Conservation Laws for Some Systems of Nonlinear Partial Differential Equations via Multiplier Approach

Abstract

The conservation laws for the integrable coupled KDV type system, complexly coupled kdv system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics are computed by multipliers approach. First of all, we calculate the multipliers depending on dependent variables, independent variables, and derivatives of dependent variables up to some fixed order. The conservation laws fluxes are computed corresponding to each conserved vector. For all understudying systems, the local conservation laws are established by utilizing the multiplier approach.

1. Introduction

The partial differential equations, which arise in the sciences, dynamics, fluid mechanics, electromagnetism, economics and so forth, express conservation of mass, momentum, energy, electric charge, or value of firm. All the conservation laws of partial differential equations may not have physical interpretation but are essential in studying the integrability of the PDE. The high number of conservation laws for a partial differential equation grantees that the partial differential equation is strongly integrable and can be linearized or explicitly solved [1]. Moreover, the conservation laws are used for analysis, particularly, development of numerical schemes, soliton solutions, study of properties such as bi-Hamiltonian structures and recursion operators, and reduction of partial differential equations.

There are different methods for the construction of conservation laws as described by Naz [2], Naz et al. [3], Bluman et al. [4], Hereman et al. [5], and references therein. Rocha Filho and Figueiredo [6] developed computer packages based on Noether's method for the variational problems. Wolf [7] and Wolf et al. [8] introduced computer programmes in REDUCE to calculate conservation laws.

In this work, the multiplier approach is used to derive the conservation laws for some systems of partial differential equations important due to physical point of view. Stuedel [9] introduced the multiplier approach and the conserved vectors were written in a characteristic form as 𝐷_𝑖𝑇^𝑖=Λ^𝛼𝐸_𝛼. The determining equations for the multipliers (characteristics) were obtained by taking the variational derivative of 𝐷_𝑖𝑇^𝑖=𝑄^𝛼𝐸_𝛼 for the arbitrary functions not only for solutions of system of partial differential equations [10]. A conserved vector is associated with each multiplier. The conservation laws for the integrable coupled kdv-type system, complexly coupled KDA system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics are computed by utilizing the multiplier approach. The conserved vectors derived here can be used in constructing the solutions of underlying systems in the following different ways. The corresponding potential system can be written for the conservation laws, and symmetry reductions [11] can be carried out. Another approach to deduce exact solutions is via the double reduction theory [12–14]. The exact solution can be derived if the conservation laws give physical conserved quantities like Naz et al. [15]. The exact solutions of systems under consideration are subject of future work.

The outline of paper is as follows. In Section 2, some definitions related with multiplier approach are presented. The conservation laws for integrable coupled kdv-type system, complexly coupled kdv system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics are constructed in Section 3. Conclusions are summarized in Section 4.

2. Preliminaries

Consider a 𝑘th-order system of partial differential equations (PDEs) with 𝑛 independent variables 𝑥≡(𝑥¹,𝑥²,…,𝑥^𝑛) and 𝑚 dependent variables 𝑢≡(𝑢¹,𝑢²,…,𝑢^𝑚) defined as, 𝐸_𝛼𝑥,𝑢,𝑢₍₁₎,𝑢₍₂₎,…,𝑢_(𝑘)=0,𝛼=1,2,…,𝑚, (2.1) where 𝑢_(𝑖) is the collection of 𝑖th-order partial derivatives of 𝑢.

(1) The Euler operator is defined by 𝛿𝛿𝑢^𝛼=𝜕𝜕𝑢^𝛼−𝐷_𝑖𝜕𝜕𝑢^𝛼_𝑖+𝐷_𝑖𝐷_𝑗𝜕𝜕𝑢^𝛼_𝑖𝑗−⋯, (2.2) where 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+⋯,𝑖=1,2, (2.3) is the total derivative operator with respect to 𝑥^𝑖.

(2) A conserved vector of (2.1) is an 𝑛-tuple 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛),𝑇^𝑖∈𝐴,𝑖=1,2…𝑛, such that 𝐷_𝑖𝑇^𝑖=0 (2.4) holds for all solutions of (2.1). Equation (2.4) is called a local conservation law.

(3) The multipliers Λ^𝛼 of system (2.1) has the property 𝐷_𝑖𝑇^𝑖=Λ^𝛼𝐸_𝛼, (2.5) for the arbitrary function 𝑢^𝛼 [9, 10].

(4) The determining equations for the multipliers are obtained by taking variational derivative of (2.5) (see [10]): 𝛿𝛿𝑢^𝛼Λ^𝛼𝐸_𝛼=0. (2.6) Equation (2.6) holds for the arbitrary functions 𝑢^𝛼 not only for the solutions of system (2.1).

Equation (2.6) yields multipliers for all local conservation laws. Then conserved vectors can be derived systematically using (2.5) as the determining equation. But in some problems it is not difficult to construct the conserved vectors by elementary manipulations once the multiplier has been determined.

