Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 871253, 13 pages
doi:10.1155/2012/871253
Research Article
Conservation Laws for Some Systems of
Nonlinear Partial Differential Equations via
Multiplier Approach
Rehana Naz
Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan
Correspondence should be addressed to Rehana Naz, rehananaz qau@yahoo.com
Received 26 July 2012; Accepted 20 September 2012
Academic Editor: Fazal M. Mahomed
Copyright q 2012 Rehana Naz. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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.
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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 Di T i Λα Eα . The determining equations for the multipliers characteristics were
obtained by taking the variational derivative of Di T i Qα Eα 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 , . . . , xn and m dependent variables u ≡ u1 , u2 , . . . , um defined as,
Eα x, u, u1 , u2 , . . . , uk 0,
α 1, 2, . . . , m,
2.1
where ui is the collection of ith-order partial derivatives of u.
1 The Euler operator is defined by
∂
∂
δ
∂
− Di α Di Dj α − · · · ,
δuα ∂uα
∂ui
∂uij
2.2
where
Di ∂
∂
∂
uαi α uαij α · · · ,
∂u
∂uj
∂xi
i 1, 2,
2.3
is the total derivative operator with respect to xi .
2 A conserved vector of 2.1 is an n-tuple T T 1 , T 2 , . . . , T n , T i ∈ A, i 1, 2 . . . n,
such that
Di T i 0
holds for all solutions of 2.1. Equation 2.4 is called a local conservation law.
2.4
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3
3 The multipliers Λα of system 2.1 has the property
Di T i Λα Eα ,
2.5
for the arbitrary function uα 9, 10.
4 The determining equations for the multipliers are obtained by taking variational
derivative of 2.5 see 10:
δ
Λα Eα 0.
δuα
2.6
Equation 2.6 holds for the arbitrary functions uα 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 ut uxx − k 3k 6u2 − k2 v2 2kk 6vux k 3uvx 0,
x
2 2
2
E2 vt vxx − kk − 3v − k 3 u
2k 3kvux k − 3uvx 0.
3.1
x
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
Λ1 E1 Λ2 E2 Dt T 1 Dx T 2 ,
3.2
for all functions ut, x and vt, x where the total derivative operators Dt and Dx from 2.3
are
Dt ∂
∂
∂
∂
∂
∂
∂
ut
vt
utt
vtt
utx
vtx
··· ,
∂t
∂u
∂v
∂ut
∂vt
∂ux
∂vx
∂
∂
∂
∂
∂
∂
∂
ux
vx
uxx
Dx vxx
uxt
vxt
··· .
∂x
∂u
∂v
∂ux
∂vx
∂ut
∂vt
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
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Λ1 and Λ2 are
δ
Λ1 E1 Λ2 E2 0,
δu
δ
Λ1 E1 Λ2 E2 0,
δv
3.4
3.5
where δ/δu and δ/δv are the standard Euler operators defined in 2.2, which annihilate
divergence expressions:
δ
∂
∂
∂
∂
∂
∂
− Dt
− Dx
Dt2
Dt Dx
Dx2
− ··· ,
δu ∂u
∂ut
∂ux
∂utt
∂utx
∂uxx
3.6
δ
∂
∂
∂
∂
∂
∂
− Dt
− Dx
Dt2
Dt Dx
Dx2
− ··· .
δv ∂v
∂vt
∂vx
∂vtt
∂vtx
∂vxx
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:
Λ2xx 0,
Λ1t −
Λ2vx 0,
12
k 3u − vk 3uΛ2x ,
k
Λ2vv 0,
Λ1x −
k 3
Λ2x ,
k
Λ2t 12k 3u − kvΛ2x ,
k 3 2k2 6k Λ2 2Λ1 k2 k 3u − vk 3u Λ2v
,
k2 u − vk 3u
−3k − k2 Λ2 − Λ1 k2 − k 3u − vk 3uΛ2v
,
Λ2u u − vk 3uk
−3k − k2 Λ2 − Λ1 k2 − k 3u − vk 3uΛ2v
.
