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Guest Editors: Maria Gandarias, Mariano Torrisi, Maria Bruzón, Rita Tracinà, and Chaudry Masood Khalique

Recent Advances in Symmetry Groups and Conservation Laws for Partial Differential Equations and Applications

Abstract and Applied Analysis

Recent Advances in Symmetry Groups andConservation Laws for Partial DifferentialEquations and Applications

Abstract and Applied Analysis

Recent Advances in Symmetry Groups andConservation Laws for Partial DifferentialEquations and Applications

Guest Editors:MariaGandarias,MarianoTorrisi,Maria Bruzón,Rita Tracinà, and Chaudry Masood Khalique

Copyright © 2014 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Abstract and Applied Analysis.” All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Ravi P. Agarwal, USABashir Ahmad, Saudi ArabiaM. O. Ahmedou, GermanyNicholas D. Alikakos, GreeceDebora Amadori, ItalyDouglas R. Anderson, USAJan Andres, Czech RepublicGiovanni Anello, ItalyStanislav Antontsev, PortugalM. K. Aouf, EgyptNarcisa C. Apreutesei, RomaniaNatig M. Atakishiyev, MexicoFerhan M. Atici, USAIvan Avramidi, USASoohyun Bae, KoreaZhanbing Bai, ChinaChuanzhi Bai, ChinaDumitru Baleanu, TurkeyJózef Banaś, PolandMartino Bardi, ItalyRoberto Barrio, SpainFeyzi Başar, TurkeyA. Bellouquid, MoroccoDaniele Bertaccini, ItalyLucio Boccardo, ItalyIgor Boglaev, New ZealandMartin J. Bohner, USAGeraldo Botelho, BrazilElena Braverman, CanadaRomeo Brunetti, ItalyJanusz Brzdek, PolandDetlev Buchholz, GermanySun-Sig Byun, KoreaFabio M. Camilli, ItalyJinde Cao, ChinaAnna Capietto, ItalyJianqing Chen, ChinaWing-Sum Cheung, Hong KongMichel Chipot, SwitzerlandChangbum Chun, KoreaSoon-Yeong Chung, KoreaJaeyoung Chung, KoreaSilvia Cingolani, ItalyJean M. Combes, FranceMonica Conti, Italy

Juan Carlos Cortés López, SpainGraziano Crasta, ItalyZhihua Cui, ChinaBernard Dacorogna, SwitzerlandVladimir Danilov, RussiaMohammad T. Darvishi, IranL. F. Pinheiro de Castro, PortugalToka Diagana, USAJesús I. Dı́az, SpainJosef Dibĺık, Czech RepublicFasma Diele, ItalyTomas Dominguez, SpainAlexander Domoshnitsky, IsraelMarco Donatelli, ItalyBoQing Dong, ChinaWei-Shih Du, TaiwanLuiz Duarte, BrazilRoman Dwilewicz, USAPaul W. Eloe, USAAhmed El-Sayed, EgyptLuca Esposito, ItalyKhalil Ezzinbi, MoroccoJulian F. Bonder, ArgentinaDashan Fan, USAAngelo Favini, ItalyMárcia Federson, BrazilS. Filippas, Equatorial GuineaAlberto Fiorenza, ItalyXianlong Fu, ChinaMassimo Furi, ItalyJesús G. Falset, SpainGiovanni P. Galdi, USAIsaac Garcia, SpainJosé A. Garćıa-Rodŕıguez, SpainLeszek Gasinski, PolandGyörgy Gát, HungaryVladimir Georgiev, ItalyLorenzo Giacomelli, ItalyJaume Giné, SpainValery Y. Glizer, IsraelJean P. Gossez, BelgiumJose L. Gracia, SpainMaurizio Grasselli, ItalyLuca Guerrini, ItalyYuxia Guo, China

Qian Guo, ChinaChaitan P. Gupta, USAUno Hämarik, EstoniaMaoan Han, ChinaFerenc Hartung, HungaryJiaxin Hu, ChinaZhongyi Huang, ChinaChengming Huang, ChinaGennaro Infante, ItalyIvan Ivanov, BulgariaHossein Jafari, South AfricaJaan Janno, EstoniaAref Jeribi, TunisiaUncig Ji, KoreaZhongxiao Jia, ChinaLucas Jódar, SpainJong S. Jung, Republic of KoreaHenrik Kalisch, NorwayHamid R. Karimi, NorwayC. Masood Khalique, South AfricaSatyanad Kichenassamy, FranceTero Kilpeläinen, FinlandSung G. Kim, Republic of KoreaLjubisa Kocinac, SerbiaAndrei Korobeinikov, SpainPekka Koskela, FinlandVictor Kovtunenko, AustriaRen-Jieh Kuo, TaiwanPavel Kurasov, SwedenMilton C. L. Filho, BrazilMiroslaw Lachowicz, PolandKunquan Lan, CanadaRuediger Landes, USAIrena Lasiecka, USAMatti Lassas, FinlandChun-Kong Law, TaiwanMing-Yi Lee, TaiwanGongbao Li, ChinaElena Litsyn, IsraelShengqiang Liu, ChinaYansheng Liu, ChinaCarlos Lizama, ChileGuozhen Lu, USAJinhu Lü, ChinaG. Lukaszewicz, Poland

Wanbiao Ma, ChinaNazim I. Mahmudov, TurkeyEberhard Malkowsky, TurkeySalvatore A. Marano, ItalyCristina Marcelli, ItalyPaolo Marcellini, ItalyJesús Maŕın-Solano, SpainJose M. Martell, SpainMieczysław Mastyło, PolandMing Mei, CanadaTaras Mel’nyk, UkraineAnna Mercaldo, ItalyStanislaw Migorski, PolandMihai Mihǎilescu, RomaniaFeliz Minhós, PortugalDumitru Motreanu, FranceMaria Grazia Naso, ItalyG. M. N’GUÉRÉKATA, USAMicah Osilike, NigeriaMitsuharu Ôtani, JapanTurgut Öziş, TurkeyN. S. Papageorgiou, GreeceSehie Park, KoreaKailash C. Patidar, South AfricaKevin R. Payne, ItalyAdemir F. Pazoto, BrazilShuangjie Peng, ChinaAntonio M. Peralta, SpainSergei V. Pereverzyev, AustriaAllan Peterson, USAAndrew Pickering, SpainCristina Pignotti, ItalySomyot Plubtieng, ThailandMilan Pokorny, Czech RepublicSergio Polidoro, ItalyZiemowit Popowicz, PolandMaria M. Porzio, ItalyEnrico Priola, ItalyVladimir S. Rabinovich, MexicoI. Rachůnková, Czech Republic

Maria A. Ragusa, ItalySimeon Reich, IsraelAbdelaziz Rhandi, ItalyHassan Riahi, MalaysiaJuan P. Rincón-Zapatero, SpainLuigi Rodino, ItalyYuriy Rogovchenko, NorwayJulio D. Rossi, ArgentinaWolfgang Ruess, GermanyBernhard Ruf, ItalySatit Saejung, ThailandStefan G. Samko, PortugalMartin Schechter, USAJavier Segura, SpainValery Serov, FinlandNaseer Shahzad, Saudi ArabiaAndrey Shishkov, UkraineStefan Siegmund, GermanyA.-M. A. Soliman, EgyptPierpaolo Soravia, ItalyMarco Squassina, ItalyHari M. Srivastava, CanadaSvatoslav Staněk, Czech RepublicStevo Stević, SerbiaAntonio Suárez, SpainWenchang Sun, ChinaWenyu Sun, ChinaRobert Szalai, UKSanyi Tang, ChinaChun-Lei Tang, ChinaGabriella Tarantello, ItalyN.-E. Tatar, Saudi ArabiaGerd Teschke, GermanySergey Tikhonov, SpainClaudia Timofte, RomaniaThanh Tran, AustraliaJuan J. Trujillo, SpainGabriel Turinici, FranceMilan Tvrdy, Czech RepublicMehmet Ünal, Turkey

Csaba Varga, RomaniaCarlos Vazquez, SpainJesus Vigo-Aguiar, SpainQing-WenWang, ChinaShawn X. Wang, CanadaYushun Wang, ChinaYouyu Wang, ChinaJing P. Wang, UKPeixuan Weng, ChinaNoemi Wolanski, ArgentinaNgai-Ching Wong, TaiwanPatricia J. Y. Wong, SingaporeZili Wu, ChinaYonghong Wu, AustraliaShi-Liang Wu, ChinaShanhe Wu, ChinaTiecheng Xia, ChinaXu Xian, ChinaYanni Xiao, ChinaFuding Xie, ChinaGongnan Xie, ChinaDaoyi Xu, ChinaZhenya Yan, ChinaXiaodong Yan, USANorio Yoshida, JapanBeong In Yun, KoreaAgacik Zafer, TurkeyJianming Zhan, ChinaWeinian Zhang, ChinaChengjian Zhang, ChinaZengqin Zhao, ChinaSining Zheng, ChinaYong Zhou, ChinaTianshou Zhou, ChinaChun-Gang Zhu, ChinaQiji J. Zhu, USAMalisa R. Zizovic, SerbiaWenming Zou, China

Contents

Recent Advances in Symmetry Groups and Conservation Laws for Partial Differential Equations andApplications, Maria Gandarias, Mariano Torrisi, Maria Bruzón, Rita Tracinà,and Chaudry Masood KhaliqueVolume 2014, Article ID 457145, 2 pages

Invariant Inhomogeneous Bianchi Type-I Cosmological Models with Electromagnetic Fields Using LieGroup Analysis in Lyra Geometry, Ahmad T. AliVolume 2014, Article ID 918927, 8 pages

Derivation of Conservation Laws for the Magma Equation Using the Multiplier Method: Power Law andExponential Law for Permeability and Viscosity, N. Mindu and D. P. MasonVolume 2014, Article ID 585167, 13 pages

Nonlinear Self-Adjoint Classification of a Burgers-KdV Family of Equations,Júlio Cesar Santos Sampaio and Igor Leite FreireVolume 2014, Article ID 804703, 7 pages

Combinatorial Properties and Characterization of Glued Semigroups, J. I. Garćıa-Garćıa,M. A. Moreno-Fŕıs, and A. Vigneron-TenorioVolume 2014, Article ID 436417, 8 pages

Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-DimensionalKadomtsev-Petviashvili Equation with Time-Dependent Coefficients, Li-hua ZhangVolume 2014, Article ID 853578, 13 pages

Nonlinearly Self-Adjoint, Conservation Laws and Solutions for a Forced BBM Equation,Maria Luz Gandarias and Chaudry Masood KhaliqueVolume 2014, Article ID 630282, 5 pages

Weak Equivalence Transformations for a Class of Models in Biomathematics,Igor Leite Freire and Mariano TorrisiVolume 2014, Article ID 546083, 9 pages

Global Existence and Large Time Behavior of Solutions to the Bipolar Nonisentropic Euler-PoissonEquations, Min Chen, Yiyou Wang, and Yeping LiVolume 2014, Article ID 213569, 13 pages

Q-Symmetry and Conditional Q-Symmetries for Boussinesq Equation,Hassan A. Zedan and Seham Sh. TantawyVolume 2014, Article ID 175982, 4 pages

Approximate Symmetry Analysis of a Class of Perturbed Nonlinear Reaction-Diffusion Equations,Mehdi Nadjafikhah and Abolhassan MahdaviVolume 2013, Article ID 395847, 7 pages

Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance, Hanze LiuVolume 2013, Article ID 156965, 10 pages

Conservation Laws of Two (2 + 1)-Dimensional Nonlinear Evolution Equations with Higher-OrderMixed Derivatives, Li-hua ZhangVolume 2013, Article ID 594975, 10 pages

Symmetry Reductions, Exact Solutions, and Conservation Laws of a Modified Hunter-Saxton Equation,Andrew Gratien Johnpillai and Chaudry Masood KhaliqueVolume 2013, Article ID 204746, 5 pages

EditorialRecent Advances in Symmetry Groups and Conservation Lawsfor Partial Differential Equations and Applications

Maria Gandarias,1 Mariano Torrisi,2 Maria Bruzón,1

Rita Tracinà,2 and Chaudry Masood Khalique3

1Departamento de Matematicas, Universidad de Cádiz, Puerto Real, 11510 Cádiz, Spain2Dipartimento di Matematica e Informatica, Università di Catania, 95125 Catania, Italy3Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling,North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

Correspondence should be addressed to Maria Gandarias; [email protected]

Received 25 August 2014; Accepted 25 August 2014; Published 22 December 2014

Copyright © 2014 Maria Gandarias et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Differential equations govern many natural phenomena andplay an important role in the progress of engineering andtechnology. Essentially a lot of the fundamental equations arenonlinear and in general such nonlinear equations are oftenvery difficult to solve explicitly. Symmetry group techniquesprovidemethods to obtain solutions of these equations.Thesemethods have several applications in the studies of partialdifferential equation. They are also useful in the search forconservation laws which arises in many fields of the appliedsciences. Recent studies have shown that infinitely manynonlocal symmetries of various integrable models are relatedto their Lax pairs. Moreover symmetry method is one of themost powerful tools that give new integrable models fromknown ones. Integrablemodels have played an important rolein applied sciences and are one of the central topics in solitontheory. In order to know if a system is integrable, it is veryimportant to study Lax pairs of the system.

