On the Semi-Discrete Davey-Stewartson System with Self-Consistent Sources

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1 Jounal of Applied Mahemaics and Physics Published Online May 25 in SciRes. hp:// hp://dx.doi.og/.4236/jamp On he Semi-Discee Davey-Sewason Sysem wih Self-Consisen Souces Gegenhasi School of Mahemaical Science Inne Mongolia Univesiy Hohho China gegen@amss.ac.cn Received Apil 25; acceped 5 May 25; published May 25 Copyigh 25 by auho and Scienific Reseach Publishing Inc. his wok is licensed unde he Ceaive Commons Aibuion Inenaional License (CC BY). hp://ceaivecommons.og/licenses/by/4./ Absac A diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces is consuced using he souce geneaion pocedue. We obseve how he esuling coupled discee sysem educes o he ideniies fo deeminan by pesening he Gam-ype deeminan soluion and Casoai-ype deeminan soluion. eywods Diffeenial-Diffeence Davey-Sewason Sysem Souce Genealiaion Pocedue Discee Gam-ype Deeminan Casoai-ype Deeminan. Inoducion he sudy of discee inegable sysem has become an acive aea of eseach fo ove hiy yeas. Vaious inegable disceiaion mehods have been poposed o poduce he discee analogues of inegable sysems. One poweful echnique o find he inegable disceiaion is he Hioa s bilinea mehod []-[6]. he adiional Hioa s disceiaion of inegable equaions elies on gauge invaiance and solion soluions while he modified Hioa s appoach [5] [6] emphasies on disceiing inegable bilnea equaions such ha he esuling discee bilinea equaions have bilinea Bäcklund ansfomaions. he Davey-Sewason sysem is an inegable ( 2 ) + -dimensional genealiaion of he nonlinea Schödinge sysem. In [7] he auhos applied he modified Hioa s appoach o he Davey-Sewason sysem o poduce an inegable diffeenial-diffeence Davey-Sewason sysem which is chaaceied by deeminan soluions bilinea Bäcklund ansfomaion and lax pai. his diffeenial-diffeence Davey-Sewason sysem also can be deived as a educion of a ( 2 ) + -dimensional genealiaion of he Ablowi-Ladik laice [8]. Since he pioneeing woks of Mel nikov [9] he solion equaions wih self consisen souces have eceived consideable aenion. Solion equaions wih self consisen souces ae inegable coupled genealiaion of he oiginal solion equaions and some of such ype of equaions have found impoan physical applicaions. A va- How o cie his pape: Gegenhasi (25) On he Semi-Discee Davey-Sewason Sysem wih Self-Consisen Souces. Jounal of Applied Mahemaics and Physics hp://dx.doi.og/.4236/jamp

2 iey of mehods have been poposed o deal wih hese solion equaions wih souces such as invese scaeing mehods [9]-[3] Daboux ansfomaion mehods [4]-[7] Hioa s bilinea mehod and Wonskian echnique [8]-[28] ec. Howeve mos esuls have been achieved in coninuous case. Compaaively less wok has been done in discee case. In view of his unsaisfacoy siuaion i would be ineesing o poduce new discee solion equaions wih self consisen souces. In [27] a diec mehod called he souce genealiaion pocedue was poposed o consuc and solve he solion equaions wih self consisen souces. In his pape we apply he souce genealiaion pocedue o consuc and solve he diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces. he ouline of his pape is as follows. In Secion 2 he diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces is poduced and is Gam-ype deeminan soluions ae pesened. In Secion 3 he Casoai-ype deeminan soluions o he diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces is deived. Finally Secion 4 is devoed o a conclusion. 2. Consucing he Diffeenial-Diffeence Davey-Sewason Sysem wih Self-Consisen Souces In [7] a diffeenial-diffeence Davey-Sewason sysem which is an inegable disceiaion of he DSI sysem is poposed and he double-casoai and Gammian deeminans soluions o his discee Davey-Sewason sysem ae deived. In his secion we fis eview he Gammian deeminan soluions fo he discee Davey- Sewason sysem and hen apply he souce geneaion pocedue o his sysem o poduce a diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces. he diffeenial-diffeence Davey-Sewason sysem eads [7] un ( ) + un ( + ) 2u uk ( ) + uk ( + ) 2u 2 2 iv + α e v n + α e v k+ α + α v = () un ( ) + un ( + ) 2u uk ( ) + uk ( + ) 2u 2 2 iw+ α e wn+ + α e wk α + α w= (2) un ( + k+ ) + u uk ( + ) un ( + ) 2 e + v k+ w n+ = (3) whee α α 2 and 2 ae consans. In Equaions ()-(3) and in he following we always use a noaional simplificaion fo f( nk ) by wiing explicily a discee vaiable only when i is shifed fom is posiion. Fo example ( ) ( ) ( ) ( ) f f n k f n+ f n+ k f k f n k f n+ k f n+ k. If we apply he dependen vaiables ansfomaions i( α + α ) i( α + α ) = ln = e = e (4) 2 2 u F v GF w HF Equaions ()-(3) can be ansfomed ino he following bilinea Equaions [7] [8]: whee as usual he bilinea opeaos Dn D k id + αe + α2e G F = (5) Dn D k id αe α2e + + F H = (6) 2( Dn Dk ) 2( Dn+ Dk ) 2( DnD k ) 2 e e F F + e G H = (7) D and exp( D ) m Dab ab exp δ [28] ae defined as: m n ( ) = ( δd ) ab an ( δ) bn ( δ) n +. he Gammian deeminan soluions fo he diffeenial-diffeence Davey-Sewason sysem (5)-(7) is given by [7]: F = C+Ω = F (8) 479

