω^{2} (v_{t} + 3 {p u v}_{x} + {p v}_{x x x}) - u_{t} ω_{1} - v_{t} ω_{2}, \\ C^{x} & = & ξ^{x} L + W^{1} [\frac{\partial L}{\partial u_{x}} + D_{x}^{2} \frac{\partial L}{\partial u_{x x x}}] \\ + D_{x} W^{1} [- D_{x} \frac{\partial L}{\partial u_{x x x}}] + D_{x}^{2} W^{1} \frac{\partial L}{\partial u_{x x x}} \\ + W^{2} [\frac{\partial L}{\partial v_{x}} + D_{x}^{2} \frac{\partial L}{\partial v_{x x x}}] \\ + D_{x} W^{2} [- D_{x} \frac{\partial L}{\partial v_{x x x}}] + D_{x}^{2} W^{2} \frac{\partial L}{\partial v_{x x x}} \\ = & u_{t} (6 p u ω^{1} + {p w}_{x x}^{1}) - {p u}_{x t} ω_{x}^{1} + {p u}_{x x t} ω^{1} \\ - v_{t} (3 p v ω^{1} - 3 p u ω^{2} + p ω_{x x}^{2}) \\ + {p v}_{x t} ω_{x}^{2} - {p v}_{x x t} ω^{2} . \end{array} \end{array}

(3.24)

For the generator X₂, we have W¹ = − u_x, W² = − v_x. Thus, formulas (3.7) yield the following conserved vector

\begin{array}{rcl} \begin{array}{rcl} C^{t} & = & - u_{x} ω^{1} - v_{x} ω^{2}, \\ C^{x} & = & ω^{1} (u_{t} - 6 {p u u}_{x} + 3 {p v v}_{x} - {p u}_{x x x}) + ω^{2} \\ \times (v_{t} + 3 {p u v}_{x} + {p v}_{x x x}) + u_{x} (6 p u ω^{1} + {p w}_{x x}^{1}) \\ - {p u}_{x x} ω_{x}^{1} + {p u}_{x x x} ω^{1} - v_{x} (3 p v ω^{1} - 3 p u ω^{2} + p ω_{x x}^{2}) \\ + {p v}_{x x} ω_{x}^{2} - {p v}_{x x x} ω^{2} . \end{array} \end{array}

(3.25)

For the generator X₃, we have

W^{1} = - (\frac{2}{3} u + \frac{x}{3} u_{x} + \frac{2 t}{3} u_{t}), W^{2} = - (\frac{2}{3} v + \frac{x}{3} v_{x} + \frac{2 t}{3} v_{t})

. Thus, formulas (3.7) yield the following conserved vector

\begin{array}{rcl} \begin{array}{rcl} C^{t} & = & t [ω^{1} (u_{t} - 6 {p u u}_{x} + 3 {p v v}_{x} - {p u}_{x x x}) \\ + ω^{2} (v_{t} + 3 {p u v}_{x} + {p v}_{x x x})] \\ - (\frac{2}{3} u + \frac{x}{3} u_{x} + \frac{2 t}{3} u_{t}) ω^{1} \\ - (\frac{2}{3} v + \frac{x}{3} v_{x} + \frac{2 t}{3} v_{t}) ω^{2}, \\ C^{x} & = & \frac{x}{3} [ω^{1} (u_{t} - 6 {p u u}_{x} + 3 {p v v}_{x} - {p u}_{x x x}) \\ + ω^{2} (v_{t} + 3 {p u v}_{x} + {p v}_{x x x})] \\ - (\frac{2}{3} u + \frac{x}{3} u_{x} + \frac{2 t}{3} u_{t}) (- 6 p u ω^{1} + p ω_{x x}^{1}) \\ - p (u_{x} + \frac{x}{3} u_{x} + \frac{2}{3} {t u}_{t x}) ω_{x}^{1} \\ - (\frac{4}{3} u_{x x} + \frac{x}{3} u_{x x x} + \frac{2}{3} {t u}_{t x x}) ω^{1} - (\frac{2}{3} v + \frac{x}{3} v_{x} \\ + \frac{2 t}{3} v_{t}) (3 p v ω^{1} - 3 p u ω^{2} + p ω_{x x}^{2}) \\ + p (v_{x} + \frac{x}{3} v_{x x} + \frac{2 t}{3} v_{t x}) ω_{x}^{2} - p (\frac{4}{3} v_{x x} \\ + \frac{x}{3} v_{x x x} + \frac{2 t}{3} v_{t x x}) ω^{2} . \end{array} \end{array}