A symmetry can be considered as an equivalence trans-formation which leaves invariant not only the differentialstructure of equation but also the form of the arbitraryelements.This fact made Ovsiannikov search for equivalencetransformations in a systematic way by using an algorithmbased on the extension of the Lie infinitesimal criterion.

When an equation contains an arbitrary function, itreflects the individual characteristic of the phenomenabelonging to a large class. In this sense, the knowledge

of equivalence transformations can provide us with certainrelations between the solutions of different phenomena of thesame class.

Nowadays several branches of the theoretical and appliedsciences such as mathematics, physics, biology, economy,and finance rely on processes which are usually modeled bynonlinear differential equations. Often it is difficult to obtainreductions and exact solutions for these models. Our aim isto highlight applications of symmetry methods to nonlinearmodels in physics, engineering, and the applied science aswell as to show recent theoretical developments in symmetrygroups and geometric methods.

The authors of this special issue had been invited tosubmit original research articles as well as review articlesin the following topics: advanced researches and theoreticalanalyses in group transformations and differential equations;Noether symmetries, applications, and conservation laws;numerical algorithms concerning the symmetry groups forpartial differential equations; new and direct methods toobtain exact explicit solutions for differential equations;applications: novel applications in sciences, including engi-neering, physics, biology, and finance; and reviews: lucidsurveys and review articles dealing withmodern and classicaltopics.

However, we received 20 papers in these research fields.After a rigorous reviewing process, twelve articles were finally

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 457145, 2 pageshttp://dx.doi.org/10.1155/2014/457145

http://dx.doi.org/10.1155/2014/457145

2 Abstract and Applied Analysis

accepted for publication.These articles contain somenew andinnovative techniques and ideas that may stimulate furtherresearches in several branches of theory and applications ofthe transformation groups.

Acknowledgments

We would like to thank all the authors who sent their worksand all the referees for the time spent in reviewing thepapers. Their contributions and their efforts have been veryimportant for the publication of this special issue.

Maria GandariasMariano TorrisiMaria BruzónRita Tracinà

Chaudry Masood Khalique

Research ArticleInvariant Inhomogeneous Bianchi Type-ICosmological Models with Electromagnetic FieldsUsing Lie Group Analysis in Lyra Geometry

Ahmad T. Ali1,2

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia2Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt

Correspondence should be addressed to Ahmad T. Ali; [email protected]

Received 28 December 2013; Accepted 3 June 2014; Published 19 June 2014

Academic Editor: Maria Bruzón

Copyright © 2014 Ahmad T. Ali.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Wefind a new class of invariant inhomogeneous Bianchi type-I cosmologicalmodels in electromagnetic fieldwith variablemagneticpermeability. For this, Lie group analysis method is used to identify the generators that leave the given system of nonlinear partialdifferential equations (NLPDEs) (Einstein field equations) invariant. With the help of canonical variables associated with thesegenerators, the assigned system of PDEs is reduced to ordinary differential equations (ODEs) whose simple solutions providenontrivial solutions of the original system. A new class of exact (invariant-similarity) solutions have been obtained by consideringthe potentials of metric and displacement field as functions of coordinates 𝑥 and 𝑡. We have assumed that 𝐹

12is only nonvanishing

component of electromagnetic field tensor𝐹𝑖𝑗.TheMaxwell equations show that 𝐹

12is the function of 𝑥 alone whereas themagnetic

permeability 𝜇 is the function of 𝑥 and 𝑡 both. The physical behavior of the obtained model is discussed.

1. Introduction

The inhomogeneous cosmological models play a significantrole in understanding some essential features of the universe,such as the formation of galaxies during the early stages ofevolution and process of homogenization. Therefore, it willbe interesting to study inhomogeneous cosmological models.The best-known inhomogeneous cosmological model is theLemaitre-Tolman model (or LT model) which deals with thestudy of structure in the universe by means of exact solu-tions of Einstein’s field equations. Some other known exactsolutions of inhomogeneous cosmological models are theSzekeresmetric, Szafronmetric, Stephani metric, Kantowski-Sachs metric, Barnes metric, Kustaanheimo-Qvist metric,and Senovilla metric [1].

Einstein’s general theory relativity is based on Rieman-nian geometry. If one modifies the Riemannian geometry,then Einstein’s field equations will be changed automati-cally from its original form. Modifications of Riemanniangeometry have developed to solve the problems such asunification of gravitation with electromagnetism, problems

arising when the gravitational field is coupled to matterfields, and singularities of standard cosmology. In recentyears, there has been considerable interest in alternativetheory of gravitation to explain the above-unsolved problems.Long ago, since 1951, Lyra [2] proposed a modification ofRiemannian geometry by introducing a gauge function intothe structureless manifold that bears a close resemblance toWeyl’s geometry.

Using the above modification of Riemannian geometrySen [3, 4] and Sen and Dunn [5] proposed a new scalartensor theory of gravitation and constructed it very similar toEinstein field equations. Based on Lyra’s geometry, the fieldequations can be written as [3, 4]

𝑅𝑖𝑗−

1

2

𝑔𝑖𝑗𝑅 +

3

2

𝜙𝑖𝜙𝑗−

3

4

𝑔𝑖𝑗𝜙𝑘𝜙𝑘= −𝜒𝑇

𝑖𝑗, (1)

where 𝜙𝑖is the displacement vector and other symbols have

their usual meaning as in Riemannian geometry.Halford [6] has argued that the nature of constant

displacement field 𝜙𝑖in Lyra’s geometry is very similar to

cosmological constant Λ in the normal general relativistic


http://dx.doi.org/10.1155/2014/918927


theory. Halford also predicted that the present theory willprovide the same effects within observational limits, as far asthe classical solar system tests are concerned, as well as testsbased on the linearized form of field equations. For a reviewon Lyra geometry, one can see [7].

Recently, Pradhan et al. [8–14], Casana et al. [15],Rahaman et al. [16], Bali and Chandnani [17, 18], Kumar andSingh [19], Yadav et al. [20], Rao et al. [21], Zia and Singh [22],and Ali and Rahaman [23] have studied cosmological modelsbased on Lyra’s geometry in various contexts.

To study the nonlinear physical phenomena [24–27], it isimportant to search the exact solutions of nonlinear PDEs.Ovsiannikov [28] is the pioneer who had observed that theusual Lie infinitesimal invariance approach could as wellbe employed in order to construct symmetry groups [29–31]. The symmetry groups of a differential equation couldbe defined as the groups of continuous transformations thatlead a given family of equations invariant [32–35] and areproved to be important to solve the nonlinear equations of themodels to describe complex physical phenomena in variousfields of science, especially in fluid mechanics, solid statephysics, plasma physics, plasma wave, and general relativity.

In this paper, we have obtained exact solutions of Ein-stein’smodified field equations in inhomogeneous space-timeBianchi type-I cosmological model within the frame work ofLyra’s geometry in the presence ofmagnetic fieldwith variablemagnetic permeability and time varying displacement vector𝛽(𝑥, 𝑡) using the so-called symmetry analysis method. Sincethe field equations are highly nonlinear differential equations,therefore symmetry analysis method can be successfullyapplied to nonlinear differential equations. The similarity(invariant) solutions help to reduce the independent variablesof the problem, and therefore we employ this method inthe investigation of exact solution of the field equations.In general, invariant solutions will transform the system ofnonlinear PDEs into a system of ODEs. We attempted tofind a new class of exact (invariant) solutions for the fieldequations based on Lyra geometry.

The scheme of the paper is as follows. Magnetizedinhomogeneous Bianchi type-I cosmological model withvariable magnetic permeability based on Lyra geometry isintroduced in Section 2. In Section 3, we have performedsymmetry analysis and have obtained isovector fields forEinstein field equations under consideration. In Section 4, wefound new class of exact (invariant) solutions for Einsteinfield equations. Section 5 is devoted to study of some physicaland geometrical properties of the model.

2. The Metric and Field Equations

We consider Bianchi type-Imetric, with the convention (𝑥0 =𝑡, 𝑥1= 𝑥, 𝑥

2= 𝑦, 𝑥

3= 𝑧), in the form

𝑑𝑠2= 𝑑𝑡2− 𝐴2𝑑𝑥2− 𝐵2𝑑𝑦2− 𝐶2𝑑𝑧2, (2)

where𝐴 is a function of 𝑡 only while 𝐵 and 𝐶 are functions of𝑥 and 𝑡. Without loss of generality, we can put the followingtransformation:

𝐵 = 𝐴𝑓, 𝐶 = 𝐴𝑔, (3)

where 𝑓 and 𝑔 are functions of 𝑥 and 𝑡. The volume elementof model (2) is given by

𝑉 = √−𝑔 = 𝐴3𝑓𝑔. (4)

The four-acceleration vector, the rotation, the expansionscalar, and the shear scalar characterizing the four-velocityvector field, 𝑢𝑖, satisfying the relation in comoving coordinatesystem

𝑔𝑖𝑗𝑢𝑖𝑢𝑗= 1, 𝑢

𝑖= 𝑢𝑖= (1, 0, 0, 0) , (5)

respectively, have the usual definitions as given by Raychaud-huri [36]

̇𝑢𝑖= 𝑢𝑖;𝑗𝑢𝑗,

𝜔𝑖𝑗

= 𝑢[𝑖;𝑗]

+ ̇𝑢[𝑖𝑢𝑗],

Θ = 𝑢𝑖

;𝑖,

𝜎2=

1

2

𝜎𝑖𝑗𝜎𝑖𝑗,

(6)

where

𝜎𝑖𝑗

= 𝑢(𝑖;𝑗)

+ ̇𝑢(𝑖𝑢𝑗)

−

1

3

Θ (𝑔𝑖𝑗+ 𝑢𝑖𝑢𝑗) . (7)

In view of metric (2), the four-acceleration vector, therotation, the expansion scalar, and the shear scalar given by(6) can be written in a comoving coordinates system as

̇𝑢𝑖= 0,

𝜔𝑖𝑗

= 0,

Θ =

3 ̇𝐴

𝐴

+

𝑓𝑡

𝑓

+

𝑔𝑡

𝑔

,

𝜎2=

̇𝐴

𝐴

(

2 ̇𝐴

𝐴

+

4𝑓𝑡

3𝑓

+

4𝑔𝑡

3𝑔

) +

5𝑓2

𝑡

9𝑓2

+

𝑓𝑡𝑔𝑡

9𝑓𝑔

+

5𝑔2

𝑡

9𝑔2,

(8)

where the nonvanishing components of the shear tensor 𝜎𝑗𝑖

are

𝜎1

1= −

𝑓𝑡

3𝑓

−

𝑔𝑡

3𝑓

, 𝜎2

2=

2𝑓𝑡

3𝑓

−

𝑔𝑡

3𝑔

,

𝜎3

3=

2𝑔𝑡

3𝑔

−

𝑓𝑡

3𝑓

, 𝜎4

4= −

2 ̇𝐴

𝐴

−

2𝑓𝑡

3𝑓

−

2𝑔𝑡

3𝑔

.

(9)

To study the cosmological model, we use the field equationsin Lyra geometry given in (1) in which the displacement fieldvector 𝜙

𝑖is given by

𝜙𝑖= (𝛽 (𝑥, 𝑡) , 0, 0, 0) . (10)

Abstract and Applied Analysis 3

𝑇𝑖𝑗is the energy momentum tensor given by

𝑇𝑖𝑗

= (𝜌 + 𝑝) 𝑢𝑖𝑢𝑗− 𝑝𝑔𝑖𝑗+ 𝐸𝑖𝑗, (11)

where 𝐸𝑖𝑗is the electromagnetic field given by Lichnerowicz

[37]:

𝐸𝑖𝑗

= 𝜇 [ℎ𝑙ℎ𝑙(𝑢𝑖𝑢𝑗−

1

2

𝑔𝑖𝑗) + ℎ𝑖ℎ𝑗] . (12)

Here 𝜌 and 𝑝 are the energy density and isotropic pressure,respectively, while 𝜇 is the magnetic permeability and ℎ

𝑖is

the magnetic flux vector defined by

ℎ𝑖=

√−𝑔

2𝜇

𝜖𝑖𝑗𝑘𝑙

𝐹𝑘𝑙𝑢𝑗. (13)

𝐹𝑖𝑗is the electromagnetic field tensor and 𝜖

𝑖𝑗𝑘𝑙is a Levi-Civita

tensor density. If we consider the current flow along 𝑧-axis,then 𝐹

12is only nonvanishing component of 𝐹

𝑖𝑗. Then the

Maxwell equations

𝐹𝑖𝑗;𝑘

+ 𝐹𝑗𝑘;𝑖

+ 𝐹𝑘𝑖;𝑗

= 0,

[

1

𝜇

𝐹𝑖𝑗]

;𝑗

= 𝐽𝑖

(14)

require 𝐹12

to be function of 𝑥 alone [38]. We assume themagnetic permeability as a function of both 𝑥 and 𝑡. Here thesemicolon represents a covariant differentiation.