3 F ( n ) G = H = α 2α2 ( n ) Φ + F Ψ whee F is a ( M N) ( M N) C = c µν is a ( M N) ( M N) c µν ( µν = 2 M + N) Ω is a ( M N) ( M N) ae M + N column vecos + + maix (9) + + maix of consan elemens + + maix wih block sucue and ΦΦΨΨ φ ( n) φ j ( n) d Ω= + ψs ψ l d ( n) ( φ ( n) φm ( n) ) ( n) ( φ ( n) φ M ( n) ) ( ψ φn ) ( ψ φ N ) Φ = ; Φ = ; Ψ = ; Ψ = ; wih φ ( n ) φ ( n ) ψ ( k ) ψ k( n ) j { M} sl { N} saisfying he following equaions: j s l ( n) ( n) φ φ j i = αφ ( n ) i = αφ j ( n) () ψs ψ l i = αψ 2 s( k ) i = αψ 2 l. () We ae now in a posiion o consuc he diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces by applying he souce geneaion pocedue. Fisly we change Gammian deeminan soluions (8)- () of Equaions (5)-(7) o he following fom: F = C +Ω = F (2) ( n ) G = H = α 2α2 ( n ) F Φ + F Ψ ( µν ) whee he ( M + N) ( M + N) maix C( ) c ( ) (3) = saisfies cµ ( ) µ = ν and µ M + N cµν ( ) = (4) cµ ohewise wih cµ ( ) being an abiay funcion of being a posiive inege and Ω ΦΦΨΨ ae defined as befoe. Using Equaions ()-() we can calculae he -deivaives of he F G H in (2)-(3) in following way: F Ψ F Φ F = c ( ) A + = ig = i c ( ) + i k + α = ( k + ) ( n ) ( n) ( k ) ( n) ( n) ( n ) F Φ + Ψ A Φ ( n+ ) Ψ (5) F Φ + Φ F Φ ( n+ ) F Φ( n) i ( k+ ) α2 α ( k+ ) ( n) (6) 48

4 A kl denoes a maix esuling fom eliminaing he k h ow and l h column fom he maix F and k n k espec- whee n ively. ( n) F Ψ + Ψ A Φ ( n+ ) ih = i c ( ) + i ( n + ) 2α2 = ( k + ) ( k ) F Ψ + Φ F Ψ F Ψ ( k + ) i n+ + α2 + α ( n+ ) ( n) ( n) Φ Ψ ( ) denoe vecos esuling fom eliminaing he h elemen fom Φ (7) F n Ohe funcions appeaing in Equaions (5)-(7) such as G( n ) F( n+ ) G( k + ) F( k ) H( n+ ) F( k+ ) H G( k + ) F( n+ k+ ) can also be expessed in ems of Gammian deeminans which ae he same as he esuls given in [7]. Subsiuing Equaions (5) (7) and G( n ) F( n+ ) G( k + ) F expessed by means of Gammian deeminans given in [7] ino he lef side of Equaion (6) and hen applying he Jacobi ideniies fo he deeminans [28] we finally obain A Φ n+ F Φ ( n + ) α = ( k + ) ( k + ) i c F A Using he Jacobi ideniies fo he deeminans again Equaion (22) is equal o A i c A n α (8) Φ ( + ) = + (9) whee A A j denoe maices esuling fom eliminaing he h ow and jh column especively fom he maix F. If we inoduce wo new fields P Q fo = 2 defined by A P ( = c Q = c A Φ n+ ) (2) + hen we have shown ha F G given in (2)-(3) and P Q = 2 given in (2) saisfy he following bilinea equaion: id G F i P Q. (2) Dn Dk + αe + α2e = α = In he same way subsiuing (5) (7) and H( n+ ) F( n ) H G k + expessed by means of Gammian deeminans given in [7] ino he lef side of he Equaion (6) and hen applying he Jacobi ideniies fo he deeminans we finally obain F Ψ ( k + ) A Φ n+ α = ( + ) ( + ) i c A F. (22) 2 2 n k Using he Jacobi ideniies fo he deeminans again Equaion (22) is equal o A i c A Ψ k+ 2α 2 = +. (23) ( n ) If we inoduce anohe wo new fields J L fo = 2 defined by A J = c ( ) L = c ( ) A Ψ ( k + ) (24) + ( n ) 48