Λ1v u − vk 3uk
Λ1u
3.8
The solution of system 3.8 yields following four multipliers:
1
Λ1 1,
2
Λ1 −
3
Λ1 1
Λ2 −
1
k 32 u,
k2
1
−kv 2k 3u,
k
4
Λ1 −
4
k3
,
k
2
Λ2 v,
3
Λ2 u,
k3
x 12k 3ut − 12kvt,
k
Λ2 x 12k 3ut − 12ktv.
3.9
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From 3.2 and 3.9, we obtained following four conserved vectors:
1
1
T1 − k 3u − kv,
k
1
1
6k 32 u2 − 12kk 3uv kvxx − kuxx − 3uxx ,
T2 k
1 2
T1 − 2 u2 k 32 − k2 v2 ,
2k
1 2
T2 2 3k 32 u2x − 3k2 vx2 4k 6k 33 u3 − 4k3 k − 3v3
6k
−12kk 33 u2 v 12k3 k 3uv2 − 6k 32 uuxx 6k2 vvxx ,
3
1
k 3u2 − kuv ,
k
1 3k 3u2x 2k 32 k 12u3
3k
T1 −
3
T2
− 2k3 v3 − 6kk 3k 9u2 v 6k2 k 6uv2
−6k 3uuxx 3kuxx v 3kuvxx − 3kux vx ,
4
T1 −
1
6k 32 tu2 6k2 tv2
k
−12kk 3tuv k 3ux − kvx ,
4
T2 1
− k 3x 12k 3tu − 12ktvuxx
k
x 12k 3ut − 12ktvkvxx 6k 32 tu2x 6tk2 vx2
− 12kk 3tux vx k 3ux − kvx 48k 33 tu3
− 48k3 tv3 − 144kk 32 tu2 v 144k2 k 3tuv2
6k 32 xu2 6k2 xv2 − 12kk 3xuv .
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
ut − 6uux − 6vvx − uxxx 0,
vt − 6uvx − 6vux − vxxx 0,
3.11
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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, ux , vx , uxx , vxx and Λ2 t, x, u, v, ux , vx , uxx , vxx from 2.6 are
δ
Λ1 ut − 6uux − 6vvx − uxxx Λ2 vt − 6uvx − 6vux − vxxx 0,
δu
3.12
δ
Λ1 ut − 6uux − 6vvx − uxxx Λ2 vt − 6uvx − 6vux − vxxx 0,
δv
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:
Λ1xx 0,
Λ2xx 0,
Λ1vx 0,
Λ2vx 0,
Λ1vxx vxx 0,
Λ1xvxx 0,
Λ1vvxx 0,
Λ2xvxx 0,
Λ2vvxx 0,
Λ2vxx vxx 0,
Λ2vv 6Λ1vxx ,
Λ1t 6Λ2x v 6Λ1x u,
Λ1u Λ2v ,
Λ2u Λ1v ,
Λ1ux 0,
Λ2ux 0,
Λ1vx 0,
Λ2vx 0,
Λ1vv 6Λ2vxx ,
Λ2t 6Λ1x v 6Λ2x u,
Λ1uxx Λ2vxx ,
Λ2uxx Λ1vxx .
3.14
The solution of system 3.14 yields
x
c4 3u2 3v2 uxx c7 6uv vxx c8 ,
6
x
Λ2 c2 vc3 c4 t uc5 c6 t c6 6uv vxx c7 3u2 3v2 uxx c8 ,
6
Λ1 c1 uc3 c4 t vc5 c6 t 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 c8
7
Λ2 6uv vxx ,
7
8
Λ2 3u2 3v2 uxx .
Λ1 3u2 3v2 uxx ,
Λ1 6uv vxx ,
8
3.16
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Equation 2.5 with multipliers given in 3.16 gives two new conservation laws
1
1
7
T1 3uv2 u3 uuxx vvxx ,
2
2
1
9
1
7
T2 −6uvvxx − 27u2 v2 − vvtx − u4 − u2xx
2
2
2
9
1 2
1
1
1
− 3uxx v2 − 3u2 uxx − v4 − vxx
− uutx vx vt ux ut ,
2
2
2
2
2
8
T1
1
1
3u v v uvxx uxx v,
2
2
2
3.17
3
1
8
T2 − uvtx − 18u3 v − uxx vxx − 18uv3 − 3v2 vxx
2
1
1
1
− 3u2 vxx − vutx − 6uvuxx ux vt vx ut .