For the line element (2), field equation (1) can be reducedto the following system of NLPDEs:

𝐸1=

𝑓𝑥𝑡

𝑓

+

𝑔𝑥𝑡

𝑔

= 0,

𝐸2=

𝑓𝑡𝑡

𝑓

+

𝑓𝑡𝑔𝑡

𝑓𝑔

+

1

𝐴2(

𝑔𝑥𝑥

𝑔

−

𝑓𝑥𝑔𝑥

𝑓𝑔

) +

3 ̇𝐴𝑓𝑡

𝐴𝑓

= 0,

(15)

𝜒𝜌 +

3

4

𝛽2=

𝑔𝑡𝑡

2𝑔

+

3𝑓𝑡𝑔𝑡

2𝑓𝑔

−

1

𝐴2(

𝑓𝑥𝑥

2𝑓

+

𝑔𝑥𝑥

𝑔

+

3𝑓𝑥𝑔𝑥

2𝑓𝑔

)

+

̇𝐴

𝐴

(

2𝑓𝑡

𝑓

+

𝑔𝑡

2𝑔

+

3 ̇𝐴

𝐴

) ,

𝜒𝑝 +

3

4

𝛽2=

1

2𝐴2(

𝑓𝑥𝑥

𝑓

+

𝑓𝑥𝑔𝑥

𝑓𝑔

) −

𝑓𝑡𝑡

𝑓

−

𝑔𝑡𝑡

2𝑔

−

𝑓𝑡𝑔𝑡

2𝑓𝑔

−

̇𝐴

𝐴

(

3𝑓𝑡

𝑓

+

̇𝐴

𝐴

) −

2 ̈𝐴

𝐴

,

𝜒𝐹2

12

𝜇𝐴4𝑓2

=

1

𝐴2(

𝑓𝑥𝑥

𝑓

−

𝑓𝑥𝑔𝑥

𝑓𝑔

) +

𝑔𝑡𝑡

𝑔

+

𝑓𝑡𝑔𝑡

𝑓𝑔

+

3 ̇𝐴𝑔𝑡

𝐴𝑔

.

(16)

3. Symmetry Analysis Method

Equations (15)–(16) are highly nonlinear PDEs and hence itis so difficult to handle since there exist no standard methodsfor obtaining analytical solution.The system (15) is nonlinear

PDEs of second order for the two unknowns 𝑓 and 𝑔. If wesolve this system, then we can get the solution of the fieldequations. In order to obtain an exact solution of the systemof nonlinear PDEs (15), we will use the symmetry analysismethod. For this we write

𝑥∗

𝑖= 𝑥𝑖+ 𝜖𝜉𝑖(𝑥𝑗, 𝑢𝛽) + o (𝜖2) ,

𝑢∗

𝛼= 𝑢𝛼+ 𝜖𝜂𝛼(𝑥𝑗, 𝑢𝛽) + o (𝜖2) ,

𝑖, 𝑗, 𝛼, 𝛽 = 1, 2, (17)

as the infinitesimal Lie point transformations. We haveassumed that the system (15) is invariant under the trans-formations given in (17). The corresponding infinitesimalgenerator of Lie groups (symmetries) is given by

𝑋 =

2

∑

𝑖=1

𝜉𝑖

𝜕

𝜕𝑥𝑖

+

2

∑

𝛼=1

𝜂𝛼

𝜕

𝜕𝑢𝛼

, (18)

where 𝑥1

= 𝑥, 𝑥2

= 𝑡, 𝑢1

= 𝑓, and 𝑢2

= 𝑔. The coefficients𝜉1, 𝜉2, 𝜂1, and 𝜂

2are the functions of 𝑥, 𝑡, 𝑓, and 𝑔. These

coefficients are the components of infinitesimals symmetriescorresponding to 𝑥, 𝑡,𝑓, and𝑔, respectively, to be determinedfrom the invariance conditions:

Pr(2)𝑋(𝐸𝑚)

𝐸𝑚=0

= 0, (19)

where 𝐸𝑚

= 0,𝑚 = 1, 2 are the system (15) under study andPr(2) is the second prolongation of the symmetries 𝑋. Sinceour equations (15) are at most of order two, therefore, weneed second order prolongation of the infinitesimal generatorin (19). It is worth noting that the 2nd order prolongation isgiven by

Pr(2)𝑋 = 𝑋 +2

∑

𝑖=1

2

∑

𝛼=1

𝜂𝛼,𝑖

𝜕

𝜕𝑢𝛼,𝑖

+

2

∑

𝑗=1

2

∑

𝑖=1

2

∑

𝛼=1

𝜂𝛼,𝑖𝑗

𝜕

𝜕𝑢𝛼,𝑖𝑗

, (20)

where

𝜂𝛼,𝑖

= 𝐷𝑖[

[

𝜂𝛼−

2

∑

𝑗=1

𝜉𝑗𝑢𝛼,𝑗

]

]

+

2

∑

𝑗=1

𝜉𝑗𝑢𝛼,𝑖𝑗

,

𝜂𝛼,𝑖𝑗

= 𝐷𝑖𝑗[𝜂𝛼−

2

∑

𝑘=1

𝜉𝑘𝑢𝛼,𝑘

] +

2

∑

𝑘=1

𝜉𝑘𝑢𝛼,𝑖𝑗𝑘

.

(21)

The operator 𝐷𝑖(𝐷𝑖𝑗) is called the total derivative (Hach

operator) and takes the following form:

𝐷𝑖=

𝜕

𝜕𝑥𝑖

+

2

∑

𝛼=1

𝑢𝛼,𝑖

𝜕

𝜕𝑢𝛼

+

2

∑

𝑗=1

2

∑

𝛼=1

𝑢𝛼,𝑖𝑗

𝜕

𝜕𝑢𝛼,𝑗

, (22)

where 𝐷𝑖𝑗

= 𝐷𝑖(𝐷𝑗) = 𝐷

𝑗(𝐷𝑖) = 𝐷

𝑗𝑖and 𝑢

𝛼,𝑖= 𝜕𝑢𝛼/𝜕𝑥𝑖.

Expanding the system (19) with the aid of Mathematicaprogram, along with the original system (15) to eliminate 𝑓

𝑥𝑡

and 𝑔𝑥𝑥

while we set the coefficients involving 𝑓𝑥, 𝑓𝑡, 𝑓𝑥𝑥, 𝑓𝑡𝑡,

𝑔𝑥,𝑔𝑡,𝑔𝑥𝑡, and𝑔

𝑡𝑡and various products equal zero, these gives

rise to the essential set of overdetermined equations. Solvingthe set of these determining equations, the components ofsymmetries take the following form:

𝜉1= 𝑐1𝑥 + 𝑐2, 𝜉

2= 𝑐3𝑡 + 𝑐4,

𝜂1= 0, 𝜂

2= 𝑐5𝑔,

(23)


where 𝑐𝑖, 𝑖 = 1, 2, . . . , 5, are arbitrary constants and the

function 𝐴(𝑡) must equal

𝐴 (𝑡) = 𝑐6𝜉2(𝑡) exp [−𝑐

1∫

𝑑𝑡

𝜉2(𝑡)

] . (24)

Therefore, 𝐴(𝑡) becomes

𝐴 (𝑡) = 𝑐6(𝑐3𝑡 + 𝑐4)1−(𝑐1/𝑐3)

, if 𝑐3

̸=0,

𝐴 (𝑡) = 𝑐7exp [−𝑐1

𝑐4

𝑡] , if 𝑐3= 0,

(25)

where 𝑐6and 𝑐7= 𝑐4𝑐6are arbitrary constants.

4. Invariant Solutions

The characteristic equations corresponding to the symme-tries (23) are given by

𝑑𝑥

𝑐1𝑥 + 𝑐2

=

𝑑𝑡

𝑐3𝑡 + 𝑐4

=

𝑑𝑓

0

=

𝑑𝑔

𝑐5𝑔

. (26)

By solving the above system, we have the following four cases.

Case 1. When 𝑐1

̸=0 and 𝑐3

̸=0, the similarity variable andsimilarity functions can be written as follows:

𝜉 =

𝑥 + 𝑎

(𝑡 + 𝑏)𝑐, 𝑓 (𝑥, 𝑡) = Ψ (𝜉) ,

𝑔 (𝑥, 𝑡) = (𝑥 + 𝑎)𝑑Φ (𝜉) ,

(27)

where 𝑎 = 𝑐2/𝑐1, 𝑏 = 𝑐

4/𝑐3, 𝑐 = 𝑐

1/𝑐3, and 𝑑 = 𝑐

5/𝑐1are

arbitrary constants. In this case, 𝐴(𝑡) = 𝑞(𝑡 + 𝑏)1−𝑐, where𝑞 = 𝑐6𝑐1−𝑐

3. Substituting the transformations (27) in the field

(15) leads to the following system of ODEs:

𝜉Ψ+ Ψ

Ψ

+

𝜉Φ+ (1 + 𝑑)Φ

Φ

= 0, (28)

(𝑐2𝑞2𝜉2− 1) 𝜉Φ

Ψ

ΦΨ

+

2𝑑Φ+ 𝜉Φ

Φ

+

𝑐2𝑞2𝜉3Ψ− [𝑑 + 2𝑐 (1 − 2𝑐) 𝑞

2𝜉2]Ψ

Ψ

=

𝑑 (1 − 𝑑)

𝜉

.

(29)

If one solves the system of second order NLPDEs (28)-(29),one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30).

Case 2. When 𝑐1

̸=0 and 𝑐3

= 0, the similarity variable andsimilarity functions can be written as follows:

𝜉 = (𝑥 + 𝑎) exp [𝑏𝑡] , 𝑓 (𝑥, 𝑡) = Ψ (𝜉) ,

𝑔 (𝑥, 𝑡) = (𝑥 + 𝑎)𝑐Φ (𝜉) ,

(30)

where 𝑎 = 𝑐2/𝑐1, 𝑏 = −𝑐

1/𝑐4, and 𝑐 = 𝑐

5/𝑐1are arbitrary

constants. In this case, 𝐴(𝑡) = 𝑑 exp[𝑏𝑡], where 𝑑 = 𝑐7.

Substituting transformations (30) in the field equations (15)leads to the following system of ODEs:

𝜉Ψ+ Ψ

Ψ

+

𝜉Φ+ (1 + 𝑐)Φ

Φ

= 0, (31)

(𝑏2𝑑2𝜉2− 1) 𝜉Φ

Ψ

ΦΨ

+

2𝑐Φ+ 𝜉Φ

Φ

+

𝑏2𝑑2𝜉3Ψ+ (4𝑏2𝑑2𝜉2− 𝑐)Ψ

Ψ

=

𝑐 (1 − 𝑐)

𝜉

.

(32)

If one solves the system of second order NLPDEs (31)-(32),one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30).

Case 3. When 𝑐1

= 0 and 𝑐3

̸=0, the similarity variable andsimilarity functions can be written as follows:

𝜉 =

exp [𝑎𝑥]𝑡 + 𝑏

, 𝑓 (𝑥, 𝑡) = Ψ (𝜉) ,

𝑔 (𝑥, 𝑡) = Φ (𝜉) exp [𝑎𝑥] ,(33)

where 𝑎 = 𝑐3/𝑐2and 𝑏 = 𝑐

4/𝑎3are arbitrary constants. In this

case we have 𝐴(𝑡) = 𝑐(𝑡 + 𝑏), where 𝑐 = 𝑐6. Substituting the

transformations (33) in the field equations (15) leads to thefollowing system of ODEs:

𝜉Ψ+ Ψ

Ψ

+

𝜉Φ+ 2Φ

Φ

= 0,

𝜉 [

(𝑐2− 𝑎2) 𝜉ΦΨ

ΦΨ

+

𝑎2(3Φ+ 𝜉Φ)

Φ

+

𝑐2𝜉Ψ− (𝑐2+ 𝑎2)Ψ

Ψ

] + 𝑎2= 0.

(34)

Without loss of generality, we can take the following usefultransformation:

𝜉 = exp [𝜃] , 𝑑Ψ𝑑𝜉

= exp [−𝜃] 𝑑Ψ𝑑𝜃

,

𝑑2Ψ

𝑑𝜉2

= exp [−2𝜃] (𝑑2Ψ

𝑑𝜃2

−

𝑑Ψ

𝑑𝜃

) .