(3.26)

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 D_xG of equation (2.6), it is found that the system consists of the adjoint symmetry determining equations on Λ

\begin{array}{rcl} - D_{t} Λ - 6 {u D}_{x} Λ + 6 u^{2} D_{x} Λ - D_{x}^{3} Λ = 0, \end{array}

(A1)

and extra determining equations on Λ,

\begin{array}{rcl} - D_{x} Λ_{u_{x}} + D_{x}^{2} Λ_{u_{x x}} = 0, \end{array}

(A2)

\begin{array}{rcl} Λ_{u_{x}} - D_{x} Λ_{u_{x x}} = 0, \end{array}

(A3)

where

D_{t} = \partial_{t} - (6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x}) \partial_{u} - {(6 {u u}_{x} - 6 u^{2} u_{x} + u_{x x x})}_{x} \partial_{u_{x}} + \dots

is the total derivative operator, which expresses t derivatives of u through the Gardner equation. By expanding equation (A3), we find that the highest order derivative term is

u_{x x x} Λ_{u_{x x} u_{x x}}

, and hence

u_{x x x} Λ_{u_{x x} u_{x x}} = 0

, so that

\begin{array}{rcl} Λ = a (t, x, u, u_{x}) u_{x x} + b (t, x, u, u_{x}) . \end{array}

(A4)

Substitute Λ in the above formula into (A3), we can get

\begin{array}{rcl} b_{u_{x}} - a_{u} u_{x} - a_{x} = 0. \end{array}

(A5)

By comparing the coefficients u_xxxx, u_xxx and so on in equation (A1), we can get the following equations

\begin{array}{rcl} \begin{array}{l} 3 Λ_{{x u}_{x}} + 3 u_{x x} Λ_{u_{x} u_{x x}} + 3 u_{x} Λ_{{u u}_{x x}} = 0, \\ 6 u_{x} Λ_{{x u u}_{x x}} + 3 u_{x}^{2} Λ_{{u u u}_{x x}} + 3 Λ_{{x u}_{x}} + 3 Λ_{{x x u}_{x}} + 3 u_{x x} Λ_{{u u}_{x x}} = 0, \\ Λ_{t} - (6 {u u}_{x} - 6 u^{2} u_{x}) Λ_{u} - (18 u_{x} u_{x x} - 12 u_{x}^{3} \\ - 36 {u u}_{x} u_{x x}) Λ_{u_{x x}} + (6 u - 6 u^{2}) (Λ_{x} + u_{x} Λ_{u}) \\ - (6 u_{x}^{2} - 12 {u u}_{x}^{2}) Λ_{u_{x}} + 3 u_{x} u_{x x} Λ_{u u} + 3 u_{x} Λ_{x x u} \\ + 3 u_{x x} Λ_{x u} + 3 u_{x}^{2} Λ_{x u u} + u_{x}^{3} Λ_{u u u} + Λ_{x x x} = 0. \end{array} \end{array}

By solving the above equations and equations (A4) and (A5), we can obtain the general form of Λ as follows:

\begin{array}{rcl} \begin{array}{l} Λ = c_{1} [(2 u^{3} - 3 u^{2} + u - u_{x x}) t \\ - \frac{x}{6} + \frac{1}{3} x u] + c_{2} [- 2 u^{3} + 3 u^{2} + u_{x x}] + c_{3} u + c_{4}, \end{array} \end{array}

(A6)

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, u_t, u_x), consisting of

\begin{array}{rcl} D_{t}^{2} Λ - D_{x}^{2} Λ + {N g}^{2} u^{N - 1} Λ = 0, \end{array}

