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

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.


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  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 i T i Λ α E α . The determining equations for the multipliers characteristics were obtained by taking the variational derivative of D i T i Q α E 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.

Preliminaries
Consider a kth-order system of partial differential equations PDEs with n independent variables x ≡ x 1 , x 2 , . . . , x n and m dependent variables u ≡ u 1 , u 2 , . . . , u m defined as, where u i is the collection of ith-order partial derivatives of u.
1 The Euler operator is defined by

Integrable Coupled System
Consider the integrable coupled system 16, 17 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 Λ 1 t, x, u, v and Λ 2 t, x, u, v . Multipliers Λ 1 and Λ 2 for the system 3.1 have the property that for all functions u t, x and v t, x where the total derivative operators D t and D x from 2.3 are 3.3 The right-hand side of 3.2 is a divergence expression and T 1 and T 2 are the components of the conserved vector T T 1 , T 2 . The determining equations for the multipliers 4 Journal of Applied Mathematics where δ/δu and δ/δv are the standard Euler operators defined in 2.2 , which annihilate divergence expressions: Separating 3.4 and 3.5 , after expansion, with respect to different combinations of derivatives of u and v, yields the following overdetermined system: The solution of system 3.8 yields following four multipliers: x 12 k 3 ut − 12ktv.

3.9
Journal of Applied Mathematics 5 From 3.2 and 3.9 , we obtained following four conserved vectors: 3.10

Higher-Order Conservation Laws for Complexly Coupled KDV System
The conservation laws of complexly coupled KDV were discussed in Naz 18 Journal of Applied Mathematics and six conserved vectors were derived by multipliers approach with multipliers of form Λ 1 t, x, u, v and Λ 2 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 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 Λ 1 , Λ 2 is obtained: 3.14 The solution of system 3.14 yields

3.17
The two new higher-order conservation laws 3.17 are obtained for the system 3.11 .

Conservation Laws for Complex-Valued KDV in Magnetized Plasma
The complexly coupled KDV 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

3.19
The conservation laws for system 3.19 are derived here by using multiplier approach. The determining equations for multipliers of the form Λ 1 t, x, u, v, u x , v x , u xx , v xx and 3.20 8

Journal of Applied Mathematics
Equation 3.20 finally results in the following overdetermined system:

3.21
The solution of system 3.21 yields following five multipliers: 3t v 3 u 2 v v xx xv.

3.22
Journal of Applied Mathematics 9 The corresponding conserved vectors conserved vectors are 3.23

Conservation Laws for Ito Integrable System
Consider the following integrable Ito coupled system 20 : where u t, x , v t, x and w t, x . For simplicity, consider multipliers of the form Λ 1 The determining equations for multipliers Λ 1 , Λ 2 , and Λ 3 from 2.6 are 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 Λ 1 , Λ 2 , Λ 3 is obtained: 3.28 The system of determining equation 3.28 yields where c 1 , c 2 , c 3 are arbitrary constants and we have three multipliers 0.

3.30
From 2.5 , the conservation laws associated with multipliers given in 3.30 are

3.31
The multiplier approach gave three nontrivial conservation laws for Ito system 3.24 .

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.