(35)

Then the system of ODEs (34) transforms to

Ψ̈

Ψ

+

Φ̈ + Φ̇

Φ

= 0, (36)

(𝑐2− 𝑎2) Φ̇Ψ̇

ΦΨ

+

𝑎2(2Φ̇ + Φ̈)

Φ

+

𝑐2Ψ̈ − (2𝑐

2+ 𝑎2) Ψ̇

Ψ

+ 𝑎2= 0.

(37)

Equation (36) can be written in the following form:

Ψ̈ = −

Ψ

Φ

(Φ̈ + Φ̇) . (38)


From the above equation, if we substitute Ψ̈ in (37), we canobtain the following form:

(𝑐2− 𝑎2) [

Φ̇Ψ̇

ΦΨ

−

Φ̈

Φ

] + (2𝑎2− 𝑐2)(

Φ̇

Φ

)

− (2𝑐2+ 𝑎2)(

Ψ̇

Ψ

) + 𝑎2= 0.

(39)

Equation (39) is a nonlinear ODE which is very difficult tosolve. However, it is worth noting that this equation is easyto solve when 𝑐 = 𝑎. In this case, we can integrate (39) andobtain the following:

Φ (𝜃) = 𝑎1Ψ3(𝜃) exp [−𝜃] , (40)

where 𝑎1is an arbitrary constant of integration. Substituting

(40) in (36), we have the following ODE of the function Ψonly:

Ψ(4Ψ̈ − 3Ψ̇) + 6Ψ̇2= 0. (41)

The general solution of the above equation is

Ψ (𝜃) = 𝑎3(𝑎2+ exp [3𝜃

4

])

2/5

, (42)

where 𝑎2and 𝑎3are arbitrary constants of integration. Now,

by using (40) and the inverse of transformations (35) and (33),we can find the solution as follows:

𝐴 (𝑡) = 𝑎 (𝑡 + 𝑏) ,

𝑓 (𝑥, 𝑡) = 𝑎3(𝑎2+ [


]

3/4

)

2/5

,

𝑔 (𝑥, 𝑡) = 𝑎1𝑎3(𝑡 + 𝑏) (𝑎

2+ [


]

3/4

)

6/5

.

(43)

It is observed from (43) and (3) that the line element (2) canbe written in the following form:

𝑑𝑠2= 𝑑𝑡2− 𝑎2(𝑡 + 𝑏)

2𝑑𝑥2− 𝑑2(𝑡 + 𝑏)

2

× (𝑎2+ [


]

3/4

)

4/5

𝑑𝑦2

− 𝑞2(𝑡 + 𝑏)

4(𝑎2+ [


]

3/4

)

12/5

𝑑𝑧2,

(44)

where 𝑎, 𝑏, 𝑑 = 𝑎𝑎3, 𝑞 = 𝑎𝑎

1𝑎3, and 𝑎

2are arbitrary constants.

Case 4. When 𝑐1

= 𝑐3

= 0, the similarity variable andsimilarity functions can be written as follows:

𝜉 = 𝑎𝑥 + 𝑏𝑡, 𝑓 (𝑥, 𝑡) = Ψ (𝜉) ,

𝑔 (𝑥, 𝑡) = Φ (𝜉) exp [𝑐𝑥] ,(45)

where 𝑎 = 𝑐4, 𝑏 = −𝑐

2, and 𝑐 = 𝑐

5/𝑐2are arbitrary constants. In

this case we have𝐴(𝑡) = 𝑐7. Substituting transformations (45)

in field equations (15) leads to the following system of ODEs:

𝑎Ψ

Ψ

+

𝑎Φ+ 𝑐Φ

Φ

= 0, (46)

(𝑏2𝑟2− 𝑎2)ΦΨ

ΦΨ

+

𝑎 (2𝑐Φ+ 𝑎Φ)

Φ

+

𝑏2𝑟2Ψ− 𝑎𝑐Ψ

Ψ

+ 𝑐2= 0.

(47)

Equation (46) can be written in the following form:

Ψ

= −

Ψ

𝑎Φ

(𝑎Φ+ 𝑐Φ) . (48)

From the above equation, if we substitute Ψ̈ in (47), we canobtain the following form:

(𝑏2𝑟2− 𝑎2) [

ΦΨ

ΦΨ

−

Φ

Φ

] + (2𝑎2− 𝑏2𝑟2)(

𝑐Φ

𝑎Φ

)

− 𝑎𝑐(

Ψ

Ψ

) + 𝑐2= 0.

(49)

Equation (49) is a nonlinear ODE which is very difficult tosolve. However, it is worth noting that this equation is easyto solve when 𝑎 = 𝑏𝑟. In this case, we can integrate (49) andobtain the following:

Φ (𝜉) = 𝑎1Ψ (𝜉) exp [−𝑐𝜉

𝑏𝑟

] , (50)

where 𝑎1is an arbitrary constant of integration. Substituting

(50) in (46), we have the following ODE of function Ψ only:

2𝑏𝑟Ψ

= 𝑐Ψ. (51)

The general solution of the above equation is

Ψ (𝜉) = 𝑎3+ 𝑎2exp [ 𝑐𝜉

2𝑏𝑟

] , (52)

where 𝑎2and 𝑎3are arbitrary constants of integration. Now,

by using (50) and (33), we can find the solution as follows:

𝐴 (𝑡) = 𝑟,

𝑓 (𝑥, 𝑡) = 𝑎3+ 𝑎2exp [𝑐 (𝑡 + 𝑟𝑥)

2𝑏𝑟

] ,

𝑔 (𝑥, 𝑡) = 𝑎1exp [−𝑐𝑡

𝑟

] (𝑎3+ 𝑎2exp [𝑐 (𝑡 + 𝑟𝑥)

2𝑏𝑟

]) .

(53)

It is observed from (53) and (3) that the line element (2) canbe written in the following form:

𝑑𝑠2= 𝑑𝑡2− 𝑟2𝑑𝑥2− 𝑟2(𝑎3+ 𝑎2exp [𝑐 (𝑡 + 𝑟𝑥)

2𝑏𝑟

])

2

× (𝑑𝑦2+ 𝑎2

1exp [−2𝑐𝑡

𝑟

] 𝑑𝑧2) ,

(54)

where 𝑟, 𝑐, 𝑏, 𝑎1, 𝑎2, and 𝑎

3are arbitrary constants.


5. Physical Properties of the Model

The field equations (15)–(16) constitute a system of fivehighly nonlinear differential equations with seven unknownsvariables, 𝐴, 𝑓, 𝑔, 𝑝, 𝜌, 𝜇, and 𝛽. The symmetries giveone condition (24) for function 𝐴. Therefore, one physicallyreasonable condition amongst these parameters is required toobtain explicit solutions of the field equations. Let us assumethat the density 𝜌 and the pressure 𝑝 are related by barotropicequation of state:

𝑝 = 𝜆𝜌, 0 ≤ 𝜆 ≤ 1. (55)

5.1. ForModel (44). Using (43) in the Einstein field equations(16), with taking into account condition (55), the expressionsfor density 𝜌, pressure 𝑝, magnetic permeability 𝜇, anddisplacement field 𝛽 are given by

𝜌 (𝑥, 𝑡) =

9

5𝜒 (1 − 𝜆)

×[

[

5𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4

(𝑡 + 𝑏)2(𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4)

]

]

,

𝑝 (𝑥, 𝑡) =

9𝜆

5𝜒 (1 − 𝜆)

×[

[

5𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4

(𝑡 + 𝑏)2(𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4)

]

]

,

𝜇 (𝑥, 𝑡) =

𝜒𝐹2

12(𝑥)

3𝑎2𝑎2𝑑2(𝑡 + 𝑏)

2(𝑎2+ [


]

3/4

)

1/5

,

𝛽2(𝑥, 𝑡) =

2

15

[

[

(5𝑎2(5 + 13𝜆)

+ 2 (2 + 7𝜆) [


]

3/4

)

× ((𝜆 − 1) (𝑡 + 𝑏)2

×(𝑎2+ [


]

3/4

))

−1

] .

(56)

For the line element (44), using (4), (8), and (9), we have thefollowing physical properties. The volume element is

𝑉 = 𝑎𝑑𝑞(𝑡 + 𝑏)4(𝑎2+ [


]

3/4

)

8/5

. (57)

The expansion scalar, which determines the volume behaviorof the fluid, is given by

Θ =

2

5

[

[

10𝑎2+ 7[exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4

(𝑡 + 𝑏) (𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4)

]

]

. (58)

The nonvanishing components of the shear tensor, 𝜎𝑗𝑖, are

𝜎1

1=

[exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4 − 5𝑎215 (𝑡 + 𝑏) (𝑎

2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4)

,

𝜎2

2= −

1

30

[

[

10𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4

(𝑡 + 𝑏) (𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4)

]

]

,

𝜎3

3=

4𝑎2+ 7[exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4

6 (𝑡 + 𝑏) (𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4)

,

𝜎4

4= −

4

15

[

[

10𝑎2+ 7[exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4

(𝑡 + 𝑏) (𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4)

]

]

.

(59)

Hence the shear scalar 𝜎 is given by

𝜎2= (3500𝑎

2

2+ 4630𝑎

2[exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4

+1607[exp [𝑎𝑥] / (𝑡 + 𝑏)]3/2)

× (900(𝑡 + 𝑏)2(𝑎2+ [exp [𝑎𝑥] / (𝑡 + 𝑏)]3/4)

2

)

−1

.

(60)

The model does not admit acceleration and rotation, sincė𝑢𝑖= 0 and 𝜔

𝑖𝑗= 0. We can see that

𝜎4

4

Θ

= −

2

3

(61)

which is a constant of proportional. We found also that

lim𝑡→∞

𝜎2

Θ

=

5√35

2

̸=0 (62)

and this means that there is no possibility that the universemay got isotropized in some later time; that is, it remainsanisotropic for all times.

5.2. ForModel (54). Using (53) in the Einstein field equations(16), with taking into account condition (55), the expressions


for density 𝜌, pressure 𝑝, magnetic permeability 𝜇, anddisplacement field 𝛽 are given by

𝜌 (𝑥, 𝑡) =

𝑐2

𝜒𝑟2(𝜆 − 1)

(

𝑐2exp [𝑐 (𝑡 + 𝑟𝑥) /2𝑟] − 𝑐3

𝑐2exp [𝑐 (𝑡 + 𝑟𝑥) /2𝑟] + 𝑐3

) ,

𝑝 (𝑥, 𝑡) =

𝜆𝑐2

𝜒𝑟2(𝜆 − 1)

(

𝑐2exp [𝑐 (𝑡 + 𝑟𝑥) /2𝑟] − 𝑐3

𝑐2exp [𝑐 (𝑡 + 𝑟𝑥) /2𝑟] + 𝑐3

) ,

𝜇 (𝑥, 𝑡) =

𝜒𝐹2

12(𝑥)

𝑐3𝑐2𝑟2

(𝑐2exp [𝑐 (𝑡 + 𝑟𝑥)

2𝑟

] + 𝑐3)

−1

,

𝛽2(𝑥, 𝑡) =

2𝑐2

3𝑟2(𝜆 − 1)

× (

𝑐3(1 + 𝜆) − 2𝜆𝑐

2exp [𝑐 (𝑡 + 𝑟𝑥) /2𝑟]

𝑐3+ 𝑐2exp [𝑐 (𝑡 + 𝑟𝑥) /2𝑟]

) .

(63)

For the line element (54), using (4), (8), and (9), we have thefollowing physical properties. The volume element is

𝑉 = 𝑐1𝑟3 exp [−𝑐𝑡

𝑟

] (𝑐3+ 𝑐2exp [𝑐 (𝑡 + 𝑟𝑥)

2𝑟

])

2

. (64)

The expansion scalar, which determines the volume behaviorof the fluid, is given by

Θ = −

𝑐𝑐3

𝑟

(𝑐3+ 𝑐2exp [𝑐 (𝑡 + 𝑟𝑥)

2𝑟

])

−1

. (65)

The nonvanishing components of the shear tensor, 𝜎𝑗𝑖, satisfy

𝜎1

1

Θ

= −

1

3

,

𝜎2

2

Θ

= −(

1

3

+

𝑐2

2𝑐3

exp [𝑐 (𝑡 + 𝑟𝑥)2𝑟

]) ,

𝜎3

3

Θ

=

2

3

+

𝑐2

2𝑐3

exp [𝑐 (𝑡 + 𝑟𝑥)2𝑟

] ,

𝜎4

4

Θ

= −

2

3

.

(66)

Hence the shear scalar 𝜎 is given by

𝜎2=

𝑐2

4𝑟2+

11𝑐2

3𝑐2

36𝑟2

(𝑐3+ 𝑐2exp [𝑐 (𝑡 + 𝑟𝑥)

2𝑟

])

−2

. (67)

The model does not admit acceleration and rotation, sincė𝑢𝑖= 0 and 𝜔

𝑖𝑗= 0.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR), King Abdulaziz University, Jeddah, under Grant no.130-016-D1434. The author, therefore, acknowledges withthanks DSR technical and financial support.