(B1)

which is the symmetry determining equation (2.14) on Λ, and

\begin{array}{rcl} 2 Λ_{u} + D_{t} Λ_{u_{t}} - D_{x} Λ_{u_{x}} = 0, \end{array}

(B2)

which is an extra determining equation on Λ. Here

D_{t} = \partial_{t} + u_{t} \partial_{u} + u_{t x} \partial_{u_{x}} + (u_{t t} - u_{x x} + g^{2} u^{N}) \partial_{u_{t}} + \dots

is the total derivative operator which expresses t derivatives through the LGH equation. And from equation (B2), we can get

\begin{array}{rcl} \begin{array}{l} 2 Λ_{u} + Λ_{{t u}_{t}} + u_{t} Λ_{{u u}_{t}} + (u_{x x} - g^{2} u^{N}) Λ_{u_{t} u_{t}} \\ - Λ_{{x u}_{x}} - u_{x} Λ_{{u u}_{x}} - u_{x x} Λ_{u_{x} u_{x}} = 0. \end{array} \end{array}

(B3)

Since Λ does not contain u_xx, we have

\begin{array}{rcl} Λ_{u_{t} u_{t}} = Λ_{u_{x} u_{x}}, \end{array}

(B4)

\begin{array}{rcl} 2 Λ_{u} + Λ_{{t u}_{t}} + u_{t} Λ_{{u u}_{t}} - g^{2} u^{N} Λ_{u_{t} u_{t}} - Λ_{{x u}_{x}} - u_{x} Λ_{{u u}_{x}} = 0. \end{array}

(B5)

We next consider equation (B1), comparison coefficient u_xx, we can get

\begin{array}{rcl} - 2 u_{x} Λ_{{u u}_{x}} - 2 Λ_{{x u}_{x}} + 2 Λ_{{t u}_{t}} + 2 u_{t} Λ_{{u u}_{t}} - 2 g^{2} u^{N} Λ_{u_{t} u_{t}} = 0. \end{array}

(B6)

Combining equations (B5) and (B6), we have

\begin{array}{rcl} 4 Λ_{u} = 0. \end{array}

(B7)

Compare the coefficients of u_xt in equation (B1), we have

\begin{array}{rcl} - 2 Λ_{{x u}_{t}} + 2 Λ_{{t u}_{x}} - 2 g^{2} u^{N} Λ_{u_{t} u_{x}} = 0. \end{array}

(B8)

By solving equations (B4), (B5), (B7) and (B8), we can get the general form of multiplier is

\begin{array}{rcl} \begin{array}{l} Λ = (u_{x} - u_{t}) F_{1} (t - x) \\ + (u_{t} + u_{x}) F_{2} (t + x) + c_{1} u_{t} + F_{3} (x, t), \end{array} \end{array}

(B9)

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 (B9) into (B1), we can get

\begin{array}{rcl} \begin{array}{rcl} - (u_{t} + u_{x}) F_{1 t t} + (u_{t} + u_{x}) F_{2 t t} + F_{3 t t} \\ + (- u_{t} + u_{x}) F_{1 x x} - (u_{t} + u_{x}) F_{2 x x} \\ - F_{3 x x} + ((- u_{t} + u_{x}) F_{1} + (u_{t} + u_{x}) F_{2} \\ + c_{1} u_{t} + F_{3}) {N g}^{2} u^{N - 1} = 0, \end{array} \end{array}

(B10)

Equation (B10) separates into

\begin{array}{rcl} F_{2}^{''} = 0, F_{1}^{'} = 0, F_{3} = 0, \end{array}

(B11)

so we can obtain F₁ = a₁ = constant, F₂ = a₂ = constant and F₃ = 0. Therefore, we can get the specific form of multipliers

\begin{array}{rcl} Λ = a_{1} (u_{x} - u_{t}) + a_{2} (u_{t} + u_{x}) + c_{1} u_{t}, \end{array}

(B12)

where a₁, a₂ and c₁ are arbitrary constants.

