Application of Ibragimov’s method and Noether’s theorem for constructing conservation laws of the linear elasticity model

Application of Ibragimov’s method and Noether’s theorem for constructing conservation laws of the linear elasticity

model

Pouya Sinaian

Thesis for the degree Master of science (two years) in Mathematical Modeling and Simulation

30 credit points (30 ECTS credits) June 2014

Blekinge Institute of Technology School of Engineering

Department of Mathematics and Natural Sciences Supervisor: Dr. Raisa Khamitova

Examiner: Dr. Claes Jogr´eus

Contact info

Author:

Pouya Sinaian: pouya.sinaian@gmail.com

Supervisor:

Dr. Raisa Khamitova: raisa.khamitova@bth.se Examiner:

Dr. Claes Jogr´eus: claes.jogreus@bth.se

Dedicated to my lovely wife who patiently supported me during this work.

Abstract

In this thesis, conservation laws of the Lam´e equation describing linear elasticity have been constructed. This has been done by the implementation of two theorems for con- structing conservation laws: the new theorem suggested by N H.Ibragimov, and the classical theorem presented by E.Noether.

As a result one can see that the Ibragimov method provides more conservation laws than the Noether theorem, which may suggest the better efficiency of the Ibragimov method.

Keywords: Conservation law; Noether’s theorem; Ibragimov’s method; Nonlinear self-adjointness;

Formal Lagrangian; Linear elasticity; Lam´e equation.

Acknowledgment

I would like to express my sincere gratitude to Dr. Raisa Khamitova for her valuable and constructive suggestions during the planning and development of this research work. Her will- ingness to give her time so generously has been very much appreciated.

My special thanks go to Prof. Nail H. Ibragimov for his courses in the field of PDE and Lie group analysis, which made me familiar with differential system analysis, and his valuable com- ments during this research work.

Advice given by Dr. Niklas Lavasson and Dr. Andrew Moss has been a great help in develop- ment of my insight in research methodology and I am grateful for that.

I thank Dr. Mattias Eriksson for his valuable comments.

Finally, I would like to express my deepest gratitude to my wife and my parents for their sup- port and encouragement throughout my study.

Karlskrona, May 2014 Pouya Sinaian

Contents

Abstract i

Acknowledgment ii

Contents iii

1 Introduction 1

1.1 Linear elasticity and the Lam´e equation . . . 2 2 Calculation of the conservation laws of the Lam´e equation using Noether’s

theorem 3

2.1 Introduction . . . 3 2.2 CaseX = ∂

∂t . . . 4 2.3 CaseX1 = ∂

∂x . . . 5 2.4 CaseX2 = ∂

∂y . . . 6 2.5 CaseX₃ = ∂

∂z . . . 7 2.6 CaseX₄ =y ∂

∂z − z ∂

∂y +u² ∂

∂u³ − u³ ∂

∂u² . . . 7 2.7 CaseX₅ =z ∂

∂x − x∂

∂z +u³ ∂

∂u¹ − u¹ ∂

∂u³ . . . 8 2.8 CaseX₆ =x ∂

∂y − y ∂

∂x+u¹ ∂

∂u² − u² ∂

∂u¹ . . . 9 2.9 CaseX7 =t∂

∂t+x ∂

∂x +y ∂

∂y+z ∂

∂z . . . 10 2.10 Case X₈ =u¹ ∂

∂u¹ +u² ∂

∂u² +u³ ∂

∂u³ . . . 11 2.11 Case X_∞=w¹ ∂

∂u¹ +w² ∂

∂u² +w³ ∂

∂u³ . . . 11 3 Investigation for nonlinear self-adjointness of the Lam´e equation 13 4 Calculation of conservation laws of the Lam´e equation using Ibragimov’s

