AppliedMathematics,2020,11,1178-1195
https://www.scirp.org/journal/am
ISSNOnline:2152-7393
ISSNPrint:2152-7385
OnSymmetryReductionofthe
(1+3)-DimensionalInhomogeneous
Monge-Amp`ereEquationtotheFirst-Order
ODEs
VasylM.Fedorchuk,VolodymyrI.Fedorchuk
PidstryhachInstituteforAppliedProblemsofMechanicsandMathematics,NationalAcademyofSciencesof
Ukraine,L’viv,Ukraine
Howtocitethispaper:Fedorchuk,V.M.
andFedorchuk,V.I.(2020)OnSymmetryRe-
ductionofthe(1+3)-DimensionalInhomoge-
neousMonge-Amp`ereEquationtotheFirst-
OrderODEs.AppliedMathematics,11,1178-
1195.
https://doi.org/10.4236/am.2020.1111080
Received:October9,2020
Accepted:November22,2020
Published:November25,2020
Copyright
c
2020byauthor(s)andScientific
ResearchPublishingInc.
ThisworkislicensedundertheCreativeCom-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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saeurisusblanditmattis.Maurisutnunc.
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egettortor.Quisquedapibuspedeinerat.
Nuncenim.Induinulla,commodoat,con-
sectetuernec,malesuadanec,elit.Aliquam
ornaretelluseuurna.Sednecmetus.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integermetus.Morbipulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehiculapurus.Fusceacnislaliquamante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdummalesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbidapibusmauriscondimentum
nulla.Cumsociisnatoquepenatibusetmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesquealiquetsodalesurna.
Nullameleifendjustoinnisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
duidignissimsuscipit.Morbimagna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sedpedesitametelitrhoncusultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptenttacitisociosquadlitoratorquen-
tperconubianostra,perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel,pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretiumnonummyfringilla,tortorlacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitaeipsumacmetusdictumtempus.Namut
wisi.Quisquetortorfelis,interdumac,sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donectempusnequevitaeest.Aeneanegestas
odiosedrisusullamcorperullamcorper.Sedin
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehiculaeget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra,lacusnibhdignissimtellus,eususcipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibuso diovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabiturullamcorper
ligulanecorci.Aliquampurusturpis,aliquam
id,ornarevitae,porttitornon,wisi.Maecenas
luctusportalorem.Donecvitaeligulaeuante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclaciniaauctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Utquisnuncsedodiomalesuadaegestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesqueeuwisi.Prointurpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justolectus,condimentumsitamet,posuerea,
fringillamollis,felis.Morbinullanibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinarblandit.Nuncsitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel,metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemp er.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sitamet,placeratac,mauris.Maecenasetelit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat,wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisqueconsectetuer.Insuscipitmaurisado-
lorpellentesqueconsectetuer.Maurisconvallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiamfermentumconvallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarculibero,malesuadavitae,ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurussedsem
temporrhoncus.Utquamdui,fringillaat,dic-
tumeget,ultriciesquis,quam.Etiamsemest,
pharetranon,vulputatein,pretiumat,ipsum.
Nuncsempersagittisorci.Sedscelerisquesus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareantenuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisiultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummyvulputateorci.Donecetvelitacar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsumanisl.Vivamusvariusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisicommodofelis,utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisqueeratelit,viverrasitamet,sagittis
eget,portasitamet,lacus.
Inhachabitasseplateadictumst.Proinatest.
Curabiturtempusvulputateelit.Pellentesque
sem.Praesenteusapien.Duiselitmagna,ali-
quetat,tempussed,vehiculanon,enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimetcommodomalesuada,tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultriciesaccumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesenttempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
tegerrutrumsemegetwisi.Aeneaneusapien.
Quisqueornaredignissimmi.Duisaurnavel
risuspharetraimperdiet.Suspendissepotenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sederos.Suspendisseconsequat.Maurisadui
ettortormattispretium.Sednullametus,vo-
lutpatid,aliquameget,ullamcorperut,ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Utblanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
risornarevelitidnibhpretiumultricies.Fusce
temporpellentesqueodio.Vivamusauguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nullainterdum,nunceusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,p osuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elittortor,feugiatquis,sagittiset,ornarenon,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quamquisquamnonmetuseleifendinterdum.
Namegetsapienacmaurismalesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibusa,odio.Uteroserat,vulputateut,in-
terdumnon,portaeu,erat.Crasfermentum,
felisinportacongue,velitleofacilisisodio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrumtellus,velposuerenibhlacusnecnibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestiedolornecmetus.Vestibulumlibero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam,urnasedinterdumporttitor,metusorci
interdumodio,sitameteuismodlectusfeliset
leo.Praesentacwisi.Namsuscipitvestibu-
lumsem.Praesenteuipsumvitaepedecursus
venenatis.Duissedodio.Vestibulumeleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissimac,venenatisvitae,consequatut,lacus.