3. Conservation Laws for Nonlinear Systems of Partial Differential Equations

3.1. Integrable Coupled System

Consider the integrable coupled system [16, 17] 𝐸₁=𝑢_𝑡+𝑢_𝑥𝑥−(𝑘+3)(𝑘+6)𝑢²−𝑘²𝑣²_𝑥+2𝑘(𝑘+6)𝑣𝑢_𝑥+(𝑘+3)𝑢𝑣_𝑥𝐸=0,₂=𝑣_𝑡+𝑣_𝑥𝑥−𝑘(𝑘−3)𝑣²−(𝑘+3)²𝑢²_𝑥+2(𝑘+3)𝑘𝑣𝑢_𝑥+(𝑘−3)𝑢𝑣_𝑥=0. (3.1) The group invariant solution of (3.1) was derived in [17]. Here we will construct conservation laws for coupled system (3.1). Consider simple multipliers of the form Λ₁(𝑡,𝑥,𝑢,𝑣) and Λ₂(𝑡,𝑥,𝑢,𝑣). Multipliers Λ₁ and Λ₂ for the system (3.1) have the property that Λ₁𝐸₁+Λ₂𝐸₂=𝐷_𝑡𝑇¹+𝐷_𝑥𝑇², (3.2) for all functions 𝑢(𝑡,𝑥) and 𝑣(𝑡,𝑥) where the total derivative operators 𝐷_𝑡 and 𝐷_𝑥 from (2.3) are 𝐷_𝑡=𝜕𝜕𝑡+𝑢_𝑡𝜕𝜕𝑢+𝑣_𝑡𝜕𝜕𝑣+𝑢_𝑡𝑡𝜕𝜕𝑢_𝑡+𝑣_𝑡𝑡𝜕𝜕𝑣_𝑡+𝑢_𝑡𝑥𝜕𝜕𝑢_𝑥+𝑣_𝑡𝑥𝜕𝜕𝑣_𝑥𝐷+⋯,_𝑥=𝜕𝜕𝑥+𝑢_𝑥𝜕𝜕𝑢+𝑣_𝑥𝜕𝜕𝑣+𝑢_𝑥𝑥𝜕𝜕𝑢_𝑥+𝑣_𝑥𝑥𝜕𝜕𝑣_𝑥+𝑢_𝑥𝑡𝜕𝜕𝑢_𝑡+𝑣_𝑥𝑡𝜕𝜕𝑣_𝑡+⋯. (3.3) The right-hand side of (3.2) is a divergence expression and 𝑇¹ and 𝑇² are the components of the conserved vector 𝑇=(𝑇¹,𝑇²). The determining equations for the multipliers Λ₁ and Λ₂ are 𝛿Λ𝛿𝑢¹𝐸₁+Λ₂𝐸₂𝛿=0,Λ𝛿𝑣¹𝐸₁+Λ₂𝐸₂=0, (3.4)(3.5) where 𝛿/𝛿𝑢 and 𝛿/𝛿𝑣 are the standard Euler operators defined in (2.2), which annihilate divergence expressions: 𝛿=𝜕𝛿𝑢𝜕𝑢−𝐷_𝑡𝜕𝜕𝑢_𝑡−𝐷_𝑥𝜕𝜕𝑢_𝑥+𝐷²_𝑡𝜕𝜕𝑢_𝑡𝑡+𝐷_𝑡𝐷_𝑥𝜕𝜕𝑢_𝑡𝑥+𝐷²_𝑥𝜕𝜕𝑢_𝑥𝑥𝛿−⋯,=𝜕𝛿𝑣𝜕𝑣−𝐷_𝑡𝜕𝜕𝑣_𝑡−𝐷_𝑥𝜕𝜕𝑣_𝑥+𝐷²_𝑡𝜕𝜕𝑣_𝑡𝑡+𝐷_𝑡𝐷_𝑥𝜕𝜕𝑣_𝑡𝑥+𝐷²_𝑥𝜕𝜕𝑣_𝑥𝑥−⋯. (3.6)(3.7) Separating (3.4) and (3.5), after expansion, with respect to different combinations of derivatives of 𝑢 and 𝑣, yields the following overdetermined system: Λ_2𝑥𝑥=0,Λ_2𝑣𝑥=0,Λ_2𝑣𝑣Λ=0,_1𝑡=−12𝑘(𝑘+3)[(𝑢−𝑣)𝑘+3𝑢]Λ_2𝑥,Λ_1𝑥()=−𝑘+3𝑘Λ_2𝑥,Λ_2𝑡=12[(𝑘+3)𝑢−𝑘𝑣]Λ_2𝑥,Λ_1𝑢=(𝑘+3)2𝑘²+6𝑘Λ₂+2Λ₁𝑘²+(𝑘+3)((𝑢−𝑣)𝑘+3𝑢)Λ_2𝑣𝑘²)],Λ[(𝑢−𝑣𝑘+3𝑢_2𝑢=−3𝑘−𝑘²Λ₂−Λ₁𝑘²−(𝑘+3)((𝑢−𝑣)𝑘+3𝑢)Λ_2𝑣)]𝑘,Λ[(𝑢−𝑣𝑘+3𝑢_1𝑣=−3𝑘−𝑘²Λ₂−Λ₁𝑘²−(𝑘+3)((𝑢−𝑣)𝑘+3𝑢)Λ_2𝑣)]𝑘.[(𝑢−𝑣𝑘+3𝑢 (3.8) The solution of system (3.8) yields following four multipliers: Λ₁⁽¹⁾=1,Λ₂⁽¹⁾=−𝑘+3𝑘,Λ₁⁽²⁾1=−𝑘²(𝑘+3)²𝑢,Λ₂⁽²⁾Λ=𝑣,₁₍₃₎=1𝑘[−𝑘𝑣+2(𝑘+3)𝑢],Λ₂⁽³⁾Λ=𝑢,₁₍₄₎=−𝑘+3𝑘[()],Λ𝑥+12𝑘+3𝑢𝑡−12𝑘𝑣𝑡₂₍₄₎=𝑥+12(𝑘+3)𝑢𝑡−12𝑘𝑡𝑣. (3.9) From (3.2) and (3.9), we obtained following four conserved vectors: 𝑇₁⁽¹⁾1=−𝑘)],𝑇[(𝑘+3𝑢−𝑘𝑣₂₍₁₎=1𝑘6(𝑘+3)²𝑢²−12𝑘(𝑘+3)𝑢𝑣+𝑘𝑣_𝑥𝑥−𝑘𝑢_𝑥𝑥−3𝑢_𝑥𝑥,𝑇₁⁽²⁾1=−2𝑘²𝑢²(𝑘+3)²−𝑘²𝑣²,𝑇₂⁽²⁾=16𝑘²3(𝑘+3)²𝑢²_𝑥−3𝑘²𝑣²_𝑥+4(𝑘+6)(𝑘+3)³𝑢³−4𝑘³(𝑘−3)𝑣³−12𝑘(𝑘+3)³𝑢²𝑣+12𝑘³(𝑘+3)𝑢𝑣²−6(𝑘+3)²𝑢𝑢_𝑥𝑥+6𝑘²𝑣𝑣_𝑥𝑥,𝑇₁⁽³⁾1=−𝑘(𝑘+3)𝑢²,𝑇−𝑘𝑢𝑣₂₍₃₎=13()𝑢3𝑘𝑘+3²_𝑥+2(𝑘+3)²(𝑘+12)𝑢³−2𝑘³𝑣³−6𝑘(𝑘+3)(𝑘+9)𝑢²𝑣+6𝑘²(𝑘+6)𝑢𝑣²−6(𝑘+3)𝑢𝑢_𝑥𝑥+3𝑘𝑢_𝑥𝑥𝑣+3𝑘𝑢𝑣_𝑥𝑥−3𝑘𝑢_𝑥𝑣_𝑥,𝑇₁⁽⁴⁾1=−𝑘6(𝑘+3)²𝑡𝑢²+6𝑘²𝑡𝑣²−12𝑘(𝑘+3)𝑡𝑢𝑣+(𝑘+3)𝑢_𝑥−𝑘𝑣_𝑥,𝑇₂⁽⁴⁾=1𝑘−(𝑘+3)(𝑥+12(𝑘+3)𝑡𝑢−12𝑘𝑡𝑣)𝑢_𝑥𝑥+(𝑥+12(𝑘+3)𝑢𝑡−12𝑘𝑡𝑣)𝑘𝑣_𝑥𝑥+6(𝑘+3)²𝑡𝑢²_𝑥+6𝑡𝑘²𝑣²_𝑥−12𝑘(𝑘+3)𝑡𝑢_𝑥𝑣_𝑥+(𝑘+3)𝑢_𝑥−𝑘𝑣_𝑥+48(𝑘+3)³𝑡𝑢³−48𝑘³𝑡𝑣³−144𝑘(𝑘+3)²𝑡𝑢²𝑣+144𝑘²(𝑘+3)𝑡𝑢𝑣²+6(𝑘+3)²𝑥𝑢²+6𝑘²𝑥𝑣²−12𝑘(𝑘+3)𝑥𝑢𝑣. (3.10)