Appendix C

Substitute equations (2.26) into (2.29) when σ is 1, 2, respectively. We have

\begin{array}{rcl} \begin{array}{rcl} - D_{t} Λ_{1} + 6 {p u D}_{x} Λ_{1} + {p D}_{x} x^{3} Λ_{1} + 3 {p v}_{x} Λ_{2} \\ + (\frac{\partial Λ_{1}}{\partial u} G^{1} - D_{x} (\frac{\partial Λ_{1}}{\partial u_{x}} G^{1}) \\ + D_{x}^{2} (\frac{\partial Λ_{1}}{\partial u_{x x}} G^{1})) + (\frac{\partial Λ_{2}}{\partial u} G^{2} - D_{x} \\ \times (\frac{\partial Λ_{2}}{\partial u_{x}} G^{2}) + D_{x}^{2} (\frac{\partial Λ_{2}}{\partial u_{x x}} G^{2})) = 0, \end{array} \end{array}

(C1)

\begin{array}{rcl} \begin{array}{rcl} - D_{t} Λ_{2} - {r v D}_{x} Λ_{1} - 3 p Λ_{2} - 3 {p u D}_{x} Λ_{2} - {p D}_{x}^{3} Λ_{2} \\ + (\frac{\partial Λ_{1}}{\partial v} G^{1} - D_{x} (\frac{\partial Λ_{1}}{\partial v_{x}} G^{1}) \\ + D_{x}^{2} (\frac{\partial Λ_{1}}{\partial v_{x x}} G^{1})) + (\frac{\partial Λ_{2}}{\partial v} G^{2} - D_{x} (\frac{\partial Λ_{2}}{\partial v_{x}} G^{2}) \\ + D_{x}^{2} (\frac{\partial Λ_{2}}{\partial v_{x x}} G^{2})) = 0. \end{array} \end{array}

(C2)

Consequently, the non-leading terms in equation (2.29) are given by

\begin{array}{rcl} - D_{t} Λ_{1} + 6 {p u D}_{x} Λ_{1} + {p D}_{x} x^{3} Λ_{1} + 3 {p v}_{x} Λ_{2} = 0, \end{array}

(C3)

\begin{array}{rcl} - D_{t} Λ_{2} - {r v D}_{x} Λ_{1} - 3 p Λ_{2} - 3 {p u D}_{x} Λ_{2} - {p D}_{x}^{3} Λ_{2} = 0. \end{array}

(C4)

Comparing the coefficients of derivatives of G and G, we can get the leading term in equation (2.29)

\begin{array}{rcl} - \frac{\partial Λ_{1}}{\partial v_{x x}} + \frac{\partial Λ_{2}}{\partial u_{x x}} = 0, \end{array}

(C5)

\begin{array}{rcl} 2 \frac{\partial Λ_{1}}{\partial u_{x}} - 2 D_{x} \frac{\partial Λ_{1}}{\partial u_{x x}} = 0, 2 \frac{\partial Λ_{2}}{\partial v_{x}} - 2 D_{x} \frac{\partial Λ_{2}}{\partial v_{x x}} = 0, \end{array}

(C6)

\begin{array}{rcl} \begin{array}{l} \frac{\partial Λ_{1}}{\partial v_{x}} + \frac{\partial Λ_{2}}{\partial u_{x}} - 2 D_{x} \frac{\partial Λ_{2}}{\partial u_{x x}} = 0, \\ \times \frac{\partial Λ_{2}}{\partial u_{x}} + \frac{\partial Λ_{1}}{\partial v_{x}} - 2 D_{x} \frac{\partial Λ_{1}}{\partial v_{x x}} = 0, \end{array} \end{array}

(C7)

\begin{array}{rcl} - D_{x} \frac{\partial Λ_{1}}{\partial u_{x}} + D_{x}^{2} \frac{\partial Λ_{1}}{\partial u_{x x}} = 0, - D_{x} \frac{\partial Λ_{2}}{\partial v_{x}} + D_{x}^{2} \frac{\partial Λ_{2}}{\partial v_{x x}} = 0, \end{array}

(C8)

\begin{array}{rcl} \begin{array}{l} - \frac{\partial Λ_{1}}{\partial v} + \frac{\partial Λ_{2}}{\partial u} - D_{x} \frac{\partial Λ_{2}}{\partial u_{x}} + D_{x}^{2} \frac{\partial Λ_{2}}{\partial u_{x x}} = 0, \\ - \frac{\partial Λ_{2}}{\partial u} + \frac{\partial Λ_{1}}{\partial v} - D_{x} \frac{\partial Λ_{2}}{\partial v_{x}} + D_{x}^{2} \frac{\partial Λ_{2}}{\partial v_{x x}} = 0. \end{array} \end{array}