theorem 17

4.1 Introduction . . . 17 4.2 Translation of time: X0 = ∂

∂t . . . 21 4.3 Translation inx direction: X₁= ∂

∂x . . . 23 4.4 Translation iny direction: X₂= ∂

∂y . . . 25

4.5 Translation inz direction: X₃= ∂

∂z . . . 27 4.6 Rotation: X₄ =y ∂

∂z − z ∂

∂y +u² ∂

∂u³ − u³ ∂

∂u² . . . 28 4.7 Rotation: X₅ =z ∂

∂x− x ∂

∂z +u³ ∂

∂u¹ − u¹ ∂

∂u³ . . . 30 4.8 Rotation: X6 =x ∂

∂y − y ∂

∂x +u¹ ∂

∂u² − u² ∂

∂u¹ . . . 33 4.9 Scaling: X7 =t∂

∂t+x ∂

∂x+y ∂

∂y +z ∂

∂z . . . 35 4.10 Scaling: X₈ =u¹ ∂

∂u¹ +u² ∂

∂u² +u³ ∂

∂u³ . . . 38 4.11 Addition of solutions: X_∞=w¹ ∂

∂u¹ +w² ∂

∂u² +w³ ∂

∂u³ . . . 39

5 Comparison of results 39

6 Summary 44

References 47

1 Introduction

Conservation laws are very important tools for investigating a physical system. They state that a particular measurable property of an isolate physical system does not change as the system evolves (see for example [11]). There are several ideas for constructing conservation laws of a system. One is the direct method used by Laplace for the first time in 1798 [7].

Let us consider a system of m differential equations

F¯α(x, u, u₍₁₎, . . . , u_(s)) = 0, ¯α = 1, . . . , m, (1.1) with n independent variables x¹, . . . , xⁿ and m dependent variables u¹, . . . , u^m, and the first, second,. . . , s-th derivatives of u with respect to x denoted as u₍₁₎ ={u^α_i}, . . . , u_(s) ={u^α_i1...is} respectively,α = 1, 2, . . . , m and other indices change from 1 to n.

A conservation law for Eqs. (1.1) is defined [4] as

Di(Cⁱ)



(1.1)= 0. (1.2)

Here D_i is the total differentiation with respect toxⁱ: D_i= ∂

∂xⁱ +u^α_i ∂

∂u^α +u^α_ij ∂

∂u^α_j +· · · . (1.3) The subscript |_(1.1) means the left-hand side of (1.2) is restricted on the solutions of Eqs.(1.1).

Here and further, the usual convention of summation in repeated indices is used.

The n-dimensional vector

C = (C¹, . . . , Cⁿ) (1.4) satisfying the equation (1.2) is called a conserved vector for the system (1.1). If its components Cⁱ =Cⁱ(x, u, u₍₁₎, . . .) are functions of x, u and derivatives u₍₁₎, . . . , of a finite order, conserva- tion vector (1.4) is called a local conserved vector and, accordingly, the system (1.1) has a local conservation law.

Vector (1.4) is called a trivial conserved vector if

D_i(Cⁱ)≡ 0, (1.5)

or componentsCⁱ vanish on the solutions of the system (1.1). Two conservation laws that only differ by a trivial conservation law are considered as equivalent [4, 7].

The direct method of constructing conservation laws means that the definition of a conser- vation law is used.

The works of Jacobi and Klein motivated Emmy Noether to investigate the connection between conservation laws and symmetries of differential equations obtained from variational principle. In 1915 Noether found such a relationship and the theorem was published in 1918 [9].

However, Noether’s theorem can be applied only for differential equations with a Lagrangian, so called Euler-Lagrange equations (see Section 2). As an illustration, Noether’s theorem is not applicable to evolution equations and to differential equations of an odd order [3]. Furthermore, symmetries of Euler-Lagrange equations should satisfy an additional condition [3, 9].

In order to overcome this problem, in 2007 N.H.Ibragimov presented a remedy based on the concept of a formal Lagrangian and proved the new conservation theorem. He introduced the

definition of nonlinear self-adjointness [3, 4, 5] of differential equations and gave the formula for constructing local conservation laws. Briefly speaking, the Ibragimov theorem state that for any system of differential equations which are nonlinearly self-adjoint, it is possible to use the formal Lagrangian for construction of local conservation laws of a system corresponding to the system’s symmetries.

In this work the application of both theorems (see also, for example, [6]) to a system which has the Lagrangian is investigated and the efficiency of the Ibragimov theorem is demonstrated.

In order to execute that, the linear elasticity theorem (which is an important and useful theory in physics and mechanics), has been considered and its mathematical representation, the Lam´e equation, has been chosen.

1.1 Linear elasticity and the Lam´ e equation

According to the definition, elasticity is the tendency of a material to temporary deform under an external force and stress and return to the normal form when the load and stress are removed.

Elasticity is a useful theory with a wide range of applications, e.g. in design and analysis of the structure of beams, plates and shells, sandwich composites and etc. Besides, it is the basis of a large proportion of the fractional mechanics. The most well-known mathematical representation of the linear elasticity is the equilibrium equations for a homogeneous, isotropic and linearly elastic medium in absence of the body force which is called Navier’s equation or, alternatively, elastostatic equation [2, 10]:

μΔu + (μ + λ)grad div u = 0,

where μ and λ are Lam´e moduli, subjected to the restrictions μ > 0, 2μ + λ > 0 and u = (u¹, u², u³) and x, y, z are spacial variables.