3.2. Higher-Order Conservation Laws for Complexly Coupled KDV System

The conservation laws of complexly coupled KDV were discussed in Naz [18]𝑢_𝑡−6𝑢𝑢_𝑥−6𝑣𝑣_𝑥−𝑢_𝑥𝑥𝑥𝑣=0,_𝑡−6𝑢𝑣_𝑥−6𝑣𝑢_𝑥−𝑣_𝑥𝑥𝑥=0, (3.11) and six conserved vectors were derived by multipliers approach with multipliers of form Λ₁(𝑡,𝑥,𝑢,𝑣) and Λ₂(𝑡,𝑥,𝑢,𝑣). Now we will consider higher-order multipliers and derive the associated conservation laws fluxes. The determining equations for multipliers of the form Λ₁(𝑡,𝑥,𝑢,𝑣,𝑢_𝑥,𝑣_𝑥,𝑢_𝑥𝑥,𝑣_𝑥𝑥) and Λ₂(𝑡,𝑥,𝑢,𝑣,𝑢_𝑥,𝑣_𝑥,𝑢_𝑥𝑥,𝑣_𝑥𝑥) from (2.6) are 𝛿Λ𝛿𝑢¹𝑢_𝑡−6𝑢𝑢_𝑥−6𝑣𝑣_𝑥−𝑢_𝑥𝑥𝑥+Λ₂𝑣_𝑡−6𝑢𝑣_𝑥−6𝑣𝑢_𝑥−𝑣_𝑥𝑥𝑥𝛿=0,Λ𝛿𝑣¹𝑢_𝑡−6𝑢𝑢_𝑥−6𝑣𝑣_𝑥−𝑢_𝑥𝑥𝑥+Λ₂𝑣_𝑡−6𝑢𝑣_𝑥−6𝑣𝑢_𝑥−𝑣_𝑥𝑥𝑥=0, (3.12)(3.13) where the standard Euler operators 𝛿/𝛿𝑢 and 𝛿/𝛿𝑣 are given by (3.6) and (3.7). Equations (3.12) and (3.13) are separated, after expansion, according to different combinations of derivatives of 𝑢 and 𝑣 and after some simplification the following system of equations for Λ₁,Λ₂ is obtained: Λ_1𝑥𝑥=0,Λ_1𝑣𝑥=0,Λ_{1𝑥𝑣𝑥𝑥}=0,Λ_{1𝑣𝑣𝑥𝑥}=0,Λ_1𝑢𝑥=0,Λ_1𝑣𝑥Λ=0,_2𝑥𝑥=0,Λ_2𝑣𝑥=0,Λ_{2𝑥𝑣𝑥𝑥}=0,Λ_{2𝑣𝑣𝑥𝑥}=0,Λ_2𝑢𝑥=0,Λ_2𝑣𝑥Λ=0,_{1𝑣𝑥𝑥𝑣𝑥𝑥}=0,Λ_{2𝑣𝑥𝑥𝑣𝑥𝑥}=0,Λ_2𝑣𝑣=6Λ_{1𝑣𝑥𝑥},Λ_1𝑣𝑣=6Λ_{2𝑣𝑥𝑥},Λ_1𝑡=6Λ_2𝑥𝑣+6Λ_1𝑥𝑢,Λ_2𝑡=6Λ_1𝑥𝑣+6Λ_2𝑥Λ𝑢,_1𝑢=Λ_2𝑣,Λ_2𝑢=Λ_1𝑣,Λ_{1𝑢𝑥𝑥}=Λ_{2𝑣𝑥𝑥},Λ_{2𝑢𝑥𝑥}=Λ_{1𝑣𝑥𝑥}. (3.14) The solution of system (3.14) yields Λ₁=𝑐₁+𝑢𝑐₃+𝑐₄𝑡+𝑣𝑐₅+𝑐₆𝑡+𝑥6𝑐₄+3𝑢²+3𝑣²+𝑢_𝑥𝑥𝑐₇+6𝑢𝑣+𝑣_𝑥𝑥𝑐₈,Λ₂=𝑐₂+𝑣𝑐₃+𝑐₄𝑡+𝑢𝑐₅+𝑐₆𝑡+𝑥6𝑐₆+6𝑢𝑣+𝑣_𝑥𝑥𝑐₇+3𝑢²+3𝑣²+𝑢_𝑥𝑥𝑐₈, (3.15) where 𝑐₁,𝑐₂,…,𝑐₈ are arbitrary constants. The first six multipliers are same as derived in [18]. The two new multipliers are actually the higher-order multipliers associated with constants 𝑐₇ and 𝑐₈Λ₁⁽⁷⁾=3𝑢²+3𝑣²+𝑢_𝑥𝑥,Λ₂⁽⁷⁾=6𝑢𝑣+𝑣_𝑥𝑥,Λ₁⁽⁸⁾=6𝑢𝑣+𝑣_𝑥𝑥,Λ₂⁽⁸⁾=3𝑢²+3𝑣²+𝑢_𝑥𝑥. (3.16) Equation (2.5) with multipliers given in (3.16) gives two new conservation laws 𝑇₁⁽⁷⁾=3𝑢𝑣²+𝑢³+12𝑢𝑢_𝑥𝑥+12𝑣𝑣_𝑥𝑥,𝑇₂⁽⁷⁾=−6𝑢𝑣𝑣_𝑥𝑥−27𝑢²𝑣²−12𝑣𝑣_𝑡𝑥−92𝑢⁴−12𝑢²_𝑥𝑥−3𝑢_𝑥𝑥𝑣²−3𝑢²𝑢_𝑥𝑥−92𝑣⁴−12𝑣²_𝑥𝑥−12𝑢𝑢_𝑡𝑥+12𝑣_𝑥𝑣_𝑡+12𝑢_𝑥𝑢_𝑡,𝑇₁⁽⁸⁾=3𝑢²𝑣+𝑣³+12𝑢𝑣_𝑥𝑥+12𝑢_𝑥𝑥𝑇𝑣,₂₍₈₎1=−2𝑢𝑣_𝑡𝑥−18𝑢³𝑣−𝑢_𝑥𝑥𝑣_𝑥𝑥−18𝑢𝑣³−3𝑣²𝑣_𝑥𝑥−3𝑢²𝑣_𝑥𝑥−12𝑣𝑢_𝑡𝑥−6𝑢𝑣𝑢_𝑥𝑥+12𝑢_𝑥𝑣_𝑡+12𝑣_𝑥𝑢_𝑡. (3.17) The two new higher-order conservation laws (3.17) are obtained for the system (3.11).