2
2
2
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
wt − |w|2 w − wxxx 0
x
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
ut − 3u2 ux − 2uvvx − v2 ux − uxxx 0,
vt − 3v2 vx − 2uvux − u2 vx − vxxx 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, ux , vx , uxx , vxx and
Λ2 t, x, u, v, ux , vx , uxx , vxx from 2.6 are
δ Λ1 ut − 3u2 ux − 2uvvx − v2 ux − uxxx Λ2 vt − 3v2 vx − 2uvux − u2 vx − vxxx 0,
δu
δ Λ1 ut − 3u2 ux − 2uvvx − v2 ux − uxxx Λ2 vt − 3v2 vx − 2uvux − u2 vx − vxxx 0.
δv
3.20
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Equation 3.20 finally results in the following overdetermined system:
Λ2xx 0,
Λ1ux 0,
Λ2ux 0,
Λ1vx 0,
Λ1uxx Λ2vxx ,
Λ2uxx 0,
Λ1vxx 0,
Λ1uxx Λ2vxx ,
Λ2uxx 0,
Λ1vxx 0,
Λ2vx Λ2x
,
v
Λ2xvxx 0,
Λ2vvxx 0,
Λ1t Λ2t Λ2vx 0,
Λ2vv 6vΛ2vxx ,
Λ2vxx vxx 0,
3
uxx u3 v2 u Λ2x ,
v
3.21
3
Λ2x vxx Λ2x u2 v Λ2x v3 ,
v
Λ1x Λ2x u
,
v
Λ1u 2Λ2vxx u2 − 2Λ2vxx v2 Λ2v ,
Λ2u 2Λ2vxx uv,
Λ1v 2Λ2vxx uv.
The solution of system 3.21 yields following five multipliers:
1
Λ2 1,
2
Λ2 0,
3
Λ2 v,
Λ1 0,
Λ1 1,
Λ1 u,
1
2
3
4
Λ1 u3 uv2 uxx ,
4
Λ2 u2 v v3 vxx ,
5
Λ1 3t u3 uv2 uxx xu,
5
Λ2 3t v3 u2 v vxx xv.
3.22
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9
The corresponding conserved vectors conserved vectors are
1
T2 −u2 v − v3 − vxx ,
2
T2 −u3 − uv2 − uxx ,
T1 v,
T1 u,
3
T1 3
1
2
u2 v2
,
2
2
u2x vx2
3
3
3
− uuxx − vvxx − u4 − v4 − u2 v2 ,
2
2
4
4
2
1 4
4
2 2
u v 2u v 2uuxx 2vvxx ,
4
1
2
− u6 v6 u2xx vxx
2uv2 uxx 2u2 vvxx − vx vt
2
T2 4
T1
4
T2
3.23
−ux ut 2u uxx 3u v 3u v 2v vxx vvtx uutx ,
3
5
4 2
2 4
3
x
3t 4
x
u v4 2u2 v2 2uuxx 2vvxx u2 v2 ,
4
2
2
3t
2
− u6 v6 u2xx vxx
2uv2 uxx 2u2 vvxx − vx vt
2
T1 5
T2
−ux ut 2u3 uxx 3u4 v2 3u2 v4 2v3 vxx vvtx uutx
1
1
3
3
3
1
1
− uux − vvx x u2x vx2 − u4 − v4 − uuxx − vvxx − u2 v2 .
2
2
2
2
4
4
2
3.4. Conservation Laws for Ito Integrable System
Consider the following integrable Ito coupled system 20:
ut vx ,
vt −2vxxx 3uvx 3vux − 12wwx ,
3.24
wt wxxx 3uwx ,
where ut, x, vt, x and wt, x. For simplicity, consider multipliers of the form Λ1 Λ1 t, x, u, v, w, Λ2 Λ2 t, x, u, v, w, and Λ3 Λ3 t, x, u, v, w.