For the first time in 1962, G¨unter [1] and Knowles & Sternbery [8] performed a limited inves- tigation of the Noether theorem applications in elasticity [10]. In 1984 Olver [10] presented a first complete systematic implementation of Noether’s theorem in elasticity. Olver used elastostatic Navier’s equation and by considering tensor calculations, classified conservation laws of the sys- tem (one can find the complete method and calculations in Olver’s triple published papers [10]).

In this work, I use the Lam´e equation which is the non-elastostatic (depends on timet) version of the Navier equation, i.e. ∂²u

∂t² = 0. Hence

μΔu + (μ + λ)grad div u = utt. Consideringβ = μ

μ + λ, the Lam´e equation has the following form [2]:

u_tt = grad div u +βΔu, (1.1.1)

where u = (u¹, u², u³),t is time, and x, y, z are spacial variables.

The equation (1.1.1) has the following symmetries [2]:

X₀= ∂

∂t, X₁ = ∂

∂x, X₂ = ∂

∂y, X₃= ∂

∂z, X4= y ∂

∂z − z ∂

∂y +u² ∂

∂u³ − u³ ∂

∂u² , X5= z ∂

∂x− x ∂

∂z +u³ ∂

∂u¹ − u¹ ∂

∂u³ , X₆= x ∂

∂y − y ∂

∂x +u¹ ∂

∂u² − u² ∂

∂u¹ , X₇= t∂

∂t+x ∂

∂x+y ∂

∂y +z ∂

∂z , X8= u¹ ∂

∂u¹ +u² ∂

∂u² +u³ ∂

∂u³ , X_∞= w¹ ∂

∂u¹ +w² ∂

∂u² +w³ ∂

∂u³ , (1.1.2)

where (w¹, w², w³) is any arbitrary solution of the Lam´e equation.

Additionally, the equation (1.1.1) has the Lagrangian L = β



(u¹_y +u²_x)²+ (u¹_z+u³_x)²+ (u²_z+u³_y)²+ 2(u¹_x)²+ 2(u²_y)²+ 2(u³_z)²



+ (1− β)(u¹_x+u²_y+u³_z)²− (u¹_t)²− (u²_t)²− (u³_t)², (1.1.3) similar to the Lagrangian on the p.342 [2].

Due to the chronological reason, the application of Noether’s theorem is presented first and the Ibragimov method and its implementation come after that.

2 Calculation of the conservation laws of the Lam´ e equation using Noether’s theorem

2.1 Introduction

Consider the system (1.1) and suppose the Lagrangian, L, of the system exists. Consider also that the system admits the infinitesimal generator

X = ξⁱ ∂

∂xⁱ +η^α ∂

∂u^α·

According to the Noether theorem, symmetries which may give a nontrivial conservation law must satisfy the following conditions:

X(L) + L D_i(ξⁱ) = 0 or, alternatively, X(L) + L D_i(ξⁱ) =D_i(Bⁱ). (2.1.1) Hence, the condition (2.1.1) should be verified for each symmetry of the system (1.1) first.

As it mentioned in Section 1, a corresponding conserved vector for the system (1.1) can be written as

C = (C¹, C², ..., Cⁿ). (2.1.2)

Its components Ci have the form Cⁱ=ξⁱL + W^α

∂L

∂u^α_i − D_j

 ∂L

∂u^α_ij



+D_jD_k

 ∂L

∂u^α_ijk



− · · ·



+D_j

W^α  ∂L

∂u^α_ij − D_k

 ∂L

∂u^α_ijk

 +· · ·



+D_jD_k

W^α  ∂L

∂u^α_ijk − · · ·



, (2.1.3)

where W^α =η^α− ξ^ju^α_j , α = 1, 2, ..., m , and i, j, k = 1, · · · , n.

If we extend the Lam´e equation (1.1.1), we will have a system of differential equations consisting of three equations:

F1 ≡ β(u¹_xx+u¹_yy+u¹_zz) +u¹_xx+u²_xy+u³_xz− u¹_tt = 0,

F2 ≡ β(u²_xx+u²_yy+u²_zz) +u¹_xy+u²_yy+u³_yz− u²_tt= 0, (2.1.4) F₃ ≡ β(u³_xx+u³_yy+u³_zz) +u¹_xz+u²_yz+u³_zz− u³_tt = 0.

