Received on October 24, 2011 / Revised on November 17, 2011

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1 Journal of ath-for-industry Vol 4 202A On some roertes of a dscrete hungry Lota-Volterra system of multlcatve tye Yosue Hama Ao Fuuda Yusau Yamamoto asash Iwasa Emo Ishwata and Yoshmasa Naamura Receved on October / Revsed on November 7 20 Abstract Two nds of dscrete hungry Lota-Volterra systems dhlv are nown as dscretzatons of the addtve tye hungry Lota-Volterra system and the multlcatve one By assocatng the dhlv of addtve tye dhlv I and the dscrete hungry Toda equaton dhtoda wth LR transformatons some of the authors gve a Bäclund transformaton between these two systems In ths aer from the dhlv of multlcatve tye dhlv II we frst derve the qd-tye dhlv II Through fndng the ostvty of the qd-tye dhlv II and the LR transformaton assocated wth the dhlv II we resent Bäclund transformatons among the dhlv I the dhlv II and the dhtoda oreover by usng one of the Bäclund transformatons we show asymtotc convergence of the qd-tye dhlv II Keywords Bäclund transformaton LR transformaton asymtotc convergence dscrete hungry Toda equaton dscrete hungry Lota-Volterra system Introducton The ntegrable Lota-Volterra system LV s nown as one of the ordnary dfferental equatons that descrbe redator-rey dynamcs n mathematcal bology In [ 2 3] one of extended LV s resented as du t u t u + t u t dt 2 m t 0 u t 0 u 0 t 0 u m +t 0 u m +t 0 and 2 are called the hungry LV hlv of addtve tye and multlcatve tye resectvely Sometmes and 2 are referred to as the Bogoyavlensy lattces The hlv and 2 are also derved from a satal dscretzaton of the Korteweg-de Vres equaton [4] The dscretzed verson of s resented n [5 6] as u n+ u n + δ n+ u n+ u n + δ n u n + 2 m n 0 0 un 0 0 u n m+ 0 un 0 m+ 3 and another extended LV s gven n [2 3] as dv t v t v + t v t dt 2 m + t 0 v t 0 v 0 t 0 v m +t 0 v m ++ t 0 2 and that of 2 s gven n [6] as v n+ + δ n+ v n+ + δ n 2 m + n 0 m vn vn m where s a ostve nteger : + and u t and v t denote the oulatons of the th seces at the contnuous tme t Eqs and 2 descrbe the cometton such that the th seces s redator of the +th the +2th the +th seces and s rey of the th the 2th the th seces In the case of both and 2 become the orgnal LV As grows larger for the th seces the number of seces of both the reys and the redators ncrease So resectvely Both 3 and 4 are called the dscrete hlv dhlv In ths aer n order to dstngush two nds of the dhlvs we smly refer to 3 and 4 as the dhlv I assocated wth the contnuous hlv of addtve tye and the dhlv II assocated wth the contnuous hlv of multlcatve one 2 resectvely In 3 and 4 δ n reresents the ste sze at the dscrete tme n The varables u n and denote the oulaton of the th seces at the 5

2 6 Journal of ath-for-industry Vol 4 202A-2 dscrete tme n The dhlv I 3 s shown n [7] to have an alcaton for comutng comlex egenvalues of a certan band matrx The dscrete Toda equaton q n+ q n+ + e n+ e n+ e n 0 0 e n q n q n + en + e n 2 m 2 m m 0 n 0 s also a famous ntegrable system Here the suerscrt n s the tme varable as n 3 and 4 and the subscrt denotes the satal varable A study on box and ball system n [8] leads to an extended verson of the dscrete Toda equaton 5 Q n+ + E n+ Q n + E n 2 m Q n+ E n+ Q n + En 2 m 6 E n 0 0 E m n 0 n 0 wth ostve nteger whch s named the dscrete hungry Toda equaton