The determining equations for multipliers Λ1 , Λ2 , and Λ3 from 2.6 are
δ
Λ1 ut − vx Λ2 vt 2vxxx 6uvx 6vux 12wwx Λ3 wt − wxxx − 3uwx 0,
δu
3.25
δ
Λ1 ut − vx Λ2 vt 2vxxx 6uvx 6vux 12wwx Λ3 wt − wxxx − 3uwx 0,
δv
3.26
δ
Λ1 ut − vx Λ2 vt 2vxxx 6uvx 6vux 12wwx Λ3 wt − wxxx − 3uwx 0,
δw
3.27
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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:
Λ1xx 0,
Λ1t 0,
Λ2x 0,
Λ1u 0,
Λ2u 0,
Λ2t Λ1x ,
Λ1v 0,
Λ2v 0,
Λ1w 0,
Λ2w 0,
3.28
Λ3 t, x, u, v, w 0.
The system of determining equation 3.28 yields
Λ1 c1 c2 x,
Λ2 c3 c2 t,
Λ3 0,
3.29
where c1 , c2 , c3 are arbitrary constants and we have three multipliers
1
Λ2 0,
2
Λ2 t,
3
Λ2 1,
Λ1 1,
Λ1 x,
Λ1 0,
1
Λ3 0,
1
2
Λ3 0,
3
Λ3 0.
2
3.30
3
From 2.5, the conservation laws associated with multipliers given in 3.30 are
1
1
T1 u,
2
T1 ux vt,
3
T1 v,
T2 −v,
2
T2 6uvt 6w2 t − vx 2tvxx ,
3.31
3
T2 6uv 6w2 2vxx .
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 ρt ρx u ρux 0,
E2 ut uux R
E3 θt uθx θρx
uxx
Rθx − μ
0,
ρ
ρ
μ u2x
k θxx
R
−
0,
θux −
cv
cv ρ
cv ρ
3.32
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11
where ut, x, ρt, x, and θt, 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, cv > 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, and Λ3 Λ3 t, x, u, ρ, θ. The determining
equations for multipliers Λ1 , Λ2 , and Λ3 from 2.6 are
δ
Λ1 E1 Λ2 E2 Λ3 E3 0,
δu
δ
Λ1 E1 Λ2 E2 Λ3 E3 0,
δρ
3.33
δ
Λ1 E1 Λ2 E2 Λ3 E3 0,
δθ
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:
Λ1xx 0,
Λ3t 0,
Λ1t −uΛ1x ,
Λ2x 0,
Λ2t −Λ1x ρ,
Λ3x 0,
Λ2u Λ3
,
c
Λ3u 0,
Λ3ρ Λ3
,
ρ
Λ1θ Λ1u Λ1 ρ 0,
Λ3
,
ρ
Λ2ρ Λ2θ 0,
R cv /
0,
Λ2
,
ρ
Λ2
,
ρ
3.34
Λ3θ 0,
−R 2cv /
0.
The adiabatic constant γ 1 R/cv has physical applications in region 1, 5/3 see e.g.,
21, 22. If R cv 0 or −R 2cv 0 then the value of γ will be outside the region of
physical significance. The assumptions R cv / 0 and −R 2cv / 0 are necessary for having
a true physical model. The solution of system 3.34 results in the following four multipliers:
1
Λ2 0,
2
Λ2 ρ,
Λ1 1,
Λ1 u,
3
Λ1 x − tu,
4
Λ1 θ u2
,
2cv
1
Λ3 0,
1
2
Λ3 0,
2
3
Λ2 −tρ,
4
Λ2 uρ
,
cv
3
Λ3 0,
4
Λ3 ρ.
3.35
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The expressions for the fluxes are
1
1
T1 ρ,
2
T1 uρ,
3
T1 ρx − tu,
4
T1 ρθ u2 ρ
,
2cv
4
T2 T2 uρ,
2
T2 u2 ρ − μux Rρθ,
3
T2 xuρ − tu2 ρ μtux − Rtρθ,
3.36
1 2cv uρθ u3 ρ 2Ruρθ − 2μuux − 2kθx .
2cv
We will get same results if we consider multipliers of the form Λi Λi t, x, u, ρ, θ, ux ,
ρx , θx , i 1, 2, 3 or Λi Λi t, x, u, ρ, θ, ut , ρt , θt , 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.
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