References

[1] A. Krasinski, Inhomogeneous Cosmological Models, CambridgeUniversity Press, Cambridge, UK, 1997.

[2] G. Lyra, “Über eine Modifikation der Riemannschen Geome-trie,”Mathematische Zeitschrift, vol. 54, pp. 52–64, 1951.

[3] D. K. Sen, “A static cosmological model,” Zeitschrift für PhysikC: Particles and Fields, vol. 149, pp. 311–323, 1957.

[4] D. K. Sen, “On geodesics of a modified Riemannian manifold,”Canadian Mathematical Society, vol. 3, pp. 255–261, 1960.

[5] D. K. Sen and K. A. Dunn, “A scalar-tensor theory of gravitationin a modified Riemannian manifold,” Journal of MathematicalPhysics, vol. 12, pp. 578–586, 1971.

[6] W. D. Halford, “Cosmological theory based on Lyra’s geometry,”Australian Journal of Physics, vol. 23, no. 5, pp. 863–870, 1970.

[7] S. S. De and F. Rahaman, Finsler Geometry of Hadrons and LyraGeometry: Cosmological Aspects, Lambert Academic Publish-ing, Saarbrücken, Germany, 2012.

[8] A. Pradhan, I. Aotemshi, and G. P. Singh, “Plane symmetricdomain wall in Lyra geometry,” Astrophysics and Space Science,vol. 288, no. 3, pp. 315–325, 2003.

[9] A. Pradhan and A. K. Vishwakarma, “A new class of LRSBianchi type-I cosmological models in Lyra geometry,” Journalof Geometry and Physics, vol. 49, no. 3-4, pp. 332–342, 2004.

[10] A. Pradhan and S. S. Kumhar, “Plane symmetric inhomoge-neous perfect fluid universe with electromagnetic field in Lyrageometry,” Astrophysics and Space Science, vol. 321, no. 2, pp.137–146, 2009.

[11] A. Pradhan and P. Ram, “A plane-symmetric magnetizedinhomogeneous cosmological models of perfect fluid distribu-tion with variable magnetic permeability in Lyra geometry,”International Journal of Theoretical Physics, vol. 48, no. 11, pp.3188–3201, 2009.

[12] A. Pradhan, H. Amirhashchi, and H. Zainuddin, “A new classof inhomogeneous cosmological models with electromagneticfield in normal gauge for Lyra's manifold,” International Journalof Theoretical Physics, vol. 50, no. 1, pp. 56–69, 2011.

[13] P. Anirudh, A. Singh, and R. S. Singh, “A plane-symmetric mag-netized inhomogeneous cosmological model of perfect fluiddistribution with variable magnetic permeability,” RomanianReports of Physics, vol. 56, no. 1-2, pp. 297–307, 2011.

[14] A. Pradhan and A. K. Singh, “Anisotropic Bianchi type-I stringcosmological models in normal Gauge for Lyra's manifoldwith constant deceleration parameter,” International Journal ofTheoretical Physics, vol. 50, no. 3, pp. 916–933, 2011.

[15] R. Casana, C. A.M. deMelo, andB.M. Pimentel, “Spinorial fieldand Lyra geometry,”Astrophysics and Space Science, vol. 305, no.2, pp. 125–132, 2006.

[16] F. Rahaman, B. C. Bhui, and G. Bag, “Can Lyra geometryexplain the singularity free as well as accelerating Universe?”Astrophysics and Space Science, vol. 295, no. 4, pp. 507–513, 2005.

[17] R. Bali and N. K. Chandnani, “Bianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometry,” Journal of


Mathematical Physics, vol. 49, no. 3, Article ID 032502, 8 pages,2008.

[18] R. Bali andN. K. Chandnani, “Bianchi typeV barotropic perfectfluid cosmological model in Lyra geometry,” InternationalJournal ofTheoretical Physics, vol. 48, no. 5, pp. 1523–1533, 2009.

[19] S. Kumar andC. P. Singh, “An exact Bianchi type-I cosmologicalmodel in Lyra’s manifold,” International Journal of ModernPhysics A, vol. 23, no. 6, pp. 813–822, 2008.

[20] A. K. Yadav, A. Pradhan, and A. Singh, “LRS Bianchi type-II massive string cosmological models in general relativity,”RomanianReports of Physics, vol. 56, no. 7-8, pp. 1019–1034, 2011.

[21] V. U. M. Rao, T. Vinutha, and M. V. Santhi, “Bianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifold,” Astrophysics and Space Science, vol. 314, no. 1–3, pp.213–216, 2008.

[22] R. Zia and R. P. Singh, “Bulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometry,”Romanian Journal of Physics, vol. 57, no. 3-4, pp. 761–778, 2012.

[23] A. T. Ali and F. Rahaman, “New class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometry,” International Journalof Theoretical Physics, vol. 52, no. 11, pp. 4055–4067, 2013.

[24] A. T. Ali, “New generalized Jacobi elliptic function rationalexpansion method,” Journal of Computational and AppliedMathematics, vol. 235, no. 14, pp. 4117–4127, 2011.

[25] A. T. Ali and E. R. Hassan, “General Expa-function methodfor nonlinear evolution equations,” Applied Mathematics andComputation, vol. 217, no. 2, pp. 451–459, 2010.

[26] M. F. El-Sabbagh and A. T. Ali, “New exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equation,” International Journalof Nonlinear Sciences andNumerical Simulation, vol. 6, no. 2, pp.151–162, 2005.

[27] M. F. El-Sabbagh and A. T. Ali, “New generalized Jacobi ellipticfunction expansion method,” Communications in NonlinearScience and Numerical Simulation, vol. 13, no. 9, pp. 1758–1766,2008.

[28] L. V. Ovsiannikov, Group Analysis of Differential Equations,translated by Y. Chapovsky, edited by W. F. Ames, AcademicPress, New York, NY, USA, 1982.

[29] G. W. Bluman and S. Kumei, Symmetries and DifferentialEquations in Applied Sciences, Springer, New York, NY, USA,1989.

[30] N. H. Ibragimov, Transformation Groups Applied to Mathemat-ical Physics, D. Reidel, Dordrecht, The Netherlands, 1985.

[31] P. J. Olver, Applications of Lie Groups to Differential Equations,vol. 107 of Graduate Texts in Mathematics, Springer, New York,NY, USA, 2nd edition, 1993.

[32] A. T. Ali, A. K. Yadav, and S. R. Mahmoud, “Some planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitation,”Astrophysics and Space Science, vol.349, no. 1, pp. 539–547, 2014.

[33] A. T. Ali, “New exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fields,”Physica Scripta, vol. 79, no.3, Article ID 035006, 2009.

[34] S. K. Attallah, M. F. El-Sabbagh, and A. T. Ali, “Isovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fields,” Communications in Nonlinear Science andNumerical Simulation, vol. 12, no. 7, pp. 1153–1161, 2007.

[35] K. S.Mekheimer, S. Z. Husseny, A. T. Ali, andA. E. Abo-Elkhair,“Lie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluid,” Physica Scripta, vol. 83, no. 1, ArticleID 015017, 2011.

[36] A. K. Raychaudhuri, Theoritical Cosmology, Oxford SciencePublications, Oxford, UK, 1979.

[37] A. Lichnerowicz, Relativistic Hydrodynamics and Magneto-Hydro-Dynamics, W. A. Benjamin, New York, NY, USA, 1967.

[38] A. Pradhan and P. Mathur, “Inhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometry,” Fizika B,vol. 18, no. 4, pp. 243–264, 2009.

Research ArticleDerivation of Conservation Laws for the MagmaEquation Using the Multiplier Method: Power Law andExponential Law for Permeability and Viscosity

N. Mindu and D. P. Mason

Programme for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics,University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

Correspondence should be addressed to D. P. Mason; [email protected]

Received 24 February 2014; Accepted 1 April 2014; Published 4 May 2014

Academic Editor: Chaudry M. Khalique

Copyright © 2014 N. Mindu and D. P. Mason. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The derivation of conservation laws for the magma equation using the multiplier method for both the power law and exponentiallaw relating the permeability and matrix viscosity to the voidage is considered. It is found that all known conserved vectors for themagma equation and the new conserved vectors for the exponential laws can be derived using multipliers which depend on thevoidage and spatial derivatives of the voidage. It is also found that the conserved vectors are associated with the Lie point symmetryof themagma equationwhich generates travellingwave solutionswhichmay explain by the double reduction theorem for associatedLie point symmetries why many of the known analytical solutions are travelling waves.

1. Introduction

The one-dimensional migration of melt upwards through themantle of the Earth is governed by the third order nonlinearpartial differential equation

𝜕𝜙

𝜕𝑡

+

𝜕

𝜕𝑧

[𝐾 (𝜙) (1 −

𝜕

𝜕𝑧

(𝐺 (𝜙)

𝜕𝜙

𝜕𝑡

))] = 0, (1)

where 𝜙(𝑡, 𝑧) is the voidage or volume fraction of melt, 𝑡 istime, 𝑧 is the vertical spatial coordinate,𝐾 is the permeabilityof the medium, and 𝐺 is the viscosity of the matrix phase.The variables 𝜙, 𝑡, and 𝑧 and the physical quantities 𝐾(𝜙)and 𝐺(𝜙) in (1) are dimensionless. The voidage 𝜙(𝑡, 𝑧) isscaled by the background voidage 𝜙

0. The background state

is therefore defined by 𝜙 = 1. The characteristic length in the𝑧-direction, which is vertically upwards, is the compactionlength 𝛿

𝑐defined by

𝛿𝑐= [

𝐾 (𝜙0) 𝐺 (𝜙

0)

𝜇

]

1/2

, (2)

where 𝜇 is the coefficient of shear viscosity of the melt. Thecharacteristic time is 𝑡

0defined by

𝑡0=

𝜙0

𝑔Δ𝜌

[

𝜇𝐺 (𝜙0)

𝐾 (𝜙0)

]

1/2

, (3)

where 𝑔 is the acceleration due to gravity and Δ𝜌 is thedifference between the density of the solid matrix and thedensity of the melt. The permeability is scaled by 𝐾(𝜙

0) and

therefore

𝐾 (1) = 1. (4)

When the voidage is zero the permeability must also be zeroand therefore

𝐾 (0) = 0. (5)

The viscosity 𝐺(𝜙) is scaled by 𝐺(𝜙0) so that

𝐺 (1) = 1 (6)

and 𝐺(0) will be infinite because the matrix viscosity isinfinite when the voidage vanishes. In the derivation of (1)it is assumed that the background voidage satisfies 𝜙

0≪ 1.


http://dx.doi.org/10.1155/2014/585167


The partially molten medium consists of a solid matrixand a fluid melt which are modelled as two immisciblefully connected fluids of constant but different densities. Thedensity of the melt is less than the density of the solid matrixand the melt migrates through the compacting medium bythe buoyancy force due to the difference in density betweenthe melt and the solid matrix. Changes of phase are notincluded in the model. It is assumed that the melting hasoccurred and only migration of the melt under gravity isdescribed by (1) [1].

In the model proposed by Scott and Stevenson [2],consider

𝐾(𝜙) = 𝜙𝑛, 𝐺 (𝜙) = 𝜙

−𝑚, (7)

where 𝑛 ≥ 0 and 𝑚 ≥ 0. Harris and Clarkson [3] haveinvestigated this model using Painleve analysis. Mindu andMason [4] showed that the magma equation also admits Liepoint symmetries other than translations in time and space ifthe permeability is in the form of an exponential law:

𝐾(𝜙) = exp [𝑛 (𝜙 − 1)] . (8)

Conservation laws for (1) when the permeability and matrixviscosity satisfy the power laws (7) have been obtained usingthe direct method by Barcilon and Richter [5] and Harris [6]and using Lie point symmetry generators by Maluleke andMason [7].

In this paper we will derive the conservation laws for thepartial differential equation (1) using the multiplier method.We will consider power laws given by (7) and also theexponential laws𝐾(𝜙) = exp [𝑛 (𝜙 − 1)] , 𝐺 (𝜙) = exp [−𝑚 (𝜙 − 1)] ,

(9)

where 𝑛 ≥ 0 and 𝑚 ≥ 0, relating the permeability andmatrix viscosity to the voidage. The permeability increasesas the voidage increases while the viscosity of the matrixdecreases as the voidage increases. The exponential laws arenot suitable models when the voidage 𝜙 is small because𝐾(0) = exp(−𝑛) ̸= 0 and 𝐺(0) = exp(𝑚) ̸=∞. They aresuitable for describing rarefaction for which 𝜙 > 1.