Namnibh.Vestibulumfringillaarcumollisar-
cu.Sedetturpis.Donecsemtellus,volut-
patet,variuseu,commodosed,lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus,nibhmassaeleifendturpis,sedpellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuervariusnul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravidavelitatelitvehiculaporttitor.Phasel-
lusnislmi,sagittisac,pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsitamet,consectetueradipiscingelit.Cur-
abituridsemelementumleorutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sedturpisquam,sollicitudina,hendreriteget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodonec,sempersitamet,elit.Nullalec-
tusrisus,condimentumut,laoreeteget,viver-
ranec,odio.Proinlobortis.Curabiturdictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorpertristique.Maurisenimsem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate,neque.Quisquelacus.Donecetipsumsit
ametelitnonummyaliquet.Sedviverranislat
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolorsitamet,consectetueradipiscingelit.Ut
pedesem,tempusut,porttitorbibendum,mo-
lestieeu,elit.Suspendissepotenti.Sedidlec-
tussitametpurusfaucibusvehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverraultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac,nunc.Maurismalesuadamalesua-
daelit.Etiamlacusmauris,pretiumvel,blan-
ditin,ultriciesid,libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplateadictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristiquequis,felis.Vestibulumanteip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
maurisportadui,vitaealiquamarcuturpisac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellusaliquam,tortornecornare
rhoncus,purusurnap osuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
nainvelit.Etiamullamcorperelementumdui.
Praesentnonurna.Sedplaceratquamnonmi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspedearcu,euismodac,tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapienurna,blanditnec,iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnonarcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbiipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueegetloremnecelitmalesuadavestibu-
lum.Quisquesollicitudinipsumvelsem.Nul-
laenim.Proinnonummyfelisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjustoidorciiaculisvarius.Quisqueet
lacus.Phaselluselementum,justoatdignissim
auctor,wisiodiolobortisarcu,sedsollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
acodionecnequevulputatesuscipit.Namnec
magna.Pellentesquehabitantmorbitristique
senectusetnetusetmalesuadafamesacturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisinequefringillasapien,velcommodolorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesquenonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nullaquammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenaslorem.Vivamusviverraconsequatenim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudinhen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tiseget,scelerisqueac,aliquetin,dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo,nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectusipsum,vulputatea,mollisut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsanpulvinarjusto.Nullamsemper,duiut
dignissimauctor,orciliberofringillamassa,b-
landitpulvinarpedetortoridmagna.Nunc
adipiscingjustosedvelittinciduntfermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristiquesenectusetnetusetmalesuadafames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulumdapibus
neque.
Mauristempuserosatnulla.Sedquisduidig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectuswisi,elementumeu,sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipitsitamet,rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringillatempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentumconsequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitantmorbitristiquesenectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lamlaoreeturnasedenim.Donecegeterosut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
egeterosaligulacommodoultricies.Doneceu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aeneantinciduntlaoreetdui.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendreritut,semperat,metus.Vivamussapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque,dignissimin,facilisisnec,hendreritvel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputatepurus.Morbivarius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nuncconsequatmassaidleo.Sedullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,nonporttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget,posuerea,felis.Quisquevellectusutme-
tustincidunteleifend.Duisutpede.Duisvelit
erat,venenatisvitae,vulputatea,pharetrasit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid,mi.Nullamnecnibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donecmattisullamcorpertellus.Phasellusvel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nuncnullapulvinarjusto,aeuismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus,eusodalesmagnarisusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,p osuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integerpurus.
Vivamusadipiscing.Curabiturimperdiettem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursusarcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfeliselit,dignissimsed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo,dapibussitamet,vestibulumet,commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus,sitametp osuerelectusneque
velaugue.Crasconsectetuerliberoaceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictumauctor.Inegetrisusluctuswisiconva-
llispulvinar.Vivamussapienrisus,temporin,
viverrain,aliquetpellentesque,eros.Aliquam
euismodliberoasem.
Nuncvelitaugue,scelerisquedignissim,lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
risimperdiet,metusattristiquevenenatis,pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligulasuscipiturna,vitaeportaodioer-
atquissapien.Proinluctusleoiderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
naeros,ornarenec,vulputateeget,cursussed,
justo.Phasellusneclorem.Nullamligulaligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fuscevehicula,tortoretgravidaporttitor,me-
tusnibhconguelorem,uttempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sitamet,turpis.Quisquelobortis.Proinvene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Namipsumelit,imperdietvitae,faucibusnec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenasquislacus.Morbifringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbitristiquesenectusetnetusetmalesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dalessemsitametmauris.Suspendisseornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit,gravidaac,pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu,eleifendsed,vehiculaat,conguesitamet,
sem.Sedsagittispedeanisl.Sedtincidunto-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nullaettellusnonfelisimperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
loremconvallishendrerit.Maecenashendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsitametnibhnecpurusdignissimmat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utconguemalesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titorpede,necaliquamturpisdiamfeugiatar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulumelit.Namligulaarcu,volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisusatmetuspulvinarmalesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumanteipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremp osueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdumporta.Nuncbibendumaugueluctus
diam.Etiamnecsem.Sederosturpis,facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajustovehiculadiam,eucondimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pedepurussollicitudinurna,acaliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orcisodaleslorem,nonornarenullaloremquis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverraid,velit.
Utlectuslectus,ultriciessitamet,semper
eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nullaornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisquenisl.Indignissimdapibusmassa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitametest.Vivamuspulvinarleonon
sapien.Aliquameratvolutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,p ortaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsitamet,mattissuscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisseutaugueacnibhullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusestidlacus.Vivamusloremdui,commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Crassodales.Quisquevestibulumpulvinardi-
am.Phasellustincidunt,leovitaetristiquefa-
cilisis,ipsumwisiinterdumsem,dapibussem-
pernullavelitvellectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
Suspendisseadolor.Namerateros,congue
eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim,sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentumat,sodalesin,ultricesvel,nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsedorcivitaedolorsollicitudinadipiscing.
Crasaneque.Utrisusdui,interdumat,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
egeteleifendposuere,augueerosporttitorlec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
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Praesentrhoncus.Curabitursagittisestsit
ametneque.Sedcommodomalesuadalectus.