3.3. Conservation Laws for Complex-Valued KDV in Magnetized Plasma

The complexly coupled KDV 𝑤_𝑡−|𝑤|²𝑤_𝑥−𝑤_𝑥𝑥𝑥=0 (3.18) arises in the study of the asymptotic investigation of electrostatic waves of a magnetized plasma [19]. The variable 𝑤 is the complex field amplitude 𝑤=𝑢+𝑖𝑣. The representation of (3.18) in real field variables 𝑢 and 𝑣 is 𝑢_𝑡−3𝑢²𝑢_𝑥−2𝑢𝑣𝑣_𝑥−𝑣²𝑢_𝑥−𝑢_𝑥𝑥𝑥𝑣=0,_𝑡−3𝑣²𝑣_𝑥−2𝑢𝑣𝑢_𝑥−𝑢²𝑣_𝑥−𝑣_𝑥𝑥𝑥=0. (3.19) The conservation laws for system (3.19) are derived here by using multiplier approach. The determining equations for multipliers of the form Λ₁(𝑡,𝑥,𝑢,𝑣,𝑢_𝑥,𝑣_𝑥,𝑢_𝑥𝑥,𝑣_𝑥𝑥) and Λ₂(𝑡,𝑥,𝑢,𝑣,𝑢_𝑥,𝑣_𝑥,𝑢_𝑥𝑥,𝑣_𝑥𝑥) from (2.6) are 𝛿Λ𝛿𝑢¹𝑢_𝑡−3𝑢²𝑢_𝑥−2𝑢𝑣𝑣_𝑥−𝑣²𝑢_𝑥−𝑢_𝑥𝑥𝑥+Λ₂𝑣_𝑡−3𝑣²𝑣_𝑥−2𝑢𝑣𝑢_𝑥−𝑢²𝑣_𝑥−𝑣_𝑥𝑥𝑥𝛿=0,Λ𝛿𝑣¹𝑢_𝑡−3𝑢²𝑢_𝑥−2𝑢𝑣𝑣_𝑥−𝑣²𝑢_𝑥−𝑢_𝑥𝑥𝑥+Λ₂𝑣_𝑡−3𝑣²𝑣_𝑥−2𝑢𝑣𝑢_𝑥−𝑢²𝑣_𝑥−𝑣_𝑥𝑥𝑥=0. (3.20) Equation (3.20) finally results in the following overdetermined system: Λ_2𝑥𝑥=0,Λ_1𝑢𝑥=0,Λ_2𝑢𝑥=0,Λ_1𝑣𝑥=0,Λ_2𝑣𝑥Λ=0,_{1𝑢𝑥𝑥}=Λ_{2𝑣𝑥𝑥},Λ_{2𝑢𝑥𝑥}=0,Λ_{1𝑣𝑥𝑥}Λ=0,_{1𝑢𝑥𝑥}=Λ_{2𝑣𝑥𝑥},Λ_{2𝑢𝑥𝑥}=0,Λ_{1𝑣𝑥𝑥}Λ=0,_2𝑣𝑥=Λ_2𝑥𝑣,Λ_{2𝑥𝑣𝑥𝑥}=0,Λ_2𝑣𝑣=6𝑣Λ_{2𝑣𝑥𝑥},Λ_{2𝑣𝑣𝑥𝑥}=0,Λ_{2𝑣𝑥𝑥𝑣𝑥𝑥}Λ=0,_1𝑡=3𝑣𝑢_𝑥𝑥+𝑢³+𝑣²𝑢Λ_2𝑥,Λ_2𝑡=3𝑣Λ_2𝑥𝑣_𝑥𝑥+Λ_2𝑥𝑢²𝑣+Λ_2𝑥𝑣³,Λ_1𝑥=Λ_2𝑥𝑢𝑣,Λ_1𝑢=2Λ_{2𝑣𝑥𝑥}𝑢²−2Λ_{2𝑣𝑥𝑥}𝑣²+Λ_2𝑣,Λ_2𝑢=2Λ_{2𝑣𝑥𝑥}Λ𝑢𝑣,_1𝑣=2Λ_{2𝑣𝑥𝑥}𝑢𝑣. (3.21) The solution of system (3.21) yields following five multipliers: Λ₁⁽¹⁾=0,Λ₂⁽¹⁾Λ=1,₁₍₂₎=1,Λ₂⁽²⁾Λ=0,₁₍₃₎=𝑢,Λ₂⁽³⁾Λ=𝑣,₁₍₄₎=𝑢³+𝑢𝑣²+𝑢_𝑥𝑥,Λ₂⁽⁴⁾=𝑢²𝑣+𝑣³+𝑣_𝑥𝑥,Λ₁⁽⁵⁾=3𝑡𝑢³+𝑢𝑣²+𝑢_𝑥𝑥Λ+𝑥𝑢,₂₍₅₎=3𝑡𝑣³+𝑢²𝑣+𝑣_𝑥𝑥+𝑥𝑣. (3.22) The corresponding conserved vectors conserved vectors are 𝑇₁⁽¹⁾=𝑣,𝑇₂⁽¹⁾=−𝑢²𝑣−𝑣³−𝑣_𝑥𝑥,𝑇₁⁽²⁾=𝑢,𝑇₂⁽²⁾=−𝑢³−𝑢𝑣²−𝑢_𝑥𝑥,𝑇₁⁽³⁾=𝑢²2+𝑣²2,𝑇₂⁽³⁾=𝑢²_𝑥2+𝑣²_𝑥2−𝑢𝑢_𝑥𝑥−𝑣𝑣_𝑥𝑥−34𝑢⁴−34𝑣⁴−32𝑢²𝑣²,𝑇₁⁽⁴⁾=14𝑢⁴+𝑣⁴+2𝑢²𝑣²+2𝑢𝑢_𝑥𝑥+2𝑣𝑣_𝑥𝑥,𝑇₂⁽⁴⁾1=−2𝑢⁶+𝑣⁶+𝑢²_𝑥𝑥+𝑣²_𝑥𝑥+2𝑢𝑣²𝑢_𝑥𝑥+2𝑢²𝑣𝑣_𝑥𝑥−𝑣_𝑥𝑣_𝑡−𝑢_𝑥𝑢_𝑡+2𝑢³𝑢_𝑥𝑥+3𝑢⁴𝑣²+3𝑢²𝑣⁴+2𝑣³𝑣_𝑥𝑥+𝑣𝑣_𝑡𝑥+𝑢𝑢_𝑡𝑥,𝑇₁⁽⁵⁾=3𝑡4𝑢⁴+𝑣⁴+2𝑢²𝑣²+2𝑢𝑢_𝑥𝑥+2𝑣𝑣_𝑥𝑥+𝑥2𝑢²+𝑥2𝑣²,𝑇₂⁽⁵⁾=−3𝑡2𝑢⁶+𝑣⁶+𝑢²_𝑥𝑥+𝑣²_𝑥𝑥+2𝑢𝑣²𝑢_𝑥𝑥+2𝑢²𝑣𝑣_𝑥𝑥−𝑣_𝑥𝑣_𝑡−𝑢_𝑥𝑢_𝑡+2𝑢³𝑢_𝑥𝑥+3𝑢⁴𝑣²+3𝑢²𝑣⁴+2𝑣³𝑣_𝑥𝑥+𝑣𝑣_𝑡𝑥+𝑢𝑢_𝑡𝑥−12𝑢𝑢_𝑥−12𝑣𝑣_𝑥1+𝑥2𝑢²_𝑥+12𝑣²_𝑥−34𝑢⁴−34𝑣⁴−𝑢𝑢_𝑥𝑥−𝑣𝑣_𝑥𝑥−32𝑢²𝑣². (3.23)