Here the variables t, x, y, z are independent and u¹, u², u³ are dependent variables.

A conservation law for the system has the form

D_i(Cⁱ) 

Fα= 0

= 0 where α = 1, 2, 3 , i = 0, 1, 2, 3; x⁰ =t, x¹ =x, x² =y, x³=z . Therefore we have

Dt(C⁰) +Dx(C¹) +Dy(C²) +Dz(C³) 

Fα= 0

= 0 where α = 1, 2, 3 . (2.1.5) Since the Lagrangian (1.1.3) of the system (2.1.4) includes only first derivatives, the formula (2.1.3) will be simplified to the following form:

Cⁱ =ξⁱL + W^α ∂L

∂u^α_i , i = 0, 1, 2, 3 . (2.1.6) Hence (2.1.5) can be used for checking the validity of calculations.

Remark. At the end of this thesis, formulas for components of conserved vectors obtained by Noether’s theorem are compared with the components obtained by Ibragimov’s method. It will be shown that the latter formulas are simpler. Therefore, I do not simplify formulas for components of conserved vectors in Sections 2.2-2.8 and Section 2.11.

2.2 Case X = ∂

∂t

In this caseξ⁰= 1, ξ¹=ξ² =ξ³ = 0, η^α= 0 and

W^α=η^α− ξⁱu^α_j =−u^α_t, α = 1, 2, 3 . (2.2.1) Invoking (2.1.6) we have

C₀⁰ =L − u^α_t ∂L

∂u^α_t ,

which gives C₀⁰=β



(u¹_y+u²_x)²+ (u¹_z+u³_x)²+ (u²_z+u³_y)²+ 2(u¹_x)²+ 2(u²_y)²+ 2(u³_z)²



+ (1− β)(u¹_x+u²_y+u³_z)²+ (u_t¹)²+ (u²_t)²+ (u³_t)². (2.2.2)

By applying the formula (2.1.6) for other components we can easily findC₀¹, C₀², and C₀³:

C₀¹=−u¹_t

4βu¹_x+ 2(1− β)(u¹_x+u²_y+u³_z)

− u²_t

2β(u²_x+u¹_y)

− u³_t

2β(u¹_z+u³_x)

, (2.2.3)

C₀² =−u¹_t

2β(u²_x+u¹_y)

− u²_t

4βu²_y+ 2(1− β)(u¹_x+u²_y+u³_z)

− u³_t

2β(u²_z+u³_y)

, (2.2.4)

C₀³ =−u¹_t

2β(u¹_z+u³_x)

− u²_t

2β(u²_z+u³_y)

− u³_t

4βu³_z+ 2(1− β)(u¹_x+u²_y+u³_z)

. (2.2.5)

Now we need to verify if the derived results satisfy the condition (2.1.5). After application of the total differentiations and doing the simplifications, we can easily show that

D_t(C⁰) +D_x(C¹) +D_y(C²) +D_z(C³) 

Fα= 0

=

− 2u¹_tF₁− 2u²_tF₂− 2u³_tF₃ 

Fα= 0

= 0, α = 1, 2, 3 . (2.2.6)

Proposition: The operatorX₀ admitted by the system (2.1.4) provides the conserved vector (C₀⁰, C₀¹, C₀², C₀³) with the components (2.2.2), (2.2.3), (2.2.4) and (2.2.5).

2.3 Case X

= ∂

∂x

Here ξ¹= 1, ξ⁰ =ξ² =ξ³ = 0, η^α = 0 and

W^α =η^α− ξⁱu^α_j =−u^α_x, α = 1, 2, 3 . (2.3.1) Invoking (2.1.6) we obtain:

C₁⁰ = 2u¹_tu¹_x+ 2u_t²u²_x+ 2u³_tu³_x. (2.3.2) Additionally, we have

C₁¹ =L − u¹_x

4βu¹_x+ 2(1− β)(u¹_x+u²_y+u³_z)

− u²_x

2β(u²_x+u¹_y)

− u³_x

2β(u¹_z+u³_x) .