An outline of the paper is as follows. In Section 2 wepresent the formulae and theory that we will use in the paper.In Section 3 conservation laws for the magma equation, withpower laws relating the permeability and viscosity to thevoidage, are derived using the multiplier method. Further inSection 4 conservation laws for the magma equation, withexponential laws relating the permeability and viscosity to thevoidage, are derived using the multiplier method. Finally theconclusions are summarized in Section 5.

2. Formulae and Theory

Consider an 𝑠th order partial differential equation𝐹 (𝑥, 𝜙, 𝜙

(1), . . . , 𝜙

(𝑠)) = 0, (10)

in the variables 𝑥 = (𝑥1, 𝑥2, . . . , 𝑥𝑝), where 𝜙(𝑝)

denotes thecollection of 𝑝th-order partial derivatives of 𝜙. The equation

𝐷𝑖𝑇𝑖= 0, (11)

evaluated on the surface given by (10), where 𝑖 runs from 1 to𝑟 and𝐷

𝑖is the total derivative defined by

𝐷𝑖=

𝜕

𝜕𝑥𝑖

+ 𝜙𝑖

𝜕

𝜕𝜙

+ 𝜙𝑘𝑖

𝜕

𝜕𝜙𝑘

+ ⋅ ⋅ ⋅ , (12)

is called a conservation law for the differential equation (10).The vector 𝑇 = (𝑇1, . . . , 𝑇𝑟) is a conserved vector for the par-tial differential equation and 𝑇1, . . . , 𝑇𝑟 are its components.Thus, a conserved vector gives rise to a conservation law. ALie point symmetry generator

𝑋 = 𝜉𝑖(𝑥, 𝑢)

𝜕

𝜕𝑥𝑖+ 𝜂 (𝑥, 𝜙)

𝜕

𝜕𝜙

, (13)

where 𝑖 runs from 1 to 𝑟, is said to be associated with theconserved vector 𝑇 = (𝑇1, . . . , 𝑇𝑟) for the partial differentialequation (10) if [8, 9]

𝑋(𝑇𝑖) + 𝑇𝑖𝐷𝑘(𝜉𝑘) − 𝑇𝑘𝐷𝑘(𝜉𝑖) = 0, 𝑖 = 1, 2, . . . 𝑟. (14)

The association of a Lie point symmetrywith a conserved vec-tor can be used to integrate the partial differential equationtwice by the double reduction theorem of Sjöberg [10].

Conserved vectors for a partial differential equation canbe generated from known conserved vectors and Lie pointsymmetries of the partial differential equation. For

𝑇𝑖

∗= 𝑋(𝑇

𝑖) + 𝑇𝑖𝐷𝑘(𝜉𝑘) − 𝑇𝑘𝐷𝑘(𝜉𝑖) , 𝑖 = 1, 2, . . . 𝑟,

(15)

where 𝑘 runs from 1 to 𝑟, is a conserved vector for the partialdifferential equation although it may be a linear combinationof known conserved vectors [8, 9].

We now present the multiplier method for the derivationof conservation laws for partial differential equations.Wewilloutline its application to the partial differential equation (1) intwo independent variables.

(1) Multiply the partial differential equation (1) by themultiplier, Λ, to obtain the conservation law

Λ𝐹 = 𝐷1𝑇1+ 𝐷2𝑇2, (16)

where 𝐹 = 0 is the partial differential equation (1), 𝑥1 = 𝑡 and𝑥2= 𝑧, and

𝐷1= 𝐷𝑡=

𝜕

𝜕𝑡

+ 𝜙𝑡

𝜕

𝜕𝜙

+ 𝜙𝑡𝑡

𝜕

𝜕𝑡

+ 𝜙𝑧𝑡

𝜕

𝜕𝑧

+ ⋅ ⋅ ⋅ ,

𝐷2= 𝐷𝑧=

𝜕

𝜕𝑧

+ 𝜙𝑧

𝜕

𝜕𝜙

+ 𝜙𝑡𝑧

𝜕

𝜕𝑡

+ 𝜙𝑧𝑧

𝜕

𝜕𝑧

+ ⋅ ⋅ ⋅ .

(17)

The multiplier depends on 𝑡, 𝑧, 𝜙, and the partial derivativesof𝜙.Themore derivatives included in themultiplier thewiderthe range of conserved vectors that can be derived.

(2) The determining equation for the multiplier isobtained by operating on (16) by the Euler operator 𝐸

𝜙

defined by [11]

𝐸𝜙=

𝜕

𝜕𝜙

− 𝐷𝑡

𝜕

𝜕𝜙𝑡

− 𝐷𝑧

𝜕

𝜕𝜙𝑧

+ 𝐷2

𝑡

𝜕

𝜕𝜙𝑡𝑡

+ 𝐷𝑡𝐷𝑧

𝜕

𝜕𝜙𝑡𝑧

+ 𝐷2

𝑧

𝜕

𝜕𝜙𝑧𝑧

− ⋅ ⋅ ⋅ .

(18)


Since the Euler operator annihilates divergence expressionsthis gives [11]

𝐸𝜙 [Λ𝐹] = 0. (19)

(3) The determining equation (19) is separated by equat-ing the coefficients of like powers and products of thederivatives of 𝜙 because 𝜙 is an arbitrary function.

(4) When 𝜙 is a solution of the partial differential equa-tion, 𝐹 = 0, (16) becomes a conservation law. The condition𝐹 = 0 is imposed on (16). The product of the multiplier andthe partial differential equation is then written in conservedform by elementary manipulations.This yields the conservedvectors by setting all the constants equal to zero except one inturn.

3. Conservation Laws for the MagmaEquation with Power Law Permeability andViscosity by the Multiplier Method

When the permeability and viscosity are related to thevoidage by the power laws (7) the magma equation becomes

𝜕𝜙

𝜕𝑡

+

𝜕

𝜕𝑧

[𝜙𝑛(1 −

𝜕

𝜕𝑧

(𝜙−𝑚 𝜕𝜙

𝜕𝑡

))] = 0. (20)

3.1. Lower Order Conservation Laws. In order to deriveconservation laws for (20) consider first a multiplier of theform

Λ = Λ (𝜙) . (21)

A multiplier for the partial differential equation has theproperty

Λ (𝜙) 𝐹 (𝜙, 𝜙𝑡, 𝜙𝑧, 𝜙𝑡𝑧, 𝜙𝑧𝑧, 𝜙𝑡𝑧𝑧) = 𝐷

1𝑇1+ 𝐷2𝑇2, (22)

where𝐹 (𝜙, 𝜙

𝑡, 𝜙𝑧, 𝜙𝑡𝑧, 𝜙𝑧𝑧, 𝜙𝑡𝑧𝑧)

= 𝜙𝑡+ 𝑛𝜙𝑛−1𝜙𝑧+ 𝑚 (𝑛 − 𝑚 − 1) 𝜙

𝑛−𝑚−2𝜙2

𝑧𝜙𝑡

+ 𝑚𝜙𝑛−𝑚−1

𝜙𝑧𝑧𝜙𝑡+ (2𝑚 − 𝑛) 𝜙

𝑛−𝑚−1𝜙𝑧𝜙𝑡𝑧

− 𝜙𝑛−𝑚

𝜙𝑡𝑧𝑧.

(23)

The determining equation for the multiplier is𝐸𝜙[Λ (𝜙) 𝐹 (𝜙, 𝜙

𝑡, 𝜙𝑧, 𝜙𝑡𝑧, 𝜙𝑧𝑧, 𝜙𝑡𝑧𝑧)] = 0, (24)

where 𝐸𝜙is defined by (18). Separating (24) with respect to

products and powers of the partial derivatives of 𝜙 we obtainthe following system of equations:

𝜙𝑧𝜙𝑡𝑧: 𝜙

𝑑2Λ

𝑑𝜙2+ (𝑚 + 𝑛)

𝑑Λ

𝑑𝜙

= 0, (25)

𝜙𝑡𝜙𝑧𝑧: 𝜙

𝑑2Λ

𝑑𝜙2+ (𝑚 + 𝑛)

𝑑Λ

𝑑𝜙

= 0, (26)

𝜙𝑡𝜙2

𝑧: 𝜙2 𝑑3Λ

𝑑𝜙3+ 2𝑛𝜙

𝑑2Λ

𝑑𝜙2− (𝑚 + 𝑛) (𝑚 − 𝑛 + 1)

𝑑Λ

𝑑𝜙

= 0.

(27)

n

m

n=m−1

( 32,1

2)

m = 1

(1, 1)

n+m−2=0

n+m−1=0

2.01.51.00.5

2.0

1.5

1.0

0.5

0.0

(1, 0)

Figure 1: The (𝑚, 𝑛)-plane. The conservation laws are constrainedto the region 𝑚 ≥ 0, 𝑛 ≥ 0. The special cases are the straight lines𝑛 + 𝑚 − 1 = 0, 𝑛 + 𝑚 − 2 = 0,𝑚 = 1, and 𝑛 = 𝑚 − 1.

Equation (26) is the same as (25). It is readily verified thatevery solution of (25) is a solution of (27). We therefore needto consider only (25). The general solution of (25) is

Λ (𝜙) = 𝑐2𝜙1−𝑚−𝑛

+ 𝑐1, if 𝑛 + 𝑚 − 1 ̸= 0, (28)

Λ (𝜙) = 𝑐2ln𝜙 + 𝑐

1, if 𝑛 + 𝑚 − 1 = 0. (29)

There are several cases to consider depending on thevalues of 𝑚 and 𝑛. The special cases are illustrated as linesand points in the (𝑚, 𝑛) plane in Figure 1.

(i) 𝑛 + 𝑚 − 1 ̸= 0, 𝑛 + 𝑚 − 2 ̸= 0,𝑚 ̸=1. From (22) and (28),

(𝑐1+ 𝑐2𝜙1−𝑚−𝑛

) (𝜙𝑡+ 𝑛𝜙𝑛−1𝜙𝑧+ 𝑚 (𝑛 − 𝑚 − 1)

× 𝜙𝑛−𝑚−2

𝜙2

𝑧𝜙𝑡+ 𝑚𝜙𝑛−𝑚−1

𝜙𝑧𝑧𝜙𝑡

+ (2𝑚 − 𝑛) 𝜙𝑛−𝑚−1

𝜙𝑧𝜙𝑡𝑧− 𝜙𝑛−𝑚

𝜙𝑡𝑧𝑧)

= 𝐷1[𝑐1𝜙 + 𝑐2(

1

2 − 𝑚 − 𝑛

(𝜙2−𝑚−𝑛

− 1)

+

(1 − 𝑚 − 𝑛)

2

𝜙−2𝑚

𝜙2

𝑧)]

+ 𝐷2[𝑐1(𝜙𝑛(1 + 𝑚𝜙

−𝑚−1𝜙𝑡𝜙𝑧− 𝜙−𝑚𝜙𝑡𝑧))

+ 𝑐2(

𝑛

𝑚 − 1

𝜙1−𝑚

− 𝜙1−2𝑚

𝜙𝑡𝑧+ 𝑚𝜙−2𝑚

𝜙𝑡𝜙𝑧)] .

(30)


Equation (30) is satisfied for arbitrary functions𝜙(𝑡, 𝑧).When𝜙(𝑡, 𝑧) is a solution of the partial differential equation (20),then

𝐷1[𝑐1𝜙+𝑐2(

1

2 − 𝑚 − 𝑛


−1)+

(1−𝑚 −𝑛)

2

𝜙−2𝑚

𝜙2

𝑧)]

+ 𝐷2[𝑐1(𝜙𝑛(1 + 𝑚𝜙

−𝑚−1𝜙𝑡𝜙𝑧− 𝜙−𝑚𝜙𝑡𝑧))

+𝑐2(

𝑛

𝑚 − 1

𝜙1−𝑚

− 𝜙1−2𝑚


𝜙𝑡𝜙𝑧)] = 0.

(31)

Hence, any conserved vector of the partial differential equa-tion (20) with 𝑚 and 𝑛 satisfying the conditions of thiscase and with multiplier of the form Λ = Λ(𝜙) is a linearcombination of the two conserved vectors

𝑇1= 𝜙, 𝑇

2= 𝜙𝑛(1 + 𝑚𝜙

−𝑚−1𝜙𝑡𝜙𝑧− 𝜙−𝑚𝜙𝑡𝑧) , (32)

𝑇1=

1

2 − 𝑚 − 𝑛


− 1) +

(1 − 𝑚 − 𝑛)

2

𝜙−2𝑚

𝜙2

𝑧,

𝑇2=

𝑛

1 − 𝑚

𝜙1−𝑚

− 𝜙1−2𝑚


𝜙𝑡𝜙𝑧.

(33)

The conserved vector (32) is the elementary conserved vector.