3.4. Conservation Laws for Ito Integrable System

Consider the following integrable Ito coupled system [20]: 𝑢_𝑡=𝑣_𝑥,𝑣_𝑡=−2𝑣_𝑥𝑥𝑥+3𝑢𝑣_𝑥+3𝑣𝑢_𝑥−12𝑤𝑤_𝑥,𝑤_𝑡=𝑤_𝑥𝑥𝑥+3𝑢𝑤_𝑥, (3.24) where 𝑢(𝑡,𝑥),𝑣(𝑡,𝑥) and 𝑤(𝑡,𝑥). For simplicity, consider multipliers of the form Λ₁=Λ₁(𝑡,𝑥,𝑢,𝑣,𝑤),Λ₂=Λ₂(𝑡,𝑥,𝑢,𝑣,𝑤),andΛ₃=Λ₃(𝑡,𝑥,𝑢,𝑣,𝑤).

The determining equations for multipliers Λ₁,Λ₂, and Λ₃ from (2.6) are 𝛿Λ𝛿𝑢¹𝑢_𝑡−𝑣_𝑥+Λ₂𝑣_𝑡+2𝑣_𝑥𝑥𝑥+6𝑢𝑣_𝑥+6𝑣𝑢_𝑥+12𝑤𝑤_𝑥+Λ₃𝑤_𝑡−𝑤_𝑥𝑥𝑥−3𝑢𝑤_𝑥𝛿=0,Λ𝛿𝑣¹𝑢_𝑡−𝑣_𝑥+Λ₂𝑣_𝑡+2𝑣_𝑥𝑥𝑥+6𝑢𝑣_𝑥+6𝑣𝑢_𝑥+12𝑤𝑤_𝑥+Λ₃𝑤_𝑡−𝑤_𝑥𝑥𝑥−3𝑢𝑤_𝑥𝛿=0,Λ𝛿𝑤¹𝑢_𝑡−𝑣_𝑥+Λ₂𝑣_𝑡+2𝑣_𝑥𝑥𝑥+6𝑢𝑣_𝑥+6𝑣𝑢_𝑥+12𝑤𝑤_𝑥+Λ₃𝑤_𝑡−𝑤_𝑥𝑥𝑥−3𝑢𝑤_𝑥=0, (3.25)(3.26)(3.27) where the standard Euler operators 𝛿/𝛿𝑢, 𝛿/𝛿𝑣, and 𝛿/𝛿𝑤 can be computed from (2.2). Separating (3.25)–(3.27), after expansion, according to different combinations of derivatives of 𝑢,𝑣, and 𝑤 and after some simplification following system of equations for Λ₁,Λ₂,Λ₃ is obtained: Λ_1𝑥𝑥=0,Λ_1𝑡=0,Λ_1𝑢=0,Λ_1𝑣=0,Λ_1𝑤Λ=0,_2𝑥=0,Λ_2𝑢=0,Λ_2𝑣=0,Λ_2𝑤Λ=0,_2𝑡=Λ_1𝑥,Λ₃(𝑡,𝑥,𝑢,𝑣,𝑤)=0. (3.28) The system of determining equation (3.28) yields Λ₁=𝑐₁+𝑐₂𝑥,Λ₂=𝑐₃+𝑐₂𝑡,Λ₃=0, (3.29) where 𝑐₁,𝑐₂,𝑐₃ are arbitrary constants and we have three multipliers Λ₁⁽¹⁾=1,Λ₂⁽¹⁾=0,Λ₃⁽¹⁾Λ=0,₁₍₂₎=𝑥,Λ₂⁽²⁾=𝑡,Λ₃⁽²⁾Λ=0,₁₍₃₎=0,Λ₂⁽³⁾=1,Λ₃⁽³⁾=0. (3.30) From (2.5), the conservation laws associated with multipliers given in (3.30) are 𝑇₁⁽¹⁾=𝑢,𝑇₂⁽¹⁾𝑇=−𝑣,₁₍₂₎=𝑢𝑥+𝑣𝑡,𝑇₂⁽²⁾=6𝑢𝑣𝑡+6𝑤²𝑡−𝑣𝑥+2𝑡𝑣_𝑥𝑥,𝑇₁⁽³⁾=𝑣,𝑇₂⁽³⁾=6𝑢𝑣+6𝑤²+2𝑣_𝑥𝑥. (3.31) The multiplier approach gave three nontrivial conservation laws for Ito system (3.24).

3.5. Conservation Laws Navier-Stokes Equations for the Compressible Flow of a Heat-Conducting, Viscous Fluid for the Ideal Gas Case