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aliquameu,quam.Aeneanquisquamquis
wisigravidavehicula.Pellentesqueamassa
aleopretiumrhoncus.Suspendisseultrices.
Doneclaciniamalesuadamassa.Classapten-
ttacitisociosquadlitoratorquentperconu-
bianostra,perinceptoshymenaeos.Donec
pretiumornaremauris.Phasellusauctorer-
ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
tis,diamarutrumfermentum,odiosapien
tristiquequam,idmollistellusquaminodi-
o.Mauriseusapien.Donecaliquamloremsit
ametlorempharetralobortis.
Donecacvelit.Sedconvallisvestibulumsapi-
en.Vivamustemporlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
tumvulputate,nequeliberovestibulumorci,
vitaefringillanequearcualiquetante.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisquevenenatislobortisaugue.Sed
tempor,tellusiaculispellentesquepharetra,
pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
hicula,vestibulumvitae,feugiateleifend,fe-
lis.Vivamuspulvinar.Aliquameratvolutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
Suspendisseeununc.Aliquamdignissimurna
sitametmauris.Crascommodo,urnautport-
titorvenenatis,arcumetussodalesrisus,vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultricesfeliseratatlectus.Inestmassa,male-
suadanon,suscipitat,ullamcorpereu,elit.
Namnullalacus,bibendumsitamet,sagit-
tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibhpretiumtellus,atdignissimleoeros
egetpede.Nullavenenatiseleifenderos.Ae-
neanutodiodignissimauguerutrumfaucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumanteipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamusvehiculaleoajusto.Quisquenecau-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenimsitametrisus.Curabiturvitaevelit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusceeunullautnullainterdumconsectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
We present theresults obtainedconcerning theclassification ofsym-
metryreductionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ere equationto first-order ODEs.Someclasses ofthe invariant
solutionsareconstructed.
Keywords
SymmetryReduction,InvariantSolutions,Monge-Amp`ere
Equation,ClassificationofLieAlgebras,Poincar´eGroupP(1,4)
1.Introduction
Itiswellknownthatthesymmetryreductionisoneofthemostuni-
versaltoolsforinvestigationofpartialdifferentialequations(PDEs).
Inparticular,forthisaim,wecanusetheclassicalLie-Ovsiannikov
method[1–4](see,also,thereferencestherein).
However,itturnedoutthatinordertoapplytheclassicalLie-
OvsiannikovmethodforPDEswithnon-trivialsymmetrygroups,we
hadtosolveapurealgebraicproblemofdescribingallnonconjugate
(nonsimilar)subalgebrasoftheLiealgebrasofsymmetrygroupsof
theequationsunderinvestigation.Moredetailsonthisthemecanbe
foundin[2,3](see,also,thereferencestherein).
In1975,Patera, Winternitz,andZassenhaus[5]proposedageneral
methodfordescribingthenonconjugatesubalgebrasofLiealgebras
withnontrivialideals.
In1984,Grundland,Harnad,andWinternitz[6]pointedoutthat
thereducedequations,obtainedwiththehelpofnonconjugatesubal-
gebras ofthe same ranks ofthe Lie algebras ofthe symmetry groupsof
some PDEs, were of differenttypes.They also investigated the similar
DOI:10.4236/am.2020.1111080Nov.25,20201178AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
phenomenon.Somedetailsonthisthemecanbefoundin[7–14](see,
also,thereferencestherein).
Theresultsobtainedcannotbeexplainedusingonlytherankof
nonconjugatesubalgebrasoftheLiealgebrasofthesymmetrygroups
ofPDEsunderinvestigation.
Totrytoexplainsomeofthedifferencesinthepropertiesofthe
reduced equations forPDEswith nontrivial symmetrygroups,we sug-
gestedto investigatetherelationship between thestructural properties
ofnonconjugatesubalgebrasofthesamerankoftheLiealgebrasof
the symmetry groups of those PDEsand the properties of the reduced
equationscorrespondingwiththem[13].
Atthepresenttime,therelationshiphasbeenstudiedbetweenthe
structural properties of the low-dimensional (dimL3) nonconjugate
subalgebrasofthesame rankoftheLiealgebraofthePoincar´egroup
P(1,4)andthepropertiesofthereducedequationsfortheEikonal
andEuler-Lagrange-Born-Infeldequations.Thedetailsonthistheme
canbefoundin[12–16].
Asolutionof manyproblems ofthe geometry, unifiedstringtheories,
geometricaloptics,elastictheoriesofshallowshells,optimaltrans-
portation,one-dimensionalgasdynamics,meteorologyandoceanog-
raphy etc.isreduced totheinvestigation oftheMonge-Amp`ere equa-
tionsinthespacesofdifferentdimensionsanddifferenttypes.Some
detailson thisthemecanbe foundin[17–34](see,also,thereferences
therein).
Thispaperisdevotedtothestudyoftherelationshipbetweenthe
structural properties of the low-dimensional (dimL3) nonconjugate
subalgebrasofthesamerankoftheLiealgebraofthegroupP(1,4)
and theproperties of thereduced equations for the(1+3)-dimensional
inhomogeneousMonge-Amp`ereequation.
Bynow,therelationshiphasbeeninvestigatedbetweenthestruc-
turalpropertiesofthethree-dimensionalnonconjugatesubalgebrasof
theLiealgebraofthegroupP(1,4)andtheproperties ofthereduced
equationsforthe(1+ 3)-dimensionalinhomogeneousMonge-Amp`ere
equation.Weobtainedthefollowingtypesofthereducedequations:
-algebraicequations,
-thefirst-orderlinearODEs,
-thefirst-ordernonlinearODEs,
-thesecond-ordernonlinearODEs,
-thepartialdifferentialequations.