(ii) 𝑛 + 𝑚 = 1,𝑚 ̸=1. Proceeding as before we obtain

𝑇1= 𝜙, 𝑇

2= 𝜙1−𝑚

(1 + 𝑚𝜙−𝑚−1

𝜙𝑡𝜙𝑧− 𝜙−𝑚𝜙𝑡𝑧) ,

(34)

𝑇1= −

1

2

𝜙−2𝑚

𝜙2

𝑧+ ln𝜙,

𝑇2= 𝜙1−2𝑚 ln𝜙 − 1

1 − 𝑚

𝜙1−𝑚

− (𝜙1−2𝑚 ln𝜙) 𝜙

𝑡𝑧+ (𝑚𝜙

−2𝑚 ln𝜙) 𝜙𝑡𝜙𝑧.

(35)

The conserved vector (34) is the elementary conserved vectorwith 𝑛 = 1 − 𝑚. The multiplier for (35) is, from (29),

Λ (𝜙) = ln𝜙. (36)

(iii) 𝑛 + 𝑚 = 2,𝑚 ̸=1.We find that

𝑇1= 𝜙, 𝑇

2= 𝜙2−𝑚

(1 + 𝑚𝜙−𝑚−1

𝜙𝑡𝜙𝑧− 𝜙−𝑚𝜙𝑡𝑧) ,

(37)

𝑇1= −

1

2

𝜙−2𝑚

𝜙2

𝑧+ ln𝜙,

𝑇2=

2 − 𝑚

1 − 𝑚

𝜙1−𝑚

− 𝜙1−2𝑚


𝜙𝑡𝜙𝑧.

(38)

The conserved vector (37) is the elementary conserved vectorwith 𝑛 = 2 − 𝑚. The multiplier for (38) is, from (28),

Λ (𝜙) =

1

𝜙

. (39)

(iv)𝑚 = 1, 𝑛 = 0.We obtain

𝑇1= 𝜙, 𝑇

2= 1 + 𝜙

−2𝜙𝑡𝜙𝑧− 𝜙−1𝜙𝑡𝑧, (40)

𝑇1= −

1

2

𝜙−2𝜙2

𝑧+ 𝜙 ln𝜙 − 𝜙,

𝑇2= − (𝜙

−1 ln𝜙) 𝜙𝑡𝑧+ (𝜙−2 ln𝜙) 𝜙

𝑡𝜙𝑧.

(41)

The conserved vector (40) is the elementary conserved vectorwith𝑚 = 1, 𝑛 = 0. The multiplier for (41) is given by (29).

(v)𝑚 = 𝑛 = 1.We obtain

𝑇1= 𝜙, 𝑇

2= 𝜙 (1 + 𝜙

−2𝜙𝑡𝜙𝑧− 𝜙−1𝜙𝑡𝑧) , (42)

𝑇1= −

1

2

𝜙−2𝜙2

𝑧+ ln𝜙,

𝑇2= ln𝜙 − 𝜙−1𝜙

𝑡𝑧+ 𝜙−2𝜙𝑡𝜙𝑧.

(43)

The conserved vector (42) is the elementary conserved vectorwith𝑚 = 𝑛 = 1. The multiplier for (43) is (39).

(vi)𝑚 = 1, 𝑛 ̸= 0, 𝑛 ̸= 1.We obtain

𝑇1= 𝜙, 𝑇

2= 𝜙𝑛(1 + 𝜙

−2𝜙𝑡𝜙𝑧− 𝜙−1𝜙𝑡𝑧) , (44)

𝑇1=

1

1 − 𝑛

𝜙1−𝑛

−

𝑛

2

𝜙−2𝜙2

𝑧,


𝑡𝑧+ 𝜙−2𝜙𝑡𝜙𝑧.

(45)

The conserved vector (44) is the elementary conserved vectorwith𝑚 = 1. The multiplier for (45) is

Λ (𝜙) = 𝜙−𝑛. (46)

3.2. The Search for Higher Order Conservation Laws. We nowconsider a multiplier of the form

Λ = Λ (𝜙, 𝜙𝑧) . (47)

As before the determining equation for the multiplier is

𝐸𝜙[Λ (𝜙, 𝜙

𝑧) 𝐹 (𝜙, 𝜙

𝑡, 𝜙𝑧, 𝜙𝑡𝑧, 𝜙𝑧𝑧, 𝜙𝑡𝑧𝑧)] = 0, (48)

where 𝐹 is given by (23). By equating the coefficient of thehighest order derivative term, 𝜙

𝑡𝑧𝑧𝑧, to zero in (48) we have

𝜕Λ

𝜕𝜙𝑧

= 0, (49)

and therefore

Λ (𝜙, 𝜙𝑧) = Λ (𝜙) . (50)

Hence, (47) does not give a newmultiplier or a new conservedvector.

Consider next the multiplier

Λ = Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) . (51)


The determining equation for the multiplier is

𝐸𝜙[Λ (𝜙, 𝜙

𝑧, 𝜙𝑧𝑧) 𝐹 (𝜙, 𝜙

𝑡, 𝜙𝑧, 𝜙𝑡𝑧, 𝜙𝑧𝑧, 𝜙𝑡𝑧𝑧)] = 0, (52)

where 𝐹 is given by (23). By Equating the coefficients of𝜙𝑡𝑧𝜙𝑧𝑧𝑧𝑧

, 𝜙𝑡𝜙𝑧𝑧𝑧𝑧

, and 𝜙2𝑧𝑧𝑧𝑧

to zero in (52), we obtain thefollowing system of equations:

𝜙𝑡𝑧𝜙𝑧𝑧𝑧𝑧

: (2𝑚 − 𝑛) 𝜙𝑧

𝜕2Λ

𝜕𝜙2

𝑧𝑧

+ 𝜙

𝜕2Λ

𝜕𝜙𝑧𝜙𝑧𝑧

= 0, (53)

𝜙𝑡𝜙𝑧𝑧𝑧𝑧

: (𝜙𝑚+2

+ 𝑚 (𝑛 − 𝑚 − 1) 𝜙𝑛+ 𝑚𝜙𝑛+1)

𝜕2Λ

𝜕𝜙2

𝑧𝑧

+ (𝑚 + 𝑛) 𝜙𝑛+1 𝜕Λ

𝜕𝜙𝑧𝑧

+ 𝜙𝑛+2 𝜕

2Λ

𝜕𝜙𝜕𝜙𝑧𝑧

= 0,

(54)

𝜙2

𝑧𝑧𝑧𝑧:

𝜕2Λ

𝜕𝜙2

𝑧𝑧

= 0. (55)

From (55) it follows that

Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) = 𝐴 (𝜙, 𝜙

𝑧) 𝜙𝑧𝑧+ 𝐵 (𝜙, 𝜙

𝑧) . (56)

Substituting (56) into (53) we find that

𝜕𝐴

𝜕𝜙𝑧

= 0, (57)

and therefore

𝐴 (𝜙, 𝜙𝑧) = 𝐴 (𝜙) . (58)

Thus, (51) becomes

Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) = 𝐴 (𝜙) 𝜙

𝑧𝑧+ 𝐵 (𝜙, 𝜙

𝑧) . (59)

Now substitute (59) into (54) which gives

𝜙

𝑑𝐴

𝑑𝜙

+ (𝑚 + 𝑛)𝐴 = 0. (60)

The solution to (60) is

𝐴 = 𝑐1𝜙−(𝑚+𝑛)

, (61)

where 𝑐1is a constant. Equation (59) becomes

Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) = 𝑐1𝜙−(𝑚+𝑛)

𝜙𝑧𝑧+ 𝐵 (𝜙, 𝜙

𝑧) . (62)

Now substitute (62) into (48) and then equate the coeffi-cient of 𝜙

𝑡𝑧𝑧𝑧in (48) to zero. This gives

𝜕𝐵

𝜕𝜙𝑧

= 𝑐1(𝑛 − 2𝑚) 𝜙

−(𝑚+𝑛+1)𝜙𝑧 (63)

and integrating (63) we have

𝐵 (𝜙, 𝜙𝑧) =

1

2

𝑐1(𝑛 − 2𝑚) 𝜙

−(𝑚+𝑛+1)𝜙2

𝑧+ 𝑃 (𝜙) . (64)

Thus, (62) becomes

Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) = 𝑐1𝜙−(𝑚+𝑛)

𝜙𝑧𝑧

+

1

2

𝑐1(𝑛 − 2𝑚) 𝜙

−(𝑚+𝑛+1)𝜙2

𝑧+ 𝑃 (𝜙) .

(65)

Lastly, substitute (65) into (48) and equate the coefficient of𝜙𝑡𝜙2

𝑧𝜙𝑧𝑧to zero, which gives

(𝑚 − 𝑛 − 1) 𝑐1= 0. (66)

It follows from (66) that there are two cases.

Case 1 (𝑐1= 0). If𝑚−𝑛−1 ̸= 0, then from (66) we have 𝑐

1= 0.

Therefore,

Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) = 𝑃 (𝜙) . (67)

Thus, Λ(𝜙, 𝜙𝑧, 𝜙𝑧𝑧) does not give a new multiplier and

therefore new conservation laws will not be derived.

Case 2 (𝑛 = 𝑚−1). If 𝑐1̸= 0, then from (66), we have 𝑛 = 𝑚−1.

Thus, (65) becomes

Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) = 𝑐1𝜙1−2𝑚

𝜙𝑧𝑧−

1

2

(𝑚 + 1) 𝑐1𝜙−2𝑚

𝜙2

𝑧+ 𝑃 (𝜙) .

(68)

Now substitute (68) into (48) and equate the coefficient of𝜙𝑧𝜙𝑡𝑧in (48) to zero, which gives

𝑑2𝑃

𝑑𝜙2+

(2𝑚 − 1)

𝜙

𝑑𝑃

𝑑𝜙

= (𝑚 − 2) 𝑐1𝜙1−2𝑚

. (69)

Solving (69) we have

𝑃 (𝜙) =

𝑚 − 2

3 − 2𝑚

𝜙3−2𝑚

𝑐1+

𝜙2(1−𝑚)

2 (1 − 𝑚)

𝑐2+ 𝑐3

(70)

provided that 𝑚 ̸=3/2 and 𝑚 ̸=1. Since 𝑛 = 𝑚 − 1, these twospecial cases correspond to the points (3/2, 1/2) and (1, 0) onthe (𝑚, 𝑛) plane in Figure 1.

Consider first the general case 𝑛 = 𝑚 − 1 excluding thepoints (3/2, 1/2) and (1, 0). Equation (68) becomes


𝜙𝑧𝑧−

1

2

(𝑚 + 1) 𝑐1𝜙−2𝑚

𝜙2

𝑧

+

𝑚 − 2

3 − 2𝑚

𝜙3−2𝑚

𝑐1+

𝜙2(1−𝑚)

2 (1 − 𝑚)

𝑐2+ 𝑐3.

(71)

Substituting (71) into (48) we find that 𝑐2= 0. Hence,


𝜙𝑧𝑧−

1

2

(𝑚 + 1) 𝑐1𝜙−2𝑚

𝜙2

𝑧

+

𝑚 − 2

3 − 2𝑚

𝜙3−2𝑚

𝑐1+ 𝑐3.

(72)

Since the multiplier (72) contains two constants, 𝑐1and 𝑐3,

it leads to two conserved vectors. The conserved vector


corresponding to 𝑐3= 1, 𝑐1= 0 is the elementary conserved

vector (32).The constants 𝑐1= 1, 𝑐3= 0 lead to the conserved

vector

𝑇1=

1

2

𝜙−2𝑚

𝜙2

𝑧𝑧−

1

6

𝑚 (𝑚 + 2) 𝜙−2𝑚−2

𝜙4

𝑧

+

1

2

(3 − 𝑚) 𝜙1−2𝑚

𝜙2

𝑧+

1

2 (3 − 2𝑚)

𝜙4−2𝑚

,

(73)

𝑇2= [

1

6

𝑚 (𝑚 + 5) 𝜙−2𝑚−2

𝜙3

𝑧−

(𝑚 − 3) (𝑚 − 1)

(3 − 2𝑚)

𝜙1−2𝑚

𝜙𝑧] 𝜙𝑡

− [

1

2

(𝑚 + 1) 𝜙−2𝑚−1

𝜙2

𝑧+

(2 − 𝑚)

(3 − 2𝑚)

𝜙2−2𝑚

] 𝜙𝑡𝑧

−

1

2

(𝑚 − 1) 𝜙−𝑚−1

𝜙2

𝑧+

(𝑚 − 1)

(3 − 2𝑚)

𝜙2−𝑚

.

(74)

The case 𝑛 = 𝑚−1with𝑚 = 1 and 𝑛 = 0 has already beenconsidered.Themultiplier is given by (29) and the conservedvectors by (40) and (41).