In ths aer for the smlcty we call 6 the dhtoda In [9] a new algorthm for comutng matrx egenvalues s desgned based on the dhtoda 6 Some of the authors n [0] found a relatonsh of deendent varables namely a Bäclund transformaton between the dhlv I 3 and the dhtoda 6 through assocatng these ntegrable systems wth a sequence of LR transformatons of matrces Bäclund transformaton s orgnally derved from the study of dfferental geometry Exlct form of the Bäclund transformaton hels us to understand ntrnsc features of an ntegrable system such as the solutons and symmetry and ts relatonsh wth another ntegrable system [] Here for a nonsngular matrx A the LR transformaton [2] s defned as 5 A LR  RL 7 The st equaton of 7 reresents the LR decomoston of A where L s a lower trangular and R s a unt uer trangular matrx It s to be noted that the LR decomoston where R has unt dagonal entres s unquely gven The 2nd equaton generates  as the matrx roduct RL Let  ˆL ˆR be the LR decomoston of  From 7 we get ˆL ˆR RL Ths tye of equaton aears n the matrx reresentaton of some dscrete ntegrable systems and s called the Lax reresentaton of them The egenvalues of  concde wth those of A So the LR transformaton 7 yelds a smlarty transformaton from A to  namely  RAR For examle n order to comute the egenvalues of a symmetrc trdagonal matrx the quotent dfference qd algorthm emloys a sequence of LR transformatons It s nterestng that the recurson formula of the qd algorthm s just equal to the dscrete Toda equaton 5 However there s no observaton that the dhlv II 4 s assocated wth a sequence of LR transformatons In ths aer we frst assocate the dhlv II 4 wth a sequence of LR transformatons Based on ths result we resent a Bäclund transformaton between the dhlv II 4 and the dhtoda 6 Addtonally a Bäclund transformaton between the dhlv I 3 and the dhlv II 4 s also resented for the case of δ n Wth the hel of the relatonsh among the dhlv I 3 the dhlv II 4 and the dhtoda 6 we next show the asymtotc behavor of the dhlv II 4 as n by usng the convergence roerty of the dhtoda 6 gven n [9] The dhlv II 4 s also shown to be alcable for matrx egenvalue comutaton Ths aer s organazed as follows In Secton 2 we derve a system called the qd-tye dhlv II from the orgnal dhlv II 4 through varable transformaton We also show the ostvty of the qd-tye dhlv II under sutable condtons In Secton 3 we gve a Lax reresentaton for the dhlv II 4 and then relate t to the LR transformaton of a band matrx We also revew the Lax reresentaton for the dhtoda 6 and the LR transformaton assocated wth t In Secton 4 by comarng two LR transformatons n Secton 2 we derve a Bäclund transformaton between the dhlv II 4 and the dhtoda 6 By tang account of the Bäclund transformaton between the dhlv I 3 and the dhtoda 6 gven n [0] we also derve a Bäclund transformaton between the dhlv I 3 and the dhlv II 4 for the case of δ n We nvestgate the asymtotc behavour of the dhlv II varables through the Bäclund transformaton between the dhlv II 4 and the dhtoda 6 The asymtotc behavour of the dhtoda varables s already shown n [9] In Secton 5 we gve numercal examles n order to demonstrate some theorems n the revous sectons Fnally n Secton 6 concluson s resented 2 The qd-tye dhlv II and ostvty of ts varables In ths secton we ntroduce the qd-tye dhlv II whch s derved from the dhlv II 4 and show the ostvty of ts varables For the smlcty we emloy the notatons Φ Φ 2 Φ 3 Φ 4 and Φ 5 defned as Φ : 2 m + } Φ 2 : 2 m + } Φ 3 : 2 m } Φ 4 : 2 m} Φ 5 : 0 } These ndex sets aear throughout ths aer frequently 2 The qd-tye dhlv II Let us ntroduce new varables ω n : + δ n γ n : δ n Φ 8 + Φ 2 9

