Wronskian Determinant Solutions for the (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation

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1 Jornal of Appled Mahemacs and Physcs Pblshed Onlne ovember 0 (hp:// hp://d.do.org/0.46/jamp Wronskan Deermnan Solons for he ( + )-Dmensonal Bo-Leon-Manna-Pempnell Eqaon Hongca Ma Yongbn Ba Deparmen of Appled Mahemacs Dongha Unversy Shangha Chna Emal: hongcama@homal.com Receved Ags 4 0; revsed Sepember 4 0; acceped Ocober 0 Copyrgh 0 Hongca Ma Yongbn Ba. Ths s an open access arcle dsrbed nder he Creave Commons Arbon Lcense whch perms nresrced se dsrbon and reprodcon n any medm provded he orgnal work s properly ced. ABSTRACT In hs paper we consder ( + )-dmensonal Bo-Leon-Manna-Pempnell eqaon. Based on he blnear form we derve eac solons of ( + )-dmensonal Bo-Leon-Manna-Pempnell (BLMP) eqaon by sng he Wronskan echnqe whch nclde raonal solons solon solons posons and negaons. Keywords: ( + )-Dmensonal Bo-Leon-Manna-Pempnell Eqaon; The Wronskan Technqe; Solon; egaon; Poson. Inrodcon The Wronskan echnqe s nrodced by Freeman and mmo []. Afer ha many researches are based on he Wronskan echnqe. The ( + )-dmensonal BLMP eqaon was frs derved n []: 0 y y y y y and sbscrps represen paral dfferenaon wh respec o he gven varable. Ths eqaon was sed o descrbe he ( + )-dmensonal neracon of he Remann wave propagaed along he y-as wh a long wave propagaed along he -as. The Panlevé analyss La pars Bäcklnd ransformaon symmery smlary redcons and new eac solons of he ( + )-dmensonal BLMP eqaon are gven n [-4]. In [5] based on he bnary Bell polynomals he blnear form for he BLMP eqaon s obaned. ew solons of ( + )-dmensonal BLMP eqaon from Wronskan formalsm and he Hroa mehod are obaned n [67]. The ( + )-dmensonal BLMP eqaon y z y z y z 0 y z whch was nrodced n [8] has he blnear form () () DDDDDD DD f f 0 () y z y z js by sbsng ln f y z no eqaon () he blnear dfferenal operaor D s defned by Hroa [9] as D D a m n m b n a s y b s y m n s y. Wronskan Formlaon s0 y0 Solons deermned by ln f o he Eqaon () are called Wronskan solons and f W (4) 0 j j. (5) j

2 H. C. MA Y. B. BAI 9 Lemma Dab Dcd Dac Dbd Dad Dbc 0 (6) here D s w mar and abcd are n-dmensonal colmn vecors. Lemma Se b j n o be an n-dmensonal j colmn vecor and b no o be zero. Then we have r j n o be a real consa n j rbb b bb rb a (7) j j rb T j rb j rb j rbj. Lemma The followng eqales hold: n j. (8) Proposon. Assmng ha yz 0 y z ) has connos de- ( rvave p o any order and sasfes he followng lnear dfferenal condons 4 (9) j j y z hen f defned by Eqaon (4) solves he bl- near Eqaon (). Proof. Usng he condons (9) we ge ha f f f z y f f f z y f f f 4 z y f f f 5 4 z y f f y fz f. Hence we have DDDDDD DD f f y z y z (0) 4 6. Wh he help of Lemma and Lemma we oban () 6. Sbsng Eqaon () no Eqaon (0) and sng lemma we ge DD y DD z DD y DD z f f 4 0. Therefore we have shown ha f solves Eqaon (4) nder he lnear dfferenal condons (9) The correspondng solon of Eqaon () s f. () f. Wronskan Solons In wha follows accordng o [0-] we wold lke o presen a few specal Wronskan solons o he ( +

3 0 H. C. MA Y. B. BAI ) -dmensonal Bo-Leon-Manna-Pempnell eqaon by solvng he lnear condons (9). I s well known ha he correspondng Jordan form of a real mar J 0 J A () 0 J m have he followng wo ype of blocks: ) ) J J 0 (4) 0 k k A 0 I A 0 I A ll 0 A I 0 (5) and are all real consans. The frs ype of blocks have he real egenvale wh alge- brac mlplcy of blocks have he comple egenvale wh algebrac mlplcy l. n k k.. Raonal Solons Sppose A have he frs ype of Jordan blocks A 0 0 and he second ype (6) In hs case f he egenvale 0 correspondng o he followng form: A (7) 0 0 from he condon (9) we ge 0 4 y z. (8) are all polynomals n yz and and a general Wronskan solon o he ( + ) dm en- sonal Bo-Leon-Manna-Pempnell Eqaon () ln W k s called a raonal Wronskan solon. From Eqaon (8) we ge y z (9) 0 4. (0) Solvng Eqaon (0) by sng Maple we ge he followng formlas: Smlarly by solvng. C yz C 0 4 y z () () hen wo specal raonal solon of lower-order are obaned afer seng some negral consans o be zero. ) Zero-order: Takng C yzc he correspondng Wronskan deermnan and he assocaed raonal Wronskan solon of zero-order read f W C y z lnw C C yz C C () (4) C C are arbrary consans. ) Frs-order: Takng C yz C we can have C 6 z y y C 6Cz6C 6Cy 6 Cy 6Cz 6Cy C4 Cz Cz. 6 (5) zy yz 4z zy Then he correspondng Wronskan deermnan and raonal Wronskan solon of frs-order are f W P ln W C yzyz y z P CC yzc P P C y z z y z y zy yz y z 4 CC yyzy y z C yz C C CC 4