FromtheinvariantsofsomenonconjugatesubalgebrasoftheLie
algebra of the group P(1,4),it is impossible to construct the ansatzes,
which reduce the (1+3) - dimensional inhomogeneous Monge-Amp`ere
equation.
Thispaperisthefirstinwhichtherelationshipisstudiedbetween
the structural properties of the three-dimensional nonconjugate subal-
gebrasoftheLiealgebraofthegroupP(1,4)andtypesofreductions
of the(1+3)-dimensionalinhomogeneousMonge-Amp`ere equation to
thefirst-orderODEs.
Institute forAppliedProblemsof Mechanics andMathematics,Na-
tionalAcademyofSciencesofUkraine,3-b,NaukovaStr.,79060,
L’viv,Ukraine
DOI:10.4236/am.2020.11110801179AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
2.LieAlgebraofthePoincar´eGroup
P(1,4)andItsNonconjugate
Subalgebras
ThegroupP(1,4)isagroupofrotationsandtranslationsofthefive-
dimensional Minkowski spaceM(1,4).It is the smallest group,which
contains,assubgroups,theextendedGalileigroup
G(1,3)[35](the
symmetrygroup ofclassicalphysics) andthePoincar´e groupP(1,3)
(thesymmetrygroupofrelativisticphysics).
Lie algebraof the groupP(1,4)isgenerated by 15 bases elements
M
µν
= M
νµ
(µ,ν= 0,1,2,3,4)andP
µ
(µ= 0,1,2,3,4),which
satisfythecommutationrelations
[P
µ
,P
ν
] = 0,[M
µν
,P
σ
] = g
νσ
P
µ
g
µσ
P
ν
,(1)
[M
µν
,M
ρσ
] = g
µσ
M
νρ
+g
νρ
M
µσ
g
µρ
M
νσ
g
νσ
M
µρ
,(2)
whereg
00
= g
11
= g
22
= g
33
= g
44
= 1,g
µν
= 0,ifµ̸= ν.
Inthispaper,weconsiderthefollowingrepresentation[36]ofthe
LiealgebraofthegroupP(1,4) :
P
0
=
∂x
0
,P
1
=
∂x
1
,P
2
=
∂x
2
,P
3
=
∂x
3
,(3)
P
4
=
∂u
,M
µν
= x
µ
P
ν
x
ν
P
µ
,x
4
u.(4)
Inthefollowing,wewillusethenextbaseselements:
G= M
04
,L
1
= M
23
,L
2
= M
13
,L
3
= M
12
,(5)
P
a
= M
a4
M
0a
,C
a
= M
a4
+M
0a
,(a= 1,2,3),(6)
X
0
=
1
2
(P
0
P
4
),X
k
= P
k
(k= 1,2,3),X
4
=
1
2
(P
0
+P
4
).(7)
NonconjugatesubalgebrasoftheLiealgebraofthegroupP(1,4)
havebeendescribedinthepapers[37–39].
TheLiealgebraoftheextendedGalileigroup
G(1,3)isgenerated
bythefollowingbaseselements:
L
1
,L
2
,L
3
,P
1
,P
2
,P
3
,X
0
,X
1
,X
2
,X
3
,X
4
.(8)
The classificationof all nonconjugatesubalgebras of theLie algebra
ofthegroupP(1,4)ofdimensions3wasperformedin[40].
3.OnSymmetryReductionofthe
(1+3)-DimensionalInhomogeneous
Monge-Amp`ereEquationtothe
First-OrderODEs
Inthissection,weconsiderequationoftheform:
det(u
µν
) = λ(1u
ν
u
ν
)
3
̸= 0,
DOI:10.4236/am.2020.11110801180AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
whereu= u(x),x= (x
0
,x
1
,x
2
,x
3
) M(1,3),
u
µν
2
u
∂x
µ
∂x
ν
,u
ν
= g
να
u
α
,u
α
∂u
∂x
α
,
g
µν
= (1,1,1,1)δ
µν
,µ,ν,α= 0,1,2,3.
Here,M(1,3)isafour-dimensionalMinkowskispace.
In1983,FushchichandSerov [36]studiedsymmetry propertiesand
constructedsomeclassesofexactsolutionsforthemultidimensional
Monge–Amp`ere equation.From this work, it follows thatthe equation
underinvestigationisinvariantwithrespecttothegroupP(1,4).
Inordertoperformsymmetryreductionaswellastoconstruc-
tclassesofindependentinvariantsolutionsfortheequationunder
consideration,weusedthestructuralpropertiesofthree-dimensional
nonconjugatesubalgebrasoftheLie algebraofthegroupP(1,4)[40].
Below,wepresentashortreviewoftheresultsobtained.
3.1.LieAlgebrasoftheType3A
1
Reductionstothefirst-orderlinearODEs
Taking into account the invariants of 11 subalgebras we constructed
theansatzes,whichreducedtheequationunderconsiderationtothe
first-orderlinearODEs.Belowwepresentsomeoftheresultsobtained.
1.P
1
⟩⊕⟨P
2
X
2
⟩⊕⟨X
3
:
Ansatz
x
2
0
x
2
1
u
2
x
0
+u
x
2
2
x
0
+u+1
= φ(ω)= x
0
+u.
Reducedequation
ω(ω+1)φ
= 0.
Solutionsofthereducedequation
φ(ω) = c,ω+1 = 0= 0.