3 Yosue Hama Ao Fuuda Yusau Yamamoto asash Iwasa Emo Ishwata and Yoshmasa Naamura 7 from the dhlv II varable and the dscrete ste sze δ n From the boundary condton of we have ω n Φ 5 } \ 0} 0 γ n 0 Φ 5 γ n m +j 0 j Φ 5 2 Then these varables satsfy the recurson formula ω n+ + γ n ωn + γ n Φ ω n+ γ n + ωn ++ γn Φ 2 \ m + } 3 Ths s easly checed as follows ω n+ + γ n v n+ ω n ω n+ γ n + [ [ v n+ + δ n+ + δ n + δ n + γ n δ n+ + δ n ω n ++ γn + δ n + + δ n v n+ + δ n + δ n + δ n v n+ ] [ + ] [ δ n ++ δ n + δ n ++ + δ n + ++ ] + Eq 3 has the form smlar to the recurson formula of the qd algorthm 5 In order to dstngush 3 from the dhlv II 4 we herenafter call 3 the qd-tye dhlv II Also we can rewrte the qd-tye dhlv II as ω n+ ω n + γ n γ n γ n ++ γn 4 If ω n + ωn ω n+ for Φ and γ n are gven we can obtan ω n+ for Φ 2 \ } by usng 4 Let for Φ and γ n us assume that ω n > 0 for Φ and γ n > 0 Then ] we can relate γ n to δ n as follows From 9 and 0 we derve + γ n δ n δ n From 5 t holds that δ n + ω n δ n ω n + ω n + δ n + ω n δ + n γ n ω n + γn ω n ω n 5 ω n Hence the condton δ n > 0 s equvalent to 0 < γ n < ω n + 22 Postvty of the qd-tye dhlv II varables We gve a theorem concernng the ostvty of the qd-tye dhlv II varables ω n and γ n Theorem Let us assume that ω n > 0 Φ and 0 < γ n < ω n + then t holds that ω n+ > 0 γ n > 0 Φ Φ 2 Proof In the dscusson for the ostvty of the qd-tye dhlv II varables t s useful to ntroduce an auxlary varable d n defned by d n ω n From and 6 t follows that γ n Φ 6 d n ω n Φ 5 } \ 0} d n + ωn + γn By combnng 7 wth the assumton we have d n > 0 Φ 5 + } \ 0} 7

4 8 Journal of ath-for-industry Vol 4 202A-2 In terms of d n we may rewrte the st equaton of 4 as ω n+ γ n + d n Φ 8 From 4 and 6 we also get the recurson formula for d n and d n ++ as follows d n ++ ωn ++ γn + ωn ++ ω n+ ωn ++ ω n+ ω n+ γ n d n 9 From 4 8 and 9 we get a dfferental form wthout subtracton as follows ω n+ γ n + d n γ n + ωn ++ γn ω n+ d n ++ ωn ++ ω n+ d n 20 By usng 20 reeatedly we can comute ω n+ γ n + and d n ++ for 2 Snce the ntal values ωn for Φ γ n and d n for Φ 5 + } \ 0} are ostve by assumton and there are no subtractons we can conclude that all the comuted varables are ostve 3 LR transformatons assocated wth the dhlv II In ths secton we gve a Lax reresentaton of the dhlv II 4 and then resent a sequence of LR transformatons assocated wth the dhlv II 4 In addton we brefly revew [0] concernng the LR transformaton assocated wth the dhtoda 6 A Lax reresentaton of the qd-tye dhlv II 3 s gven by ˆL n+ ˆRn ˆR n ˆLn 2 where 0 ω n 0 ω n 2 0 ˆL n 0 ω n 0 } 0 } m + }} 0 0 γ n n 0 0 γ 2 ˆR n γ n 0 0 m By focusng on the entres n the both sdes of 2 we can get the qd-tye dhlv II 3 Ths means that 2 s a Lax reresentaton of the qd-tye dhlv II 3 Of course the Lax reresentaton 2 wth 8 and 9 s just equal to that of the dhlv II 4 n [6] It s remarable here that the Lax reresentaton s not always unquely gven In the followng theorem we resent a new Lax reresentaton for the qd-tye dhlv II 3 Theorem 2 As δ n a Lax reresentaton of the qd-tye dhlv II 3 becomes where L n j R n j L n+ j+ R n j+ Rn j L n ω n +j ω n 2 +j γ n +j γ n m +j j+ j Φ 5 22 ω n m+j Proof In 20 let us assume that ω n > 0 for Φ 0 < c γ n ω n + where c s some ostve constant We frst show that the qd-tye dhlv II varables satsfy the followng nequalty d d n d Φ \ + } Φ 3 d n d + } Φ 3 γ γ n γ Φ 2 ω ω n+ ω Φ 25 where d d γ γ ω and ω are some ostve constants that do not deend on γ n We frst consder the case where Φ 5 + } \ 0} From 7 we have d d n d Φ 5 } \ 0} 26