(C9)

Consider equation (C6). Its highest-order term is

u_{x x x} Λ_{1 u_{x x} u_{x x}}

and

v_{x x x} Λ_{2 v_{x x} v_{x x}}

, hence

Λ_{1 u_{x x} u_{x x}} = 0

and

Λ_{2 v_{x x} v_{x x}} = 0

. Similarly, we have

Λ_{1 v_{x x} v_{x x}} = 0

and

u_{x x x} Λ_{1 u_{x x} u_{x x}}

from equation (C7). This means that Λ₁ and Λ₂ are linear with respect to u_xx and v_xx, combining equation (C5), we have the general form of Λ₁ and Λ₂

\begin{array}{rcl} \begin{array}{rcl} Λ_{1} & = & a_{1} (t, x, u, u_{x}) u_{x x} + b_{1} (t, x, u, u_{x}) v_{x x} \\ + c_{1} (t, x, u, u_{x}), \\ Λ_{2} & = & b_{1} (t, x, u, u_{x}) u_{x x} + b_{2} (t, x, u, u_{x}) v_{x x} \\ + c_{2} (t, x, u, u_{x}), \end{array} \end{array}

(C10)

Then the remaining terms in equation (C6), after some cancellations,

\begin{array}{rcl} \begin{array}{l} b_{1 u_{x}} v_{x x} + c_{1 u_{x}} - (a_{1 x} + a_{1 u} u_{x} + a_{1 v} v_{x} + a_{1 v_{x}} v_{x x}) = 0, \\ b_{1 v_{x}} v_{x x} + c_{2 v_{x}} - (b_{2 x} + b_{2 u} u_{x} + b_{2 v} v_{x} + b_{2 u_{x}} u_{x x}) = 0. \end{array} \end{array}

(C11)

According to the coefficients of u_xxxx and v_xxxx of equations (C3) and (C4), we have

\begin{array}{rcl} \begin{array}{l} a_{1 v_{x}} u_{x x} + b_{1 v_{x}} v_{x x} + c_{1 v_{x}} = 0, \\ b_{1 u_{x}} u_{x x} + b_{2 u_{x}} u_{x x} + c_{2 u_{x}} = 0. \end{array} \end{array}

(C12)

Clearly, equation (C12) separates into

\begin{array}{rcl} a_{1 v_{x}} = b_{1 v_{x}} = c_{1 v_{x}} = 0, b_{1 u_{x}} = b_{2 u_{x}} = c_{2 u_{x}} = 0. \end{array}

(C13)

Hence a_1x = a_1u = a_1v = 0, b_2x = b_2u = b_2v = 0 from equation (C11). Similarly, we have

a_{1 u_{x}} = b_{1 x} = b_{1 u} = b_{1 v} = b_{2 v_{x}} = 0

. That is to say, a₁, b₁ and b₂ are independent of x, u, u_x, equation (C10) becomes

\begin{array}{rcl} \begin{array}{rcl} Λ_{1} & = & a_{1} (t) u_{x x} + b_{1} (t) v_{x x} + c_{1} (t, x, u), \\ Λ_{2} & = & b_{1} (t) u_{x x} + b_{2} (t) v_{x x} + c_{2} (t, x, u), \end{array} \end{array}

(C14)

Next, we solve the concrete forms of a₁, b₁, b₂, c₁, c₂. The coefficient of u_xxxxx from equation (C3), we have

\begin{array}{rcl} 2 {p b}_{1} (t) = 0, \end{array}

(C15)

hence b₁(t) = 0. Comparing the coefficient v_xxx in equation (C3) with the coefficient u_xxx in equation (C4), we have

\begin{array}{rcl} 2 {p c}_{1 v} + {r v a}_{1} (t) = 0, 2 {p c}_{2 u} + {r v a}_{1} (t) = 0. \end{array}

(C16)