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
2
0
x
2
1
u
2
x
0
+u
x
2
2
x
0
+u+1
= c,x
0
+u+1 = 0,
x
0
+u= 0.
2.P
1
⟩⊕⟨P
2
⟩⊕⟨X
3
:
Ansatz
x
2
0
x
2
1
x
2
2
u
2
= φ(ω)= x
0
+u.
Reducedequation
ωφ
φ= 0.
Solutionofthereducedequation
φ(ω) = cω.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
2
0
x
2
1
x
2
2
u
2
= c(x
0
+u).
3.P
3
⟩⊕⟨X
1
⟩⊕⟨X
2
:
Ansatz
x
2
0
x
2
3
u
2
= φ(ω)= x
0
+u.
DOI:10.4236/am.2020.11110801181AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
Reducedequation
ωφ
φ= 0.
Solutionofthereducedequation
φ(ω) = cω.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
2
0
x
2
3
u
2
= c(x
0
+u).
Itshouldbenotedthatsubalgebras(1)–(3)belongtotheLie
algebraoftheextendedGalileigroup
G(1,3) P(1,4).
Reductionstothefirst-ordernonlinearODEs.
Fromtheinvariantsofsevensubalgebrasweconstructedthe
ansatzes,which reducedtheequation underconsiderationto thefirst-
ordernonlinearODEs.Belowwepresenttheresultobtained.
1.P
3
2X
0
⟩⊕⟨X
1
⟩⊕⟨X
2
:
Ansatz
1
6
(x
0
+u)
3
+x
3
(x
0
+u)+x
0
u= φ(ω),
ω= (x
0
+u)
2
+4x
3
.
Reducedequation
16(φ
)
2
ω= 0.
Solutionofthereducedequation
φ(ω) =
ε
6
ω
3/2
+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
1
6
(x
0
+u)
3
+x
3
(x
0
+u)+x
0
u
=
ε
6
(x
0
+u)
2
+4x
3
3/2
+c,ε= ±1.
2.P
3
2X
0
⟩⊕⟨X
1
⟩⊕⟨X
4
:
Ansatz
(x
0
+u)
2
+4x
3
= φ(ω)= x
2
.
Reducedequation
(φ
)
2
+16 = 0.
Solutionofthereducedequation
φ(ω) = 4iεω+c.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
0
+u)
2
+4x
3
= 4iεx
2
+c.
3.G+αX
3
,α>0⟩⊕⟨X
1
⟩⊕⟨X
2
:
Ansatz
x
3
αln(x
0
+u) = φ(ω)= x
2
0
u
2
.
Reducedequation
4ω(φ
)
2
+4αφ
1 = 0.
Solutionofthereducedequation
DOI:10.4236/am.2020.11110801182AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
φ(ω) = ε
ω+α
2
+
ε
2
αln
α
2
+ωα
α
2
+ω+α
α
2
ln(ω)+c.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
3
αln(x
0
+u) = ε
x
2
0
u
2
+α
2
+
ε
2
αln
x
2
0
u
2
+α
2
α
x
2
0
u
2
+α
2
+α
α
2
ln
x
2
0
u
2
+c.
4.L
3
⟩⊕⟨P
3
+C
3
⟩⊕⟨X
0
+X
4
:
Ansatz
(x
2
3
+u
2
)
1/2
= φ(ω)= (x
2
1
+x
2
2
)
1/2
.
Reducedequation
(φ
)
2
+1
φ= 0.
Solutionsofthereducedequation
φ(ω) = iεω+c,φ(ω) = 0.
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
2
3
+u
2
)
1/2
= (x
2
1
+x
2
2
)
1/2
+c,u
2
+x
2
3
= 0.
5.L
3
+α(X
0
+X
4
),α>0⟩⊕⟨X
3
⟩⊕⟨X
4
:
Ansatz
x
0
+u+αarctan
x
2
x
1
= φ(ω)= (x
2
1
+x
2
2
)
1/2
.
Reducedequation
ω
2
(φ
)
2
+α
2
= 0.
Solutionofthereducedequation
φ(ω) = (αln(ω)+ic).
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
u=
α
2
ln(x
2
1
+x
2
2
)+ic
αarctan
x
2
x
1
x
0
.
6.G⟩⊕⟨X
2
⟩⊕⟨X
1
:
Ansatz
(x
2
0
u
2
)
1/2
= φ(ω)= x
3
.
Reducedequation
(φ
1)(φ
+1) = 0.
Solutionofthereducedequation
φ(ω) = εω+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
2
0
u
2
)
1/2
= εx
3
+c,ε= ±1.
7.G⟩⊕⟨L
3
⟩⊕⟨X
3
:
Ansatz
(x
2
0
u
2
)
1/2
= φ(ω)= (x
2
1
+x
2
2
)
1/2
.
Reducedequation
DOI:10.4236/am.2020.11110801183AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
(φ
1)(φ
+1) = 0.
Solutionofthereducedequation
φ(ω) = εω+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
2
0
u
2
)
1/2
= ε(x
2
1
+x
2
2
)
1/2
+c,ε= ±1.
3.2.LieAlgebrasoftheTypeA
2
A
1
Reductionstothefirst-ordernonlinearODEs
Takingintoaccounttheinvariantsofsixsubalgebrasweconstruct-
edtheansatzes,whichreducedtheequationunderconsiderationto
thefirst-ordernonlinearODEs.Belowwepresentsomeoftheresult
obtained.
1.⟨−(G+αX
2
),X
4
,α>0⟩⊕⟨X
1
:
Ansatz
x
2
αln(x
0
+u) = φ(ω)= x
3
.
Reducedequation
(φ
)
2
+1 = 0.