5 Yosue Hama Ao Fuuda Yusau Yamamoto asash Iwasa Emo Ishwata and Yoshmasa Naamura 9 wth d d ω n Smlary from 7 we get d n + d + 27 wth d + ω n + Obvously the assumton leads to γ γ n γ 28 wth γ c and γ ω n + By combnng wth 20 we can rove the followng nequaltes by nducton γ γ n γ Φ 5 + } \ 0 } ω ω n+ ω Φ 5 + } \ 0} 29 In the case where + + } Φ 5 + } \ 0} from the 3rd equaton of and the 2nd equaton of 29 t holds that d d n d + + } Φ 5 } \ 0} 30 d n 2+ d 2+ 3 oreover from 29 t follows that Eqs lead to γ +2 γ n +2 γ γ γ n γ + + } Φ 5 + } \ 0 } ω ω n+ ω + + } Φ 5 + } \ 0} Smlary for + + j + } Φ 3 \ 2} j Φ 5 t follows that d d n d γ γ n γ ω ω n+ ω and for + } Φ 3 \ 2} we have d n d 33 We next consder the case where m } Φ 5 By combnng the 3rd equaton of wth 2 we have d + d n + d + Φ 5 34 From 34 and the st equaton of 20 we have ω + ω n+ + ω + To sum u we obtan 25 Φ 5 By usng 25 we dscuss the behavor of varables n 3 as δ n We frst consder the case of + Φ 3 By usng 9 reeatedly we derve d n + 2 ω n + ω n+ + Eqs 5 and 7 lead to d n + ωn + γn ω n + δ n d n + Φ 3 \ } 35 ω n + + ω n δ + n ω n + ω n 36 As δ n from 25 t follows that d n + 0 n 36 From 35 we derve d n 0 for Φ 3 \ } By + combnng them wth 6 we get lm ω n δ n + γn 0 Φ 3 37 oreover from 8 and d n + 0 for Φ 3 we have lm ω n+ δ n + γn + 0 Φ 3 38 Thus as δ n 3 becomes the trval equaltes ω n+ + + γn ω n + + γn + Φ 3 and ω n+ + γn + ω n + + γn + Φ 3 \ m } Next we consder the cases excet for + Φ 3 n the st and 2nd equatons of 3 We here focus on the roduct of L n+ j+ and R n j+ The and + entres of L n+ j+ R n j+ are gven as resectvely L n+ j+ R n j+ ω n+ +j + γn +j Φ 4 L n+ j+ R n j+ + ω n+ +j γn +j+ Φ 3 Smlarly t follows that R n j L n j+ ω n +j + γn +j Φ 4 R n j L n j+ + ω n ++j+ γn +j Φ 3 j Φ 5 39 j Φ 5 40 j Φ 5 4 j Φ 5 42 Eqs and 4 brng to the st equaton of 3 Also and 42 lead to the 2nd one Ths ndcates that 22 s a Lax reresentaton for the qd-tye dhlv II 3

6 0 Journal of ath-for-industry Vol 4 202A-2 oreover we gve a lemma concernng the relatonsh of the Lax matrces R n+ 0 and R n as δn Lemma As δ n t holds that R n+ 0 R n 43 Proof Obvously from 37 and 38 ω n+ + and ω n+ + γ n+ γn+ γn as + δn So t holds that γ n + as δn Ths leads to 43 Let us ntroduce the matrx gven by the roduct of the Lax matrces L n Ln 2 Ln n 23 and Rn 0 n 24 A n L n Ln 2 L n Rn 0 44 Let us consder R n 0 An R n 0 as a smlarty transformaton of A n by R n 0 Then wth the hel of Theorem 2 we derve R n 0 An R n 0 R n 0 Ln Ln 2 L n L n+ R n Ln 2 Ln 3 L n L n+ L n+ 2 R n 2 Ln 3 L n L n+ L n+ 2 L n+ Rn Ln L n+ L n+ 2 L n+ By combnng t wth Lemma we see that Ln+ Rn 45 R n 0 An R n 0 A n+ 46 Ths means that the egenvalues of A n are nvarant under the tme evoluton from n to n + Eqs 45 and 46 also lead to A n L n Ln 2 L n Rn 0 A n+ R n 0 Ln Ln 2 L n 47 Hence we now that A n+ s gven through the LR transformaton of A n Accordng to 45 the LR transformaton n 