It yields that

\begin{array}{rcl} \begin{array}{rcl} c_{1} & = & - \frac{r}{4 p} v^{2} a_{1} (t) + k_{1} (x, t, u), \\ c_{2} & = & - \frac{r}{2 p} {u v a}_{1} (t) + l_{1} (x, t, v) . \end{array} \end{array}

(C17)

Furthermore, by comparing the coefficients of u_x, u_xx, and constant terms in equations (C3) and (C4), we can obtain the following equations:

\begin{array}{rcl} - a'_{1} (t) - 18 a_{1} (t) {p u}_{x} + 3 {p u}_{x} c_{1 u u} + 3 {p c}_{1 x u} = 0, \end{array}

(C18)

\begin{array}{rcl} \frac{r}{2} a_{1} (t) + {p b}_{2} (t) = 0, \end{array}

(C19)

\begin{array}{rcl} b'_{2} (t) - 3 {p b}_{2} (t) + 3 {p c}_{2 u v} = 0, \end{array}

(C20)

\begin{array}{rcl} 9 {p u c}_{2 u} + {r v c}_{1 u} + 3 {p c}_{2} = 0, \end{array}

(C21)

\begin{array}{rcl} - c'_{1} (t) + 6 {p u c}_{1 x} + {p c}_{1 x x x} = 0, \end{array}

(C22)

\begin{array}{rcl} c_{2 t} + {r v c}_{1 x} + 3 {p u c}_{2 x} + {p c}_{2 x x x} = 0, \end{array}

(C23)

\begin{array}{rcl} c_{1 x v} = c_{2 x u} = c_{2 v v} = 0. \end{array}

(C24)

Simultaneous equations (C17)–(C24), we can get

\begin{array}{rcl} \begin{array}{rcl} c_{1} & = & - \frac{r}{4 p} a_{1} v^{2} + 3 a_{1} u^{2} + k_{1}, \\ c_{2} & = & - \frac{r}{2 p} a_{1} u v, b_{2} = - \frac{r}{2 p} a_{1}, \end{array} \end{array}

(C25)

where a₁ and k₁ are arbitrary constants. Therefore, we can get the specific form of multipliers

\begin{array}{rcl} \begin{array}{rcl} Λ_{1} & = & a_{1} u_{x x} - \frac{r}{4 p} a_{1} v^{2} + 3 a_{1} u^{2} + k_{1}, \\ Λ_{2} & = & - \frac{r}{2 p} a_{1} v_{x x} - \frac{r}{2 p} a_{1} u v, \end{array} \end{array}

(C26)

where a₁ and k₁ are arbitrary constants.

References

List( Publishing order | Descend order by publishing year | Descend order by cited within ) Chart analysis

1	Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Method for solving the Korteweg-de Vries equation Phys. Rev. Lett. 19 1095 https://doi.org/10.1103/PhysRevLett.19.1095 Cited in this article [1]

2	Zhang Y F, Zhang X Z, Dong H H 2017 Infinite conservation laws, continuous symmetries and invariant solutions of some discrete integrable equations Commun. Theor. Phys. 68 755 https://doi.org/10.1088/0253-6102/68/6/755

3	Zhang Y F, Tam H 2013 Discussion on integrable properties for higher-dimensional variable-coefficient nonlinear partial differential equations J. Math. Phys. 54 013516 https://doi.org/10.1063/1.4788665

4	Ma W X, Zhou Y 2018 Lump solutions to nonlinear partial differential equations via Hirota bilinear forms J. Differ. Equ. 264 2633 2659 https://doi.org/10.1016/j.jde.2017.10.033 Cited in this article [1]

5	Babcock G T, Wikström M 1992 Oxygen activation and the conservation of energy in cell respiration Nature 356 301 309 https://doi.org/10.1038/356301a0 Cited in this article [1]

6	Shirani E, Ashgriz N, Mostaghimi J 2005 Interface pressure calculation based on conservation of momentum for front capturing methods J. Comput. Phys. 203 154 175 https://doi.org/10.1016/j.jcp.2004.08.017 Cited in this article [1]

7	Wolfenstein L 1951 Conservation of angular momentum in the statistical theory of nuclear reactions Phys. Rev. 82 690 https://doi.org/10.1103/PhysRev.82.690 Cited in this article [1]