Solutionofthereducedequation
φ(ω) = iεω+c.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
2
αln(x
0
+u) = iεx
3
+c.
2.⟨−(G+αX
2
),P
3
,α>0⟩⊕⟨X
1
:
Ansatz
x
2
αln(x
0
+u) = φ(ω)= (x
2
0
x
2
3
u
2
)
1/2
.
Reducedequation
ω
ω(φ
)
2
+2αφ
ω
= 0.
Solutionsofthereducedequation
φ(ω) = εαln
2
α
ω
2
+α
2
+α
2
ω
αln(ω)
ε
ω
2
+α
2
+c,ε= ±1;ω= 0.
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
2
= εαln
2
α
x
2
0
x
2
3
u
2
+α
2
+α
2
x
2
0
x
2
3
u
2
α
2
ln(x
2
0
x
2
3
u
2
)ε
x
2
0
x
2
3
u
2
+α
2
+αln(x
0
+u)+c,ε= ±1;x
2
0
x
2
3
u
2
= 0.
3.⟨−(G+αX
3
),X
4
,α>0⟩⊕⟨L
3
+βX
3
,β>0:
Ansatz
x
3
αln(x
0
+u)+βarctan
x
1
x
2
= φ(ω),
ω= (x
2
1
+x
2
2
)
1/2
.
DOI:10.4236/am.2020.11110801184AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
Reducedequation
ω
2
(φ
)
2
+ω
2
+β
2
= 0.
Solutionofthereducedequation
φ(ω) = iεβln
2
β
ω
2
+β
2
+β
2
ω
ω
2
+β
2
+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
3
αln(x
0
+u)+βarctan
x
1
x
2
= iεβln
2
β
x
2
1
+x
2
2
+β
2
+β
2
x
2
1
+x
2
2
x
2
1
+x
2
2
+β
2
+c,ε= ±1.
4.⟨−(G+αX
3
),X
4
,α>0⟩⊕⟨L
3
:
Ansatz
x
3
αln(x
0
+u) = φ(ω)= (x
2
1
+x
2
2
)
1/2
.
Reducedequation
(φ
)
2
+1 = 0.
Solutionofthereducedequation
φ(ω) = iεω+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
3
αln(x
0
+u) = (x
2
1
+x
2
2
)
1/2
+c,ε= ±1.
5.⟨−G,P
3
⟩⊕⟨X
1
:
Ansatz
(x
2
0
x
2
3
u
2
)
1/2
= φ(ω)= x
2
.
Reducedequation
(φ
1)(φ
+1)φ= 0.
Solutionsofthereducedequation
φ(ω) = εω+c,φ(ω) = 0= ±1.
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
2
0
x
2
3
u
2
)
1/2
= εx
2
+c,ε= ±1,x
2
0
x
2
3
u
2
= 0.
3.3.LieAlgebrasoftheTypeA
3,1
Reductionstothefirst-ordernonlinearODEs
Takingintoaccounttheinvariantsofthreenonconjugatesubalge-
bras,weconstructedtheansatzes,whichreducedtheequationunder
consideration to the first-order nonlinear ODEs.Below we present the
resultsobtained.
DOI:10.4236/am.2020.11110801185AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
1.2µX
4
,P
3
2X
0
,X
1
+µX
3
,µ>0:
Ansatz
(x
0
+u)
2
+4x
3
4µx
1
= φ(ω)= x
2
.
Reducedequation
(φ
)
2
+16(µ
2
+1) = 0.
Solutionofthereducedequation
φ(ω) = 4
µ
2
+1ω+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
0
+u)
2
+4x
3
4µx
1
= 4
µ
2
+1x
2
+c,ε= ±1.
2.2X
4
,P
3
L
3
2αX
0
,X
3
,α>0:
Ansatz
2αarctan
x
1
x
2
x
0
u= φ(ω)= (x
2
1
+x
2
2
)
1/2
.
Reducedequation
ω
2
(φ
)
2
+4α
2
= 0.
Solutionofthereducedequation
φ(ω) = 2iεαln(ω)+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
2αarctan
x
1
x
2
x
0
u= iεαln(x
2
1
+x
2
2
)+c,ε= ±1.
3.⟨−2βX
4
,L
3
+βX
3
,P
3
2X
0
,β>0:
Ansatz
βarctan
x
1
x
2
+
1
4
(x
0
+u)
2
+x
3
= φ(ω)= (x
2
1
+x
2
2
)
1/2
.
Reducedequation
ω
2
(φ
)
2
+ω
2
+β
2
= 0.
Solutionofthereducedequation
φ(ω) = iεβln
2
β
ω
2
+β
2
+β
2
ω
ω
2
+β
2
+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
βarctan
x
1
x
2
+
1
4
(x
0
+u)
2
+x
3
= iεβln
2
β
x
2
1
+x
2
2
+β
2
+β
2
x
2
1
+x
2
2
x
2
1
+x
2
2
+β
2
+c,ε= ±1.
DOI:10.4236/am.2020.11110801186AppliedMathematics
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3.4.LieAlgebrasoftheTypeA
3,2
Reductiontothefirst-ordernonlinearODEs
From the invariants of two nonconjugatesubalgebras, we construct-
edtheansatzes,whichreducedtheequationunderconsiderationto
thefirst-order nonlinearODEs.Belowwepresentthe resultsobtained.
1.2βX
4
,P
3
,G+αX
1
+βX
3
,α>0,β>0:
Ansatz
x
1
αln(x
0
+u) = φ(ω)= x
2
.
Reducedequation
(φ
)
2
+1 = 0.