Hence using the expression for Lagrangian we obtain:

C₁¹ = 3 α=1

 β

(u^α_y)²+ (u^α_z)²− (u^α_x)²

− (u^α_t)²



+ 2β(u²_zu³_y− u²_yu³_z) + 2u²_yu³_z

− (u¹_x)²+ (u²_y)²+ (u³_z)². (2.3.3)

Other components are:

C₁²=− u¹_x

2β(u²_x+u¹_y)

− u²_x

4βu²_y+ 2(1− β)(u¹_x+u²_y+u³_z)

− u³_x

2β(u²_z+u³_y) ,

C₁³=− u¹_x

2β(u¹_z+u³_x)

− u²_x

2β(u²_z+u³_y)

− u³_x

4βu³_z+ 2(1− β)(u¹_x+u²_y+u³_z)

. (2.3.4)

Checking (2.1.5) we obtain:

D_t(C⁰) +D_x(C¹) +D_y(C²) +D_z(C³) 

Fα= 0

=

− 2u¹_xF₁− 2u²_xF₂− 2u³_xF₃ 

Fα= 0

= 0, α = 1, 2, 3. (2.3.5)

Proposition: The operatorX₁ admitted by the system (2.1.4) provides the conserved vector (C₁⁰, C₁¹, C₁², C₁³) with the components (2.3.2) and (2.3.4).

2.4 Case X

= ∂

∂y

In this caseξ²= 1, ξ⁰=ξ¹ =ξ³ = 0, η^α= 0 and

W^α=η^α− ξⁱu^α_j =−u^α_y, α = 1, 2, 3. (2.4.1) Invoking (2.1.6) we have:

C₂⁰ = 2u¹_tu¹_y+ 2u²_tu²_y+ 2u³_tu³_y,

C₂¹ = − u¹_y

4βu¹_x+ 2(1− β)(u¹_x+u²_y +u³_z)

− u²_y

2β(u²_x+u¹_y)

− u³_y

2β(u¹_z+u³_x) ,

C₂² = 3 α=1

 β

(u^α_x)²+ (u^α_z)²− (u^α_y)²

− (u^α_t)²



+ 2β(u¹_zu³_x− u¹_xu³_z) + 2u¹_xu³_z + (u¹_x)²+ (u_z³)²− (u²_y)²,

C₂³ =− u¹_y

2β(u¹_z+u³_x)

− u²_y

2β(u²_z+u³_y)

− u³_y

4βu³_z+ 2(1− β)(u¹_x+u²_y+u³_z)

. (2.4.2)

Now we need to verify if the derived results satisfy the condition (2.1.5). After application of the total differentiations and doing the simplifications, we can easily show that

D_t(C⁰) +D_x(C¹) +D_y(C²) +D_z(C³) 

Fα= 0

=

− 2u¹_yF1− 2u²_yF2− 2u³_yF3

Fα= 0

= 0, α = 1, 2, 3 .

Proposition: The operator X₂ admitted by the system (2.1.4) provides the conserved vector (C₂⁰, C₂¹, C₂², C₂³) with the components (2.4.2).

2.5 Case X

= ∂

∂z

We have ξ³= 1, ξ¹ =ξ² =ξ⁰ = 0, η^α= 0 and

W^α=η^α− ξⁱu^α_j =−u^α_z, α = 1, 2, 3. (2.5.1) Invoking (2.1.6) we obtain

C₃⁰ = 2u¹_tu¹_z+ 2u²_tu²_z+ 2u³_tu³_z,

C₃¹ =− u¹_z

4βu¹_x+ 2(1− β)(u¹_x+u²_y+u³_z)

− u²_z

2β(u²_x+u¹_y)

− u³_z

2β(u¹_z+u³_x) ,

C₃² =− u¹_z

2β(u²_x+u¹_y)

− u²_z

4βu²_y+ 2(1− β)(u¹_x+u²_y+u³_z)

− u³_z

2β(u²_z+u³_y) ,

C₃³ = 3 α=1

 β

(u^α_x)²+ (u^α_y)²− (u^α_z)²

− (u^α_t)²



+ 2β(u¹_yu²_x− u¹_xu²_y) + 2u¹_xu²_y

+ (u¹_x)²+ (u_y²)²− (u³_z)². (2.5.2) Accordingly, the equation (2.1.5) has the form:

D_t(C⁰) +D_x(C¹) +D_y(C²) +D_z(C³) 

Fα= 0

=

− 2u¹_zF₁− 2u²_zF₂− 2u³_zF₃ 

Fα= 0

= 0, α = 1, 2, 3. (2.5.3)

Proposition: The operatorX₃ admitted by the system (2.1.4) provides the conserved vector (C₃⁰, C₃¹, C₃², C₃³) with the components (2.5.2).