The one-dimensional compressible flow of a heat-conducting, viscous fluid for the ideal gas case is represented by the Navier-stokes equations [21] 𝐸₁=𝜌_𝑡+𝜌_𝑥𝑢+𝜌𝑢_𝑥𝐸=0,₂=𝑢_𝑡+𝑢𝑢_𝑥+𝑅𝜃𝜌_𝑥𝜌+𝑅𝜃_𝑥𝑢−𝜇^𝑥𝑥𝜌𝐸=0,₃=𝜃_𝑡+𝑢𝜃_𝑥+𝑅𝑐_𝑣𝜃𝑢_𝑥−𝜇𝑐_𝑣𝑢²_𝑥𝜌−𝑘𝑐_𝑣𝜃_𝑥𝑥𝜌=0, (3.32) where 𝑢(𝑡,𝑥),𝜌(𝑡,𝑥), and 𝜃(𝑡,𝑥) are the density, velocity, and temperature, 𝜇>0 is the viscosity coefficient, 𝑘>0 is the heat conductivity coefficient, 𝑅>0 is called the ideal gas constant, 𝑐_𝑣>0 represents the specific heat at constant volume. Consider multipliers of the form Λ₁=Λ₁(𝑡,𝑥,𝑢,𝜌,𝜃),Λ₂=Λ₂(𝑡,𝑥,𝑢,𝜌,𝜃),andΛ₃=Λ₃(𝑡,𝑥,𝑢,𝜌,𝜃). The determining equations for multipliers Λ₁,Λ₂, and Λ₃ from (2.6) are 𝛿Λ𝛿𝑢¹𝐸₁+Λ₂𝐸₂+Λ₃𝐸₃𝛿=0,Λ𝛿𝜌¹𝐸₁+Λ₂𝐸₂+Λ₃𝐸₃𝛿=0,Λ𝛿𝜃¹𝐸₁+Λ₂𝐸₂+Λ₃𝐸₃=0, (3.33) where the standard Euler operators 𝛿/𝛿𝑢, 𝛿/𝛿𝜌, and 𝛿/𝛿𝜃 can be calculated from (2.2). Expanding (3.33) and then separating according to different combinations of derivatives of 𝑢,𝜌, and 𝜃 and after some simplification following system of equations for Λ₁,Λ₂,Λ₃ is obtained: Λ_1𝑥𝑥=0,Λ_1𝑡=−𝑢Λ_1𝑥,Λ_2𝑡=−Λ_1𝑥Λ𝜌,_3𝑡=0,Λ_2𝑥=0,Λ_3𝑥=0,Λ_1𝑢=Λ₂𝜌,Λ_2𝑢=Λ₃𝑐,Λ_3𝑢=0,Λ₁𝜌=0,Λ_2𝜌=Λ₂𝜌,Λ_3𝜌=Λ₃𝜌,Λ_1𝜃=Λ3𝜌,Λ_2𝜃=0,Λ_3𝜃=0,𝑅+𝑐_𝑣≠0,−𝑅+2𝑐_𝑣≠0. (3.34) The adiabatic constant 𝛾=1+𝑅/𝑐_𝑣 has physical applications in region (1,(5/3)] (see e.g., [21, 22]). If 𝑅+𝑐_𝑣=0 or −𝑅+2𝑐_𝑣=0 then the value of 𝛾 will be outside the region of physical significance. The assumptions 𝑅+𝑐_𝑣≠0 and −𝑅+2𝑐_𝑣≠0 are necessary for having a true physical model. The solution of system (3.34) results in the following four multipliers: Λ₁⁽¹⁾=1,Λ₂⁽¹⁾=0,Λ₃⁽¹⁾Λ=0,₁₍₂₎=𝑢,Λ₂⁽²⁾=𝜌,Λ₃⁽²⁾Λ=0,₁₍₃₎=𝑥−𝑡𝑢,Λ₂⁽³⁾=−𝑡𝜌,Λ₃⁽³⁾Λ=0,₁₍₄₎𝑢=𝜃+²2𝑐_𝑣,Λ₂⁽⁴⁾=𝑢𝜌𝑐_𝑣,Λ₃⁽⁴⁾=𝜌. (3.35) The expressions for the fluxes are 𝑇₁⁽¹⁾=𝜌,𝑇₂⁽¹⁾𝑇=𝑢𝜌,₁₍₂₎=𝑢𝜌,𝑇₂⁽²⁾=𝑢²𝜌−𝜇𝑢_𝑥𝑇+𝑅𝜌𝜃,₁₍₃₎=𝜌(𝑥−𝑡𝑢),𝑇₂⁽³⁾=𝑥𝑢𝜌−𝑡𝑢²𝜌+𝜇𝑡𝑢_𝑥𝑇−𝑅𝑡𝜌𝜃,₁₍₄₎𝑢=𝜌𝜃+²𝜌2𝑐_𝑣,𝑇₂⁽⁴⁾=12𝑐_𝑣2𝑐_𝑣𝑢𝜌𝜃+𝑢³𝜌+2𝑅𝑢𝜌𝜃−2𝜇𝑢𝑢_𝑥−2𝑘𝜃_𝑥. (3.36) We will get same results if we consider multipliers of the form Λ_𝑖=Λ_𝑖(𝑡,𝑥,𝑢,𝜌,𝜃,𝑢_𝑥,𝜌_𝑥,𝜃_𝑥),  𝑖=1,2,3 or Λ_𝑖=Λ_𝑖(𝑡,𝑥,𝑢,𝜌,𝜃,𝑢_𝑡,𝜌_𝑡,𝜃_𝑡),𝑖=1,2,3.

4. Conclusions

The conservation laws for the integrable coupled kdv type system, complexly coupled kdv system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics were computed by multipliers approach. The multipliers having dependence on dependent variables, independent variables, and derivatives of dependent variables up to some fixed order were constructed. After computing multipliers, the conservation laws fluxes were derived.

First of all, we considered integrable coupled kdv-type system and multiplier approach yielded four local conserved vectors. For the complexly coupled KDV system, total eight multipliers were obtained. The two new conserved vectors corresponding to second-order multipliers were obtained and were not found in [18]. The multiplier approach on coupled system arising from complex-valued KDV in magnetized plasma gave five conserved vectors. For Ito integrable system three and for Navier stokes equations of gas dynamics four, nontrivial conserved vectors were derived.

The conserved vectors derived here can be used in constructing the solutions of underlying PDE systems and will be considered in the future work.