Solutionofthereducedequation
φ(ω) = iεω+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
1
αln(x
0
+u) = iεx
2
+c,ε= ±1.
2.
2αX
4
1
P
3
,
1
λ
1
L
3
+G+
α
λ
1
X
3
,α>0
1
>0
:
Ansatz
ln(x
0
+u)+λ
1
arctan
x
1
x
2
= φ(ω)= (x
2
1
+x
2
2
)
1/2
.
Reducedequation
ω
2
(φ
)
2
+λ
2
1
= 0.
Solutionofthereducedequation
φ(ω) = iελ
1
ln(ω)+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
ln(x
0
+u)+λ
1
arctan
x
1
x
2
=
λ
1
2
ln(x
2
1
+x
2
2
)+c,ε= ±1.
3.5.LieAlgebrasoftheTypeA
3,3
Reductionstothefirst-ordernonlinearODEs
Taking into account the invariantsof two nonconjugatesubalgebras,
weconstructedtheansatzes,whichreducedtheequationundercon-
siderationtothefirst-ordernonlinearODEs.Belowwepresentthe
resultsobtained.
1.P
3
,X
4
,G+αX
1
,α>0:
Ansatz
x
1
αln(x
0
+u) = φ(ω)= x
2
.
Reducedequation
(φ
)
2
+1 = 0.
Solutionofthereducedequation
φ(ω) = iεω+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
1
αln(x
0
+u) = iεx
2
+c,ε= ±1.
DOI:10.4236/am.2020.11110801187AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
2.
P
3
,X
4
,
1
λ
1
L
3
+G,λ
1
>0
:
Ansatz
ln(x
0
+u)+λ
1
arctan
x
1
x
2
= φ(ω)= (x
2
1
+x
2
2
)
1/2
.
Reducedequation
ω
2
(φ
)
2
+λ
2
1
= 0.
Solutionofthereducedequation
φ(ω) = iελ
1
ln(ω)+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
ln(x
0
+u)+λ
1
arctan
x
1
x
2
=
λ
1
2
ln(x
2
1
+x
2
2
)+c,ε= ±1.
3.6.LieAlgebrasoftheTypeA
3,6
Reductionstothefirst-orderlinearODEs
Takingintoaccounttheinvariantsoffournonconjugatesubalge-
bras,weconstructedtheansatzes,whichreducedtheequationunder
considerationtothefirst-orderlinearODEs.Belowwepresentsome
oftheresultsobtained.
1.X
1
,X
2
,P
3
L
3
:
Ansatz
x
2
0
x
2
3
u
2
= φ(ω)= x
0
+u.
Reducedequation
ωφ
φ= 0.
Solutionofthereducedequation
φ(ω) = cω.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
2
0
x
2
3
u
2
= c(x
0
+u).
2.P
1
,P
2
,(L
3
+αX
3
),α>0:
Ansatz
x
2
0
x
2
1
x
2
2
u
2
= φ(ω)= x
0
+u.
Reducedequation
ωφ
φ= 0.
Solutionofthereducedequation
φ(ω) = cω.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
x
2
0
x
2
1
x
2
2
u
2
= c(x
0
+u).
Itshouldbenotedthatsubalgebras(1)and(2)belongtotheLie
algebraoftheextendedGalileigroup
G(1,3) P(1,4).
Reductionstothefirst-ordernonlinearODEs
Fromtheinvariantsofeightsubalgebrasweconstructedtheansa-
tzes, which reducedthe equation underconsideration to the first-order
nonlinearODEs.Belowwepresenttheresultsobtained.
DOI:10.4236/am.2020.11110801188AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
1.X
1
,X
2
,P
3
L
3
2αX
0
,α>0:
Ansatz
(x
0
+u)
3
+6αx
3
(x
0
+u)+6α
2
(x
0
u) = φ(ω),
ω= (x
0
+u)
2
+4αx
3
.
Reducedequation
4(φ
)
2
9ω= 0.
Solutionofthereducedequation
φ(ω) = εω
3/2
+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
0
+u)
3
+6αx
3
(x
0
+u)+6α
2
(x
0
u)
= ε
(x
0
+u)
2
+4αx
3
3/2
+c,ε= ±1.
2.
X
1
,X
2
,L
3
1
2
(P
3
+C
3
)α(X
0
+X
4
),α>0
:
Ansatz
αarctan
x
3
u
x
0
= φ(ω)= (x
2
3
+u
2
)
1/2
.
Reducedequation
ω
ω
2
(φ
)
2
ω
2
+α
2
= 0.
Solutionsofthereducedequation
φ(ω)=iεαln
2(
ω
2
α
2
α
2
)
ω
ε
ω
2
α
2
+ c,ε=
±1;
ω= 0.
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
αarctan
x
3
u
x
0
= ε
x
2
3
+u
2
α
2
iεαln
2(
x
2
3
+u
2
α
2
α
2
)
x
2
3
+u
2
+c,ε= ±1,
x
2
3
+u
2
= 0.
3.
X
1
,X
2
,L
3
+
λ
1
2
(P
3
+C
3
)+α(X
0
+X
4
),α>0,
0
1
<1:
Ansatz
αarctan
x
3
u
λ
1
x
0
= φ(ω)= (x
2
3
+u
2
)
1/2
.
Reducedequation
ω
ω
2
(φ
)
2
λ
2
1
ω
2
+α
2
= 0.
Solutionsofthereducedequation
φ(ω) = iεαln
2(
λ
2
1
ω
2
α
2
α
2
)
ω
ε
λ
2
1
ω
2
α
2
+c,ε= ±1;ω= 0.