Consider 𝑛 = 𝑚 − 1 with 𝑚 = 3/2 and 𝑛 = 1/2. Thedifferential equation (69) becomes

𝑑2𝑃

𝑑𝜙2+

2

𝜙

𝑑𝑃

𝑑𝜙

= −

1

2

𝑐1

𝜙2. (75)

The general solution of (75) is

𝑃 (𝜙) = −

1

2

𝑐1ln𝜙 +

𝑐3

𝜙

+ 𝑐4. (76)

When𝑚 = 3/2 the multiplier (68) becomes

Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) = 𝑐1𝜙−2𝜙𝑧𝑧−

5

4

𝑐1𝜙−3𝜙2

𝑧−

1

2

𝑐1ln𝜙 + 𝜙−1𝑐

3+ 𝑐4.

(77)

On substituting (77) into the determining equation (48) wefind that 𝑐

3= 0 and the multiplier reduces to

Λ = 𝑐1𝜙−2𝜙2

𝑧𝑧−

5

4

𝑐1𝜙−3𝜙2

𝑧−

1

2

𝑐1ln𝜙 + 𝑐

4. (78)

The multiplier (78) again contains two arbitrary constants,𝑐1and 𝑐4. Setting the constants 𝑐

4= 1, 𝑐

1= 0 gives the

elementary conserved vector (32). Setting 𝑐1= 1, 𝑐4= 0 leads

to the conserved vector

𝑇1=

1

2

𝜙−3𝜙2

𝑧𝑧−

7

8

𝜙−5𝜙4

𝑧+

3

4

𝜙−2𝜙2

𝑧

1

2

𝜙 ln𝜙 − 12

𝜙, (79)

𝑇2= [

13

8

𝜙−5𝜙3

𝑧− 𝜙−2𝜙𝑧+

3

4

(𝜙−2 ln𝜙) 𝜙

𝑧] 𝜙𝑡

− [

5

4

𝜙−4𝜙2

𝑧+

1

2

𝜙−1 ln𝜙] 𝜙

𝑡𝑧−

1

4

𝜙−5/2

𝜙2

𝑧+

1

2

𝜙1/2 ln𝜙.

(80)

Consider next multipliers of the form

Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧, 𝜙𝑧𝑧𝑧) . (81)

The determining equation for the multiplier is

𝐸𝜙(Λ (𝜙, 𝜙

𝑧, 𝜙𝑧𝑧, 𝜙𝑧𝑧𝑧) 𝐹 (𝜙, 𝜙

𝑡, 𝜙𝑧, 𝜙𝑡𝑧, 𝜙𝑧𝑧, 𝜙𝑡𝑧𝑧)] = 0,

(82)

where 𝐹 is given by (23). Equating the coefficient of 𝜙𝑡𝑧𝑧𝑧𝑧𝑧

in(82) to zero, we find that

𝜕Λ

𝜕𝜙𝑧𝑧𝑧

= 0 (83)

and therefore

Λ = Λ (𝜙, 𝜙𝑧, 𝜙𝑧𝑧) , (84)

which has already been considered.Harris [6] proved that, except possibly for the two special

cases 𝑛 = 𝑚− 1 with𝑚 ̸=1 and𝑚 = 1 with 𝑛 ̸= 0, there are nomore independent conserved vectors. She proved this resultusing the direct method for conservation laws.

The multipliers and the corresponding conserved vectorsfor the partial differential equation (20) are listed in Table 1.This table was presented by Maluleke and Mason [7] withoutthemultipliers.These conserved vectors agreewith the resultsobtained by Barcilon and Richter [5] and Harris [6].

3.3. Association of Lie Point Symmetries with ConservedVectors. The Lie point symmetries for the partial differentialequation (20) are listed inTable 2.These Lie point symmetrieswere derived by Maluleke and Mason [7, 12]. Using (14) wewill investigate which of the Lie point symmetries are associ-ated with the conserved vectors for theMagma equation (20).

(i) 0 ≤ 𝑛 < ∞, 0 ≤ 𝑚 < ∞. Consider first the Lie symmetrygenerator

𝑋 = (𝑐1+ (2 − 𝑚 − 𝑛) 𝑐

3𝑡)

𝜕

𝜕𝑡

+ (𝑐2+ (𝑛 − 𝑚) 𝑐

3𝑧)

𝜕

𝜕𝑧

+ 2𝑐3𝜙

𝜕

𝜕𝜙

(85)

and the elementary conserved vector (32). Applying (14) wefind that (85) is associated with the conserved vector (32)provided that 𝑐

3= 0, that is, provided that

𝑋 = 𝑐1

𝜕

𝜕𝑡

+ 𝑐2

𝜕

𝜕𝑧

. (86)

(ii) 𝑛 + 𝑚 ̸= 2, 𝑚 ̸=1, 𝑛 + 𝑚 ̸= 1. Consider next the Liepoint symmetry generator (85), with the conserved vector(33). Applying (14) we find that (85) is associated with (33)provided that 𝑐

3= 0, that is, provided𝑋 is given by (86).

(iii)𝑚 = 1, 𝑛 ̸= 0, 𝑛 ̸= 1. Now consider

𝑋 = (−𝑐3𝑡 + 𝑐1)

𝜕

𝜕𝑡

+ (𝑐3𝑧 + 𝑐2)

𝜕

𝜕𝑧

+

2𝑐3𝜙

𝑛 − 1

𝜕

𝜕𝜙

(87)


Table 1: Multipliers and conserved vectors for the partial differential equation (20).

Case A. 0 ≤ 𝑛 < ∞, 0 ≤ 𝑚 < ∞ Case B.5. 𝑛 + 𝑚 = 1,𝑚 ̸= 1Multiplier: Λ = 1 Multiplier: Λ = 𝜙−2(𝑛−2)

𝑇1= 𝜙 𝑇

1= −

1

2

𝜙−2𝑚

𝜙2

𝑧+ ln𝜙

𝑇2= 𝜙𝑛(1 + 𝑚𝜙

−𝑚−1𝜙𝑡𝜙𝑧− 𝜙−𝑚𝜙𝑡𝑧) 𝑇

2= 𝜙1−𝑚 ln𝜙 − 1

1 − 𝑚

𝜙1−𝑚

− (𝜙1−2𝑚 ln𝜙)𝜙

𝑡𝑧

+(𝑚𝜙−2𝑚 ln𝜙)𝜙

𝑡𝜙𝑧

Case B.1.𝑚 ̸=1,𝑚 + 𝑛 ̸= 1,𝑚 + 𝑛 ̸= 2 Case B.6. 𝑛 + 𝑚 = 2,𝑚 ̸=1

Multiplier: Λ = 𝜙1−𝑚−𝑛 Multiplier: Λ =1

𝜙

𝑇1=

1

2 − 𝑚 − 𝑛


− 1) +

1

2

(1 − 𝑚 − 𝑛)𝜙−2𝑚

𝜙2

𝑧𝑇1= −

1

2

𝜙−2𝑚

𝜙2

𝑧+ ln𝜙

𝑇2=

𝑛

1 − 𝑚

𝜙1−𝑚

− 𝜙1−2𝑚

𝜙𝑡𝑧+ 𝑚𝜙

−2𝑚𝜙𝑡𝜙𝑧

𝑇2=

2 − 𝑚

1 − 𝑚

𝜙1−𝑚

− 𝜙1−2𝑚

𝜙𝑡𝑧+ 𝑚𝜙

−2𝑚𝜙𝑡𝜙𝑧

Case B.2.𝑚 = 1, 𝑛 ̸= 0, 𝑛 ̸= 1 Case C.1. 𝑛 = 𝑚 − 1,𝑚 ̸=3

2

,𝑚 ̸= 1

Multiplier: Λ = 𝜙−𝑛 Multiplier: Λ = 𝜙1−2𝑚𝜙𝑧𝑧 −1

2

(𝑚 + 1)𝜙−2𝑚

𝜙2

𝑧

𝑇1=

1

1 − 𝑛

−

𝑛

2

𝜙−2𝜙2

𝑧+

𝑚 − 2

3 − 2𝑚

𝜙3−2𝑚

𝑇2= 𝑛 ln𝜙 − 𝜙−1𝜙

𝑡𝑧+ 𝜙−2𝜙𝑡𝜙𝑧

𝑇1=

1

1 − 𝑛

𝜙1−𝑛

−

𝑛

2

𝜙−2𝜙2

𝑧

Case B.3.𝑚 = 𝑛 = 1 +1

2

(3 − 𝑚)𝜙1−2𝑚

𝜙2

𝑧+

1

2(3 − 2𝑚)

𝜙4−2𝑚

Multiplier: Λ = 1𝜙

𝑇2= [

1

6

𝑚 (𝑚 + 5) 𝜙−2𝑚−2

𝜙3

𝑧−

(𝑚 − 3) (𝑚 − 1)

(3 − 2𝑚)

𝜙1−2𝑚

𝜙𝑧] 𝜙𝑡

𝑇1= −

1

2

𝜙−2𝜙2

𝑧+ ln𝜙 − [

1

2

(𝑚 + 1)𝜙−2𝑚−1

𝜙2

𝑧+

(2 − 𝑚)

(3 − 2𝑚)

𝜙2−2𝑚

] 𝜙𝑡𝑧


𝑡𝑧+ 𝜙−2𝜙𝑡𝜙𝑧

−

1

2

(𝑚 − 1)𝜙−𝑚−1

𝜙2

𝑧+

(𝑚 − 1)

(3 − 2𝑚)

𝜙2−𝑚

Case B.4.𝑚 = 1, 𝑛 = 0 Case C.2. 𝑛 = 𝑚 − 1,𝑚 =3

2

, 𝑛 =

1

2

Multiplier: Λ = 𝜙2 Multiplier: Λ = 𝜙−2𝜙2𝑧𝑧 −5

4

𝜙−3𝜙2

𝑧−

1

2

ln𝜙

𝑇1=

1

2

𝜙−2𝜙2

𝑧+ 𝜙 ln𝜙 − 𝜙 𝑇1 =

1

2

𝜙−3𝜙2

𝑧𝑧−

7

8

𝜙−5𝜙4

𝑧+

3

4

𝜙−2𝜙2

𝑧

𝑇2= −(𝜙

−1 ln𝜙)𝜙𝑡𝑧+ (𝜙−2 ln𝜙)𝜙

𝑡𝜙𝑧

+

1

2

𝜙 ln𝜙 − 12

𝜙

𝑇2= [

13

8

𝜙−5𝜙3

𝑧− 𝜙−2𝜙𝑧+

3

4

(𝜙−2 ln𝜙) 𝜙

𝑧] 𝜙𝑡− [

5

4

𝜙−4𝜙2

𝑧+

1

2

𝜙−1 ln𝜙] 𝜙

𝑡𝑧

−

1

4

𝜙−(5/2)

𝜙2

𝑧+

1

2

𝜙1/2 ln𝜙

and the conserved vector (45). Applying (14) we find that (87)is associated with (45) provided that 𝑐

3= 0, that is, provided

that𝑋 is given by (86).

(iv)𝑚 = 𝑛 = 1. Consider next

𝑋 = 𝑐1

𝜕

𝜕𝑡

+ 𝑐2

𝜕

𝜕𝑧

+ 2𝑐3𝜙

𝜕

𝜕𝜙

(88)



that𝑋 is (86).

(v)𝑚 = 1, 𝑛 = 0. Consider

𝑋 = 𝜉1(𝑡, 𝑧)

𝜕

𝜕𝑡

+ (𝑐2+ 𝑐3𝑧)

𝜕

𝜕𝑧

− 2𝑐3𝜙

𝜕

𝜕𝜙

(89)



that

𝑋 = 𝜉1(𝑡, 𝑧)

𝜕

𝜕𝑡

+ 𝑐2

𝜕

𝜕𝑧

. (90)

(vi) 𝑛 + 𝑚 = 1,𝑚 ̸=1. Consider next

𝑋 = (𝑐1+ 𝑐3𝑡)

𝜕

𝜕𝑡

+ (𝑐2+ (1 − 2𝑚) 𝑐

3𝑧)

𝜕

𝜕𝑧

+ 2𝑐3𝜙

𝜕

𝜕𝜙

(91)





(vii) 𝑛 + 𝑚 = 2,𝑚 ̸=1. Consider

𝑋 = 𝑐1

𝜕

𝜕𝑡

+ (𝑐2+ 2 (1 − 𝑚) 𝑐

3𝑧)

𝜕

𝜕𝑧

+ 2𝑐3𝜙

𝜕

𝜕𝜙

(92)

with the conserved vector (38). Applying (14)we find that (92)is associated with (38) provided that 𝑐



(viii) 𝑛 = 𝑚 − 1,𝑚 ̸=3/2,𝑚 ̸=1. Consider

𝑋 = (𝑐1+ (3 − 2𝑚) 𝑐

3)

𝜕

𝜕𝑡

+ (𝑐2− 𝑐3𝑧)

𝜕

𝜕𝑧

+ 2𝑐3𝜙

𝜕

𝜕𝜙

(93)

with the conserved vector given by (73) and (74). Applying(14) we find that (93) is associated with the conserved vectorwith components (73) and (74) provided that 𝑐

3= 0, that is,

provided

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