8	Naz R, Mason D P, Mahomed F M 2009 Conservation laws and conserved quantities for laminar two-dimensional and radial jets Nonlinear Anal. Real. World Appl. 10 2641 2651 https://doi.org/10.1016/j.nonrwa.2008.07.003 Cited in this article [1]

9	Yang J Y, Ma W X 2016 Conservation laws of a perturbed Kaup-Newell equation Mod. Phys. Lett. B 30 1650381 https://doi.org/10.1142/S0217984916503814 Cited in this article [1]

10	Han Z, Zhang Y F, Zhao Z L 2013 Double reduction and exact solutions of Zakharov–Kuznetsov modified equal width equation with power law nonlinearity via conservation laws Commun. Theor. Phys. 60 699 https://doi.org/10.1088/0253-6102/60/6/12

11	Zhang Y F, Zhang X Z, Dong H H 2017 Infinite conservation laws, continuous symmetries and invariant solutions of some discrete integrable equations Commun. Theor. Phys. 68 755 https://doi.org/10.1088/0253-6102/68/6/755 Cited in this article [1]

12	Noether E 1971 Invariant variation problems Transp. Theory Stat. Phys. 2 186 207 https://doi.org/10.1080/00411457108231446 Cited in this article [1]

13	Anco S C, Bluman G 2002 Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications Eur. J. Appl. Math. 13 545 566 https://doi.org/10.1017/S095679250100465X Cited in this article [1]

14	Anco S C, Bluman G 2002 Direct construction method for conservation laws of partial differential equations Part II: General treatment Eur. J. Appl. Math. 13 567 585 https://doi.org/10.1017/S0956792501004661 Cited in this article [1]

15	Ibragimov N H 2011 Nonlinear self-adjointness in constructing conservation laws J. Phys. A: Math. Theor. 44 432002 https://doi.org/10.1088/1751-8113/44/43/432002 Cited in this article [3]

16	Atherton R W, Homsy G M 1975 On the existence and formulation of variational principles for nonlinear differential equations Studies Appl. Math. 54 31 60 https://doi.org/10.1002/sapm197554131 Cited in this article [1]

17	Olver P J 1993 Applications of Lie Groups to Differential Equations Berlin Springer https://doi.org/10.1007/978-1-4612-4350-2 Cited in this article [1]

18	Kara A H, Mahomed F M 2006 Noether-type symmetries and conservation laws via partial Lagragians Nonlinear Dyn. 45 367 383 https://doi.org/10.1007/s11071-005-9013-9 Cited in this article [1]

19	Kara A H, Mahomed F M 2000 Relationship between symmetries and conservation laws Int. J. Theor. Phys. 39 23 40 https://doi.org/10.1023/A:1003686831523 Cited in this article [1]

20	Atherton R W, Homsy G M 1975 On the existence and formulation of variational principles for nonlinear differential equations Studies Appl. Math. 54 31 60 https://doi.org/10.1002/sapm197554131

21	Ibragimov N H 2007 A new conservation theorem J. Math. Anal. Appl. 333 311 328 https://doi.org/10.1016/j.jmaa.2006.10.078 Cited in this article [1]

22	Fu Z, Liu S, Liu S 2004 New kinds of solutions to Gardner equation Chaos Solitons Fractals. 20 301 309 https://doi.org/10.1016/S0960-0779(03)00383-7 Cited in this article [1]

23	Daghan D, Donmez O 2016 Exact solutions of the Gardner equation and their applications to the different physical plasmas Braz. J. Phys. 46 321 333 https://doi.org/10.1007/s13538-016-0420-9

24	Yan Z 2003 Jacobi elliptic solutions of nonlinear wave equations via the new sinh-Gordon equation expansion method J. Phys. A: Math. Theor. 36 1961 https://doi.org/10.1088/0305-4470/36/7/311 Cited in this article [1]

25	Olver P J 1986 Applications of Lie Groups to Differential Equations Berlin Springer https://doi.org/10.1007/978-1-4684-0274-2 Cited in this article [2]

26	Bluman G, Kumei S 1989 Symmetries and Differential Equations Berlin Springer https://doi.org/10.1007/978-1-4757-4307-4 Cited in this article [2]