47 concdes wth tmes LR transformatons n 22 Let us recall here that 22 s a Lax reresentaton for the dhlv II 4 wth δ n We therefore have the followng theorem Theorem 3 The dhlv II 4 wth δ n generates the LR transformaton from A n to A n+ as n 47 Accordng to [8] the dhtoda 6 satsfes the Lax reresentaton where Q n L n+ R n+ R n L n 48 Q n Q n L n 2 49 Q n m E n R n E n 50 m > 0 Φ 4 and E n > 0 Φ 3 The Lax reresentaton 48 may loo dfferent from that n [8] Actually we can easly get the same Lax reresentaton as n [8] through matrx transoston on both sdes of 48 Let A n be the roduct of the Lax matrces L n L n+ L n+ n 49 and R n n 50 namely A n L n L n+ L n+ R n 5 Then from 48 t follows that R n A n R n R n L n L n+ L n+ L n+ R n+ L n+ L n+2 L n+ L n+ L n++ R n+2 L n+2 L n+ L n+ L n++ L n+2 R n+ A n+ 52 Obvously from 52 the dhtoda 6 gves the smlarty transformaton from A n to A n+ Eq 52 s also rewrtten as A n L n L n+ L n+ R n A n+ R n L n L n+ L n+ 53 Thus the dhtoda 6 has a relatonsh wth the LR transformaton as follows Theorem 4 [0] The dhtoda 6 generates the LR transformaton from A n to A n+ as n 53 4 Bäclund transformatons among the dscrete hungry systems In ths secton by consderng the relatonsh between the two LR transformatons assocated wth the dhlv II 4 and the dhtoda 6 we gve a Bäclund transfomaton between the dhlv II 4 and the dhtoda 6 By referrng to [0] we establsh a Bäclund transformaton between the dhlv I 3 and the dhlv II 4 We also nvestgate the asymtotc behavor of the qd-tye dhlv II 3 wth the hel of the obtaned Bäclund transformaton

7 Yosue Hama Ao Fuuda Yusau Yamamoto asash Iwasa Emo Ishwata and Yoshmasa Naamura 4 The Bäclund transformaton between the dhlv II and the dhtoda We frst show the relatonsh of the matrces n two LR transformatons assocated wth the dhlv II 4 and the dhtoda 6 Lemma 2 For some fxed n let R n 0 R n and L n j+ L n+j j Φ 5 Then t holds that L n+ j+ L n++j R n j+ Rn+j+ j Φ 5 Proof The assumton leads to R n 0 Ln R n L n Let us recall that R n 0 Ln L n+ R n n 22 and R n L n L n+ R n+ n 48 So t follows that L n+ R n L n+ R n+ Recall that the uer bdagonal matrces R n and R n+ have n every dagonal entry Hence by tang account of the unqueness of LR decomoston we get L n+ L n+ R n R n+ Smlarly t s easly roved by nducton for j 2 that L n+ j+ L n++j and R n j+ R n+j+ From 44 and 5 t s obvous that A n A n f L n j+ Ln+j j Φ 5 and R n 0 R n So by usng Lemma 2 we see that A n+ A n+ snce R n j+ R n+j+ j Φ 5 and L n+ j+ L n++j j Φ 5 In other words the evoluton from n to n+ of the dhlv II 4 can generate the LR transformaton gven by the evoluton from n to n + of the dhtoda 6 Let us relace n wth l + j n the suerscrts of the dhlv II and the dhtoda varables Herenafter we consder the evoluton from l to l + by the dhlv II 4 and the dhtoda 6 Let us assume that for some fxed l L l j+ Ll+j j Φ 5 R l 54 0 R l Then from Lemma 2 t follows that L l+ j+ L l++j R l j+ Rl+j+ j Φ 5 55 From the 2nd equaton of 55 and Lemma t holds that R l+ 0 R l+ By focusng on the entres of matrces n 54 we derve E l γ l Q l+j Φ 3 ω l +j Φ 4 j Φ 5 for l 0 oreover from 55 we obtan E l+j+ γ l +j+ Φ 3 j Φ 5 for l 0 To sum u we derve a theorem on the relatonsh of the varables namely the Bäclund transformaton between the qd-tye dhlv II 3 and the dhtoda 6 Theorem 5 A Bäclund transformaton between the qdtye dhlv II 3 wth δ l and the dhtoda 