DOI:10.4236/am.2020.11110801189AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
αarctan
x
3
u
λ
1
x
0
= ε
λ
2
1
(x
2
3
+u
2
)α
2
iεαln
2(
λ
2
1
(x
2
3
+u
2
)α
2
α
2
)
x
2
3
+u
2
+c,
ε= ±1,x
2
3
+u
2
= 0.
4.X
1
,X
2
,L
3
+λ
1
G+αX
3
,α>0
1
>0:
Ansatz
λ
1
x
3
αln(x
0
+u) = φ(ω)= (x
2
0
u
2
)
1/2
.
Reducedequation
ω
ω(φ
)
2
+2αφ
λ
2
1
ω
= 0.
Solutionsofthereducedequation
φ(ω) = εαln
2(α
λ
2
1
ω
2
+α
2
+α
2
)
ω
αln(ω)
ε
λ
2
1
ω
2
+α
2
+c,ε= ±1= 0.
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
λ
1
x
3
αln(x
0
+u) = ε
λ
2
1
(x
2
0
u
2
)+α
2
εαln
2(α
λ
2
1
(x
2
0
u
2
)+α
2
+α
2
)
x
2
0
u
2
α
2
ln(x
2
0
u
2
)+c,ε= ±1,x
2
0
u
2
= 0.
5.X
1
,X
2
,(L
3
+αX
3
),α>0:
Ansatz
u= φ(ω)= x
0
.
Reducedequation
(φ
+1)(φ
1) = 0.
Solutionofthereducedequation
φ(ω) = εω+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
u= εx
0
+c,ε= ±1.
6.
X
1
,X
2
,L
3
+
1
2
(P
3
+C
3
)
:
Ansatz
(x
2
3
+u
2
)
1/2
= φ(ω)= x
0
.
Reducedequation
(φ
+1)(φ
1)φ= 0.
Solutionsofthereducedequation
φ(ω) = εω+c,φ(ω) = 0= ±1.
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
2
3
+u
2
)
1/2
= εx
0
+c,x
2
3
+u
2
= 0= ±1.
DOI:10.4236/am.2020.11110801190AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
7.
X
1
,X
2
,L
3
λ
1
2
(P
3
+C
3
),0
1
<1
:
Ansatz
(x
2
3
+u
2
)
1/2
= φ(ω)= x
0
.
Reducedequation
(φ
+1)(φ
1)φ= 0.
Solutionsofthereducedequation
φ(ω) = εω+c,φ(ω) = 0= ±1.
Solutionsofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
2
3
+u
2
)
1/2
= εx
0
+c,x
2
3
+u
2
= 0= ±1.
8.⟨−X
1
,X
2
,(L
3
+λ
1
G)
1
>0:
Ansatz
(x
2
0
u
2
)
1/2
= φ(ω)= x
3
.
Reducedequation
(φ
+1)(φ
1) = 0.
Solutionofthereducedequation
φ(ω) = εω+c,ε= ±1.
Solutionofthe(1+3)-dimensionalinhomogeneousMonge-
Amp`ereequation
(x
2
0
u
2
)
1/2
= εx
3
+c,ε= ±1.
4.Conclusions
Westudy therelationship betweenthe structuralpropertiesof thelow-
dimensional(dimL3)nonconjugatesubalgebrasofthesamerank
oftheLiealgebraofthePoincar´egroupP(1,4)andtheproperties
ofthereducedequationsforthe(1+3)-dimensionalinhomogeneous
Monge-Amp`ereequation.
Atthepresenttime,theconnectionhasbeeninvestigatedbetween
thestructuralpropertiesofthree-dimensionalnonconjugatesubalge-
brasoftheLie algebraofthe groupP(1,4)and theresultsofsymmetry
reductionforthe(1+3)-dimensionalinhomogeneousMonge-Amp`ere
equation.Weobtaineddifferenttypesofthereducedequations.
Inthispaper,wepresentedashortreviewofourresultsconcern-
ingtherelationshipbetweenthestructuralpropertiesofthethree-
dimensionalnonconjugatesubalgebrasoftheLiealgebra ofthe group
P(1,4)andreductionsofthe(1+3)-dimensionalinhomogeneous
Monge-Amp`ereequationtothefirst-orderODEs.
Itisknown[40]thattheLiealgebraofthegroupP(1,4)contain-
sthree-dimensionalnonconjugatesubalgebrasofthefollowingtypes:
3A
1
,A
2
A
1
,A
3,1
,A
3,2
,A
3,3
,A
3,4
,A
3,6
,A
a
3,7
,A
3,8
,A
3,9
.
Fromtheresultsobtaineditfollowsthat
—thereductionstothefirst-orderlinearODEscanbeobtained
usingsomesubalgebrasofthefollowingtypes:3A
1
,A
3,6
.
—thereductionstothefirst-ordernonlinearODEscanbeobtained
usingsomesubalgebrasofthefollowingtypes:3A
1
,A
2
A
1
,A
3,1
,
A
3,2
,A
3,3
,A
3,6
.
Thefutureworkbasedonourcurrentstudywillbedevotedto
theclassificationofsymmetryreductionsofthe(1 + 3)-dimensional
inhomogeneous Monge-Amp`ere equation toalgebraic equations.Some
classesoftheinvariantsolutionswillalsobeconstructed.
DOI:10.4236/am.2020.11110801191AppliedMathematics
V.M.Fedorchuk,V.I.Fedorchuk
ConflictsofInterest
Theauthorsdeclarenoconflictsofinterestregardingthepublication
ofthispaper.
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