Iftikhar

, Ghafoor

, Zubair

, Firdous

, Mohyud-Din

S T

2013 (GÂ´/G, 1/G)-expansion method for traveling wave solutions of (2+1) dimensional generalized KdV, Sin Gordon and Landau–Ginzburg–Higgs Equations Sci. Res. Essays. 8 1349 1359

https://doi.org/10.5897/SREX2013.5555

Cited in this article [1]

28	Cyrot M 1973 Ginzburg-Landau theory for superconductors Rep. Prog. Phys. 36 103 158 https://doi.org/10.1088/0034-4885/36/2/001 Cited in this article [1]

29	Hu W P, Deng Z C, Han S M, Fa W 2009 Multi-symplectic Runge–Kutta methods for Landau–Ginzburg–Higgs equation Appl. Math. Mech. 30 1027 1034 https://doi.org/10.1007/s10483-009-0809-x Cited in this article [1]

30	Halim A A, Kshevetskii S P, Leble S B 2003 Numerical integration of a coupled Korteweg-de Vries system Comput. Math. Appl. 45 581 591 https://doi.org/10.1016/S0898-1221(03)00018-X Cited in this article [1]

31	Halim A A, Leble S B 2004 Analytical and numerical solution of a coupled KdV-MKdV system Chaos Solitons Fractals. 19 99 108 https://doi.org/10.1016/S0960-0779(03)00085-7

32	Ma W X 2019 Interaction solutions to Hirota–Satsuma–Ito equation in (2+1)-dimensions Front Math. China. 14 619 629 https://doi.org/10.1007/s11464-019-0771-y Cited in this article [1]

33	Baldwin D 2004 Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs J. Symb. Comput. 37 669 705 https://doi.org/10.1016/j.jsc.2003.09.004 Cited in this article [1]

34	Hereman W, Nuseir A 1997 Symbolic methods to construct exact solutions of nonlinear partial differential equations Math. Comput. Simul. 43 13 27 https://doi.org/10.1016/S0378-4754(96)00053-5 Cited in this article [1]

35	Anco S C, Mohiuddin M, Wolf T 2012 Traveling waves and conservation laws for complex mKdV-type equations Appl. Math. Comput. 219 679 698 https://doi.org/10.1016/j.amc.2012.06.061 Cited in this article [1]

Acknowledgments

The authors wish to thank the anonymous referees for their valuable suggestions.

RIGHTS & PERMISSIONS

PDF(367 KB)

Collection(s)

Mathematical Physics

Accesses

Citation

Detail

Sections

Recommended

Abstract
Key words
Cite this article
1. Introduction
2. Direct construction method for conservation laws of partial differential equations
2.1. Gardner equation
2.2. Landau–Ginzburg–Higgs equation
2.3. Hirota–Satsuma equation
3. Nonlinear self-adjointness and conservation laws
4. Conclusion and discussion
Funding
Appendix A
Appendix B
Appendix C
References
Acknowledgments
RIGHTS & PERMISSIONS

Received	Revised	Accepted	Published
2023-05-27	2023-12-07	2024-01-02	2024-05-01
Issue Date
2024-04-22

Please choose a citation manager

Content to export

Abstract

Key words

Cite this article

1. Introduction

2. Direct construction method for conservation laws of partial differential equations

2.1. Gardner equation

2.2. Landau–Ginzburg–Higgs equation

2.3. Hirota–Satsuma equation

3. Nonlinear self-adjointness and conservation laws

4. Conclusion and discussion

Funding

Appendix A

Appendix B

Appendix C

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

Acknowledgments

{{custom_ack.title_en}}

RIGHTS & PERMISSIONS

模态框（Modal）标题

Please choose a citation manager

Content to export

Abstract

Key words

Cite this article

1. Introduction

2. Direct construction method for conservation laws of partial differential equations

2.1. Gardner equation

2.2. Landau–Ginzburg–Higgs equation

2.3. Hirota–Satsuma equation

3. Nonlinear self-adjointness and conservation laws

4. Conclusion and discussion

Funding

Appendix A

Appendix B

Appendix C

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

Acknowledgments

{{custom_ack.title_en}}

RIGHTS & PERMISSIONS