6 s gven by E l+j γ l Q l+j for l 0 +j Φ 3 j Φ 5 ω l +j Φ 4 j Φ 5 It s observed that 8 and 9 are the Bäclund transformaton between the qd-tye dhlv II 3 and the orgnal dhlv II 4 So by combnng t wth Theorem 5 we have a man theorem n ths aer Theorem 6 A Bäclund transformaton between the dhlv II 4 wth δ l and the dhtoda 6 s gven by E l+j Q l+j δ l v l +j+ Φ 3 j Φ 5 + δ l v l +j Φ 4 j Φ 5 for l 0 v l +j 42 The Bäclund transformaton between the dhlv I and the dhlv II Let us ntroduce the new varables U n V n u n δ n + δ n u n + δ n u n Φ 2 m } Φ m + } 56 n terms of the dhlv I varable u n Then the dhlv I 3 can be rewrtten as U n+ U n+ V n U n m +j+ + V n ++ U n ++ + V n U n : 0 V + Φ 2 V n + Φ 2 m } n m ++j+ : δ j Φ n 5 57 Eq 57 s named the qd-tye dhlv I n [3] Eq 56 s a Bäclund transformaton between the orgnal dhlv I 3 and the qd-tye dhlv I 57 Some of the authors n [0] gve a Bäclund transformaton between the qd-tye

8 2 Journal of ath-for-industry Vol 4 202A-2 dhlv I 57 wth δ n and the dhtoda 6 as follows U l U l +j+ Q l+ 2 Φ 3 j Φ 5 Φ 4 Q l++j+ E l+j 58 By combnng the st equaton of 58 wth Theorem 5 for l 0 we get U l ω l + Φ 4 59 Smlarly from Theorem 5 and the 2nd of 58 we derve U l +j+ Q l+j+ Q l+ Q l+j+2 Q l++j Q l+ Q l++ Q l++2 E l+j ω l +j+ ωl +j+2 ωl + ω l+ ω l+ + ωl+ ω l+ +j γl +j +2 ωl+ +j 2 Q l++j 2 By tang account of the 2nd equaton of 3 we successvely rewrte U l +j+ as U l +j+ ωl +j+ ωl +j+2 ωl + ω l+ ω l ω l ω l+ + ωl+ γ l +j ωl ++j +j+ ωl ω l+ +j+2 ωl γ l + +2 ωl+ + ω l ++j ωl ++j +j+ ωl Note here that γ l that U l +j+ +j 2 ω l ++2 ωl ++j 2 +j+2 ωl + γ l ω l ++ ω l ++2 ωl ++j 2 ω l ++j ωl ++j From 59 and 60 we have ω l + as δn So t follows ω l ++j+ Φ 3 j Φ 5 60 U n ω n + Φ 2 m } 6 By combng 6 wth 8 and 56 t follows that u n + δ n u n δ n r + r + δ n r + δ n r0 + r +r 62 From the boundary condton of u n and the case where n 62 leads to u n + Smlarly by consderng the cases where 2 3 m we have u n vn + The above dscusson leads to the followng theorem Theorem 7 As δ n a Bäclund transformaton between the dhlv I 3 and the dhlv II 4 s gven by for n 0 u n + Φ 2 m } 43 The asymtotc behavor of the qd-tye dhlv II varables We next clarfy the asymtotc behavor of the qd-tye dhlv II varables by combnng Theorem 5 wth the asymtotc behavor of the dhtoda varables gven n [9] Let us agan relace n wth l + j j Φ 5 n the suerscrt of the dhtoda varable Of course the lmt of l s equvalent to that of n A mnor change of the lmt n [9] brngs to the followng theorem wth resect to the convergence of the dhtoda varables Theorem 8 cf[9] Let Q 0 > 0 Q > 0 Q > 0 Φ 4 and E 0 > 0 Φ 3 As l the lmts of Q l+j and E l+j lm l are gven by Q l+ C lm l El+j 0 Φ 3 Φ 4 j Φ 5 63 where C s some constant and C C 2 C m > 0 In [9] t s also shown that Q l+j } l0 s a Cauchy sequence Ths mles that Q l+j converges to some constant C j > 0 as l namely lm l Ql+j C j Φ 4 j Φ 5 64