4 H. C. MA Y. B. BAI and C C C C 4 are arbrary real consans. Smlarly we can oban more hgher order raonal Wronskan solons... Solons egaons and Posons... Solons If A becomes o he followng form A 0 0 (6) he egenvalce 0. Sbsng he form of epresson (6) no Eqao n (9) he followng sysem of dfferenal eqaons s obaned 4 y z (7) By solvng sysem (7) we ge he n-solon solon of Eqaon () ln W (8) wh beng defned by cosh 4 y z odd (9) snh 4 y z even 0 are arbrary consans. We presen he -solon and -solon solons ln cosh y z4 anh y z4 lnw cosh y z4 snh y z4 P Q P coh y z4 Q anh y z4 Smlarly we can oban -solon 4-solon solon and n-solon.... egaons and Posons 0 J becomes o he foll- If he egenvale owng form J We sar from he egenfcon 0 0 k k (0) whch s dee- rmned by 4 y z () General solon o hs sysem n wo cases of 0 and 0 are Ccosh y z4 C snh y z4 0 ( ) Ccos y z4 C4sn y z 4 0 () respecvely C C C and C4 are arbrary real consans. When 0 we ge negaon solon and when 0 we ge poson solons. To consrc Wronskan solons correspondng o Jordan blocks of hgher-order we se he basc dea developed for he KdV eqaon [0]. Dfferenang (9) wh respec o we can fnd ha he vecor fncon sasfes k! k! 0 0 kk T () (4)

5 H. C. MA Y. B. BAI 4 (5) y z denoes he dervave wh respec o and k s an ar brary nonnegave neger. Therefore hrogh hs se of egenfncons and Eqaon () a Wronskan solon of order k o Eqaon () s presened as: (6) k lnw! k! whch corresponds o he frs ype of Jordan blocks wh a nonzero real egenvale. In wha follows several eac solons of lower-order are presened o he ( + )-dmensonal Bo-Leon- Manna-Pempnell eqaon as y z 4. y z4. -negaon -negaon ln cosh anh y z4 -poson lncos y z4 an y z4 lnw cosh cosh y z4 W 4 cosh 4 cosh cosh snh y z ln cos cos -poson cos sn y z.. Ineracon Solons We are now presenng eamples of Wronskan neracon solons among dfferen knds of Wronskan solons o he ( + )-dmensonal Bo-Leon-Manna- Pempnell eqaon. Le s assme ha here are wo ses of egenfncons ; k l (7) assocaed wh wo dfferen egenvales and respecvely. A Wronskan solon k l ln W ; (8) s sad o be a Wronskan neracon solon beween wo solons deermned by he wo ses of egenfnc- ons n (7). In fac we ca n have more general Wronskan neracon solons among hree or more knds of solons sch as raonal solons posons solons negaons breahers and compleons. In wha follows we wold lke o show a few specal Wronskan neracon solons dependng on raonal solon posons and solons. Frsly we choose hree dfferen ses of specal egenfncons: raonal yz solon cosh y z4 poson cos y z4 0 0 are consans. Three Wronskan neracon deermnans beween any wo of a raonal solon a sngle solon and a sngle poson are obaned as

6 H. C. MA Y. B. BAI W W solon y z solon y zsn cos raonal snh cosh W raonal poson poson cosh snh cos snh 4 y z y z4. Frher he correspondng Wronskan neracon solons are rs rp Wraonal solon yzcosh yzsnh cosh ln ln W raonal poson yz y zsn cos cos ln W sp solon poson cosh cos cosh snh cos snh 4 y z y z4. The followng s one Wronskan neracon deermnan and solon nvolvng he hree egenfncons. The Wronskan deermnan s W raonal solon poson yz snh cos sn cosh so ha s correspondng Wronskan solon reads as rsp q ln W raonal solon poson p snh cos sn cosh snh sn snh cos cosh sn p yz q yz wh 4 y z y z4. 4. Conclson In hs paper by sng he Wronskan echnqe we have derved he Wronskan deermnan solon for he ( + )-dmensonal Bo-Leon-Manna-Pempnell eqaon whch descrbes he fld propagang and can be consdered as a model for an ncompressble fld. Moreover we obaned some raonal solons solon solons posons and negaons of hs eqaon by solvng he reslan sysems of lnear paral dfferenal eqaons whch garanee ha he Wronskan deermnan solves he eqaon n he blnear form. The presened solons show he remarkable rchness of he solon space of he ( + )-dmensonal Bo-Leon-Manna-Pempnell eqaon. 5. Acknowledgemens The work s sppored by aonal aral Scence Fondaon of Chna (projec o. 7086) he Fnd of Scence and Technology Commsson of Shangha M- (projec o. ZX ) and he Fn- ncpaly damenal Research Fnds for he Cenral Unverses. REFERECES []. C. Freeman and J. J. C. mmo Solon Solons of he Koreweg-de Vres and Kadomsev-Pevashvl Eqaons: The Wronskan Technqe Physcs Leers A Vol. 95 o. 98 pp. -. hp://d.do.org/0.06/ (8) [] M. Bo J. J.-P. Leon and F. Pempnell On he Specral Transform of a Koreweg-de Vres Eqaon n Two Spaal Dmensons Inverse Problems Vol. o. 986 pp hp://d.do.org/0.088/066-56///005 [] C.-J. Ba and H. Zhao ew Solary Wave and Jacob Perodc Wave Ecaons n (+)-Dmensonal Bo- Leon-Manna-Pempnell Sysem Inernaonal Jornal of Modern Physcs B Vol. o pp hp://d.do.org/0.4/s x [4] Y. L and D. L ew Eac Solons for he (+)- Dmensonal Bo-Leon-Manna-Pempnell Eqaon Ap-

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