2.6 Case X

= y ∂

∂z − z ∂

∂y + u

∂

∂u

− u

∂

∂u

In this case ξ⁰ =ξ¹ = 0, ξ² =−z , ξ³ =y, and η¹ = 0, η² =−u³, η³ =u². Firstly, we need to verify if the condition (2.1.1) is satisfied. Hence we need to prolong the generator:

X˜4=X4+ζ_i^α ∂

∂u^α_i , where

ζ_i^α =Di(η^α)− u^α_jDi(ξ^j). (2.6.1) After applying the prolongation formula (2.6.1) we obtain

X˜₄=y ∂

∂z − z ∂

∂y+u² ∂

∂u³ − u³ ∂

∂u² − u¹_z ∂

∂u¹_y +u¹_y ∂

∂u¹_z − u³_t ∂

∂u²_t − u³_x ∂

∂u²_x − (u³_y+u²_z) ∂

∂u²_y

− (u³_z− u²_y) ∂

∂u²_z +u²_t ∂

∂u³_t +u²_x ∂

∂u³_x + (u²_y− u³_z) ∂

∂u³_y + (u²_z+u³_y) ∂

∂u³_z · (2.6.2)

Using the prolonged operator for verification of the condition (2.1.1) we get X˜4(L) + L Di(ξⁱ) = ˜X4(L) = 0.

Hence the symmetry X₄ satisfies the condition (2.1.1) and, accordingly, there exists a conser- vation law corresponding to this symmetry. In this case we have

W^α=η^α− ξⁱu^α_j =η^α+zu^α_y − yu^α_z, α = 1, 2, 3. (2.6.3) Accordingly, using the formulae (2.1.6) for components of the conserved vector, we obtain

C₄⁰ = − 2(zu¹_y− yu¹_z)u¹_t − 2(−u³+zu²_y− yu²_z)u²_t − 2(u²+zu³_y− yu³_z)u³_t, C₄¹ = (zu¹_y− yu¹_z)



4βu¹_x+ 2(1− β)(u¹_x+u²_y+u³_z)

+ (−u³+zu²_y− yu²_z)



2β(u²_x+u¹_y)

+ (u²+zu³_y− yu³_z)



2β(u¹_z+u³_x) ,

C₄² = 3 α=1

β

 z

(u^α_y)²− (u^α_x)²− (u^α_z)²

− 2yu^α_yu^α_z



+z(u^α_t)² 

+z(u²_y)²− z(u¹_x)²− z(u³_z)² + 2β

z(u¹_xu³_z− u¹_zu³_x)− y(u¹_zu²_x− u¹_xu²_z) +u³(u¹_x− u²_y+u_z³) +u²(u²_z+u³_y)

− 2zu¹_xu³_z− 2(u³+yu²_z)(u¹_x+u²_y+u³_z),

C₄³ = 3 α=1

β

 y

(u^α_y)²+ (u^α_x)²− (u^α_z)²



+ 2zu^α_yu^α_z



− y(u^α_t)² 

+y(u¹_x)²+y(u²_y)²− y(u³_z)² + 2β

y(u¹_yu²_x− u¹_xu²_y) +z(u¹_yu³_x− u¹_xu³_y)− u²(u¹_x+u²_y − u_z³)− u³(u²_z+u³_y)

+ 2yzu¹_xu²_y+ 2(u²+zu³_y)(u¹_x+u²_y+u³_z). (2.6.4) Checking the equation (2.1.5) we obtain

Dt( ˜C⁰) +Dx( ˜C¹) +Dy( ˜C²) +Dz( ˜C³) 

Fα= 0

=

2W^αFα

Fα= 0

= 0, α = 1, 2, 3. (2.6.5)

Proposition: The operatorX4 admitted by the system (2.1.4) provides the conserved vector (C₄⁰, C₄¹, C₄², C₄³) with the components (2.6.4).

2.7 Case X

= z ∂

∂x − x ∂

∂z + u

∂

∂u

− u

∂

∂u

Here ξ⁰=ξ²= 0, ξ¹ =z, ξ³ =−x, and η² = 0, η¹=u³, η³=−u¹. Firstly, we need to verify if the condition (2.1.1) is satisfied. Hence we need to prolong the generator:

X˜5 =X5+ζ_i^α ∂

∂u^α_i , where

ζ_i^α =D_i(η^α)− u^α_jD_i(ξ^j). (2.7.1)