9 Yosue Hama Ao Fuuda Yusau Yamamoto asash Iwasa Emo Ishwata and Yoshmasa Naamura 3 Snce t s shown n Theorem 5 that Q l+j usng t n 64 we get lm l ωl +j C j γ l +j shown n Theo- By tang account that E l+j rem 5 from 63 we derve Φ 4 lm l γl +j 0 Φ 3 ω l +j by j Φ 5 65 j Φ 5 66 It s remarable that 65 and 66 show the convergence of ω n and γ n excet for + Φ 3 as l We next study the convergence of ω l + and γl + Eqs 37 and 66 wth j 0 leads to lm l ωl + 0 Φ 3 67 From 38 and 67 t follows that lm l γl + 0 Φ 3 We summarze the asymtotc behavor of the qd-tye dhlv II varables as follows Theorem 9 Let us assume that ω 0 > 0 Φ Then the lmts of the qd-tye dhlv II varables as δ n are lm n ωn +j C j Φ 4 lm n ωn + 0 Φ 3 lm n γn 0 Φ 2 j Φ 5 5 Numercal Examles In ths secton we numercally observe some roertes of the qd-tye dhlv II 3 shown n the revous sectons We easly realze numercal roertes of the orgnal dhlv II 4 through those of the qd-tye dhlv II We frst demonstrate the asymtotc behavor of the qdtye dhlv II 3 shown n Theorem 9 numercally Let ω 0 ω ω 0 3 ω 0 4 ω ω 0 6 ω 0 7 ω 0 8 and 2 m 3 δ n 0 2 resectvely n the qd-tye dhlv II 3 It s emhaszed here that the qdtye dhlv II varables excet for ω 0 deend on the value of δ n through γ n as shown n 5 Fgures and 2 show the behavor of ω n and ωn 3 6 γ n for n 0 9 resectvely We see from Fgures and 2 that as n becomes lager ω n and ωn 3 6 aroach some ostve constants and zero resectvely Ths numercal result agrees wth Theorem 9 We next gve a numercal examle n order to confrm the Bäclund transformaton shown n Theorem 5 between the dhlv II 4 and the dhtoda 6 as δ n Let Q and E wth 3 and m 4 n the dhtoda 6 oreover let Fgure : A grah of the teraton number n x-axs and the values of ω n ω n 2 ω n 4 ω n 5 ω n 7 and ω n 8 y-axs : ω n : ω n 2 dotted lne: ω n 4 dashed lne: ω n 5 : ω n 7 and : ω n Fgure 2: A grah of the teraton number n x-axs and the values of ω n 3 ω n 6 and γ n for yaxs : ω n 3 : ω n 6 : γ n dashed lne: γ n 2 dotted lne: γ n 3 + : γ n 4 : γ n 5 and : γ n 6 ω 0 +j j 0 2 and ω wth 3 and m 4 n the qd-tye dhlv II 3 We consder two cases where δ l 05 and δ l 0 2 for l 0 In Tables and 2 the st the 2nd and the 3rd columns denote the teraton number l the dhtoda varables and the qd-tye dhlv II ones resectvely Table llustrates that n the case where δ l 05 l the dhtoda and dhlv II varables concde wth each other n only a few dgts On the other hand Table 2 shows that there are lttle dfference

10 4 Journal of ath-for-industry Vol 4 202A-2 Table : Values of Q l ω l and E l γ l n the case where δ l 05 for l 0 5 and 50 l Q l ω l Table 2: Values of Q l ω l and E l γ l n the case where δ l 0 2 for l 0 5 and 50 l Q l ω l l E l γ l E E 2 l E l γ l E E 2 between Q l E l and ω l γ l resectvely Tables and 2 show that Theorem 5 aroxmately holds for a suffcently large δ l In [9] some of the authors have roosed an algorthm for comutng egenvalues of totally nonnegatve matrces for whch all the mnors are nonnegatve So from the Bäclund transformaton among the dhlv II 4 the qdtye dhlv II 3 and the dhtoda 6 shown n Secton 4 t s easly exected that the dhlv II 4 and the qdtye dhlv II 3 are alcable for comutng the egenvalues of a totally nonnegatve matrx In artcular lm n ωn + Φ 4 gve the egenvalues of the totally nonnegatve matrx A 0 6 Concluson In ths aer we frst ntroduce the qd-tye dhlv II and show the ostvty of ts varables We also gve a new Lax reresentaton for the dhlv II As δ n t s observed that the Lax reresentaton for the dhlv II s related to the LR transformaton for a band matrx In other words the tme evoluton of the dhlv II wth δ n corresonds to the LR transformaton We next exlan how to assocate the dhtoda wth the LR transformaton Remarably the dhlv II wth δ n s assocated wth the same form of LR transformaton assocated wth the dhtoda By dentfyng two these LR transformatons we fnally obtan a Bäclund transformaton between the dhlv II and the dhtoda Addtonally through consderng a Bäclund transformaton between the dhlv I and the dhtoda n [0] we establsh a Bäclund transformaton between the dhlv I and the dhlv II for the case of δ n We therefore have Bäclund transformatons among the dhlv I the dhlv II and the dhtoda Wth the hel of the Bäclund transformaton between the dhlv II and the dhtoda we nvestgate the asymtotc convergence of the qd-tye dhlv II varables as n through that of the dhtoda We fnally gve some numercal examles whch demonstrate our theoretcal results We gve a comment that the dhlv II and the qd-tye dhlv II are alcable for comutng egenvalues of a totally nonnegatve matrx A future wor s to derve the Bäclund transformatons among the dhlv I the dhlv II and the dhtoda n the case where δ n s fnte Acnowledgements Ths wor s artally suorted by Grants-n-Ad for Young Scentst B No Scentfc Research C No and Scentfc Research A No of Jaan Socety for the Promoton of Scence JSPS References [] K Narta: Solton soluton to extended Volterra equatons J Phys Soc Jn [2] O I Bogoyavlensy: Some constructons of ntegrable dynamcal systems ath USSR Izv

11 Yosue Hama Ao Fuuda Yusau Yamamoto asash Iwasa Emo Ishwata and Yoshmasa Naamura 5 [3] Y Itoh: Integrals of a Lota-Volterra system of odd number of varables Prog Theor Phys [4] V G Paageorgou and F W Njhoff: On some ntegrable dscrete-tme systems assocated wth the Bogoyavlensy lattces Physca A [5] R Hrota and S Tsujmoto: Conserved quanttes of dscrete Lota-Volterra equaton RIS Koyurou [6] Y B Surs: Integrable dscretzatons of the Bogoyavlensy lattces J ath Phys [7] A Fuuda E Ishwata Iwasa and Y Naamura: The dscrete hungry Lota-Volterra system and a new algorthm for comutng matrx egenvalues Inverse Problems [8] T Tohro A Naga and J Satsuma: Proof of soltoncal nature of box and ball systems by means of nverse ultra-dscretzaton Inverse Problems [9] A Fuuda E Ishwata Y Yamamoto Iwasa and Y Naamura: The ntegrable dscrete hungry systems and ther related matrx egenvalues Tech Re Kyoto Unv [0] A Fuuda Y Yamamoto Iwasa E Ishwata and Y Naamura: A Bäclund transformaton between two ntegrable dscrete hungry systems Phys Lett A [] C Rogers and W K Schef: Bäclund and Darboux Transformatons Geometry and ordern Alcatons n Solton Theory Cambrdge Unversty Press 2002 [2] H Rutshauser: Soluton of egenvalue roblems wth the LR-transformaton Natl Bur Stand Al ath [3] A Fuuda E Ishwata Iwasa and Y Naamura: On the qd-tye dscrete hungry Lota-Volterra system and ts alcaton to the matrx egenvalue algorthm JSIA Lett asash Iwasa Deartment of Informatcs and Envronmental Scence Kyoto Prefectural Unversty -5 Naarag-cho Shmogamo Sayo-u Kyoto Jaan E-mal: masaatuacj Emo Ishwata Deartment of athematcal Informaton Scence Toyo Unversty of Scence -3 Kagurazaa Shnjuu-u Toyo Jaan E-mal: shwataatrstusacj Yoshmasa Naamura Graduate School of Informatcs Kyoto Unversty Yoshda-Honmach Sayo-u Kyoto Jaan E-mal: ynaaatyoto-uacj Yosue Hama Graduate School of Scence Toyo Unversty of Scence -3 Kagurazaa Shnjuu-u Toyo Jaan E-mal: j4062atedagutusacj Ao Fuuda Deartment of athematcal Informaton Scence Toyo Unversty of Scence -3 Kagurazaa Shnjuu-u Toyo Jaan E-mal: afuudaatrstusacj Yusau Yamamoto Graduate School of System Informatcs Kobe Unversty - Rooda-cho Nada-u Kobe Jaan E-mal: yamamotoatcsobe-uacj