5 hen we have shown ha F H given in (2)-(3) and J L = 2 given in (24) saisfy he following bilinea equaion: id F H i J L. (25) Dn Dk + αe + α2e = 2α2 = hee ae moe quadaic elaions beween he fields inoduced. Fo example he deeminan ideniies ( n) ( n ) A Φ + F Φ ( n+ ) A F Φ ( n+ ) A + F + + = (26) ( n) ( k + ) ( k + ) Φ n Φ and ( n) ( n) ( n ) F Φ + A Ψ k+ Φ n+ A Ψ ( k+ ) + F Φ ( n) F Ψ + + A Φ ( n+ ) = fo = 2 yield he bilinea equaions and and Similaly bilinea equaions e 2 D n e 2 D n e 2 D n (27) F P + i G J = (28) α e e F L + iα e H Q =. (29) 2 D n 2 D n 2 D n D k 2 D k 2 D k 2 α e e F Q iα e G L = (3) 2 D k 2 D k 2 D k 2 e e F J i e H P = (3) fo = 2 can be deived fom he deeminan ideniies and F Ψ A Φ ( n+ ) + F + ( n ) Ψ + F Φ + + A Ψ = + F Ψ A ( n ) A ( n ) A Φ n+ Ψ k F Ψ A ( n ) + F Φ + + =. (33) he deeminan ideniies (26)-(27) and (32)-(33) ae special cases of he pfaffian ideniy [28] ( a a2 an αβγ )( a a2 an δ) ( a a2 an αβδ )( a a2 an γ) ( a a a αγδ)( a a a β) ( a a a βγδ)( a a a α) + =. 2 N 2 N 2 N 2 N Ψ So bilinea Equaions (7) (2) (25) and (28)-(3) fo = 2 consuc he diffeenial-diffeence Da- (32) (34) 482

6 vey-sewason sysem wih self-consisen souces and funcions F G H and P Q J L fo = 2 in Equaions (3) (2) (2) (24) ae he Gam-ype deeminan soluions of he diffeenial-diffeence Davey- Sewason sysem wih self-consisen souces. Unde he dependen vaiable ansfomaions G H P Q L J u = ln F v = w= P = Q = L = J = F F F F F F he bilinea Equaions (7) (2) (25) and (28)-(3) fo = 2 ae ansfomed ino he following nonlinea equaions: un ( + k+ ) + u un ( + ) uk ( + ) 2 e + v k+ w n+ = (35) iv v n v k i PQ (36) un ( ) + un ( + ) 2u uk ( ) + uk ( + ) 2u + α ( e ) + α2 ( + e ) = α = iw wn+ wk = i JL (37) un ( ) + un ( + ) 2u uk ( ) + uk ( + ) 2u α ( e ) α2 ( e ) 2α2 = α ivj n + + P P n + = (38) ( L L ( n )) iα Qw ( n ) = (39) 2 2 ( ) α α Q Q k i vl k = (4) 2 i2p w k + J k J =. (4) 3. Casoai-ype Deeminan Soluions of he Diffeenial-Diffeence Davey-Sewason Sysem wih Self-Consisen Souces I is shown in [7] ha he diffeenial-diffeence Davey-Sewason sysem exhibis N-solion soluions expessed by means of wo ypes of deeminans double-casoai and Gammian deeminans. I is naual o conside if he diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces have wo ypes of deeminan soluions. In his secion we shall deive anohe class of deeminan soluions Casoai-ype deeminan soluions o he diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces (7) (2) (25) and (28)-(3) fo = 2. Le us inoduce he following double-casoai deeminan: whee fo = 2 2N in which C saisfies wih φ ( n ) φ ( n ) ψ ( k ) ( n) ( n) ( n+ m ) ; ( k+ 2N m) ( n) ( n+ m ) ; ( k + 2N m) φ φ ψ ψ φ φ ψ ψ ( n) ( n+ m ) ; ( k+ 2N m) φ φ ψ ψ 2N 2N 2N 2N ( n) ( n) i C ( ) ( n) 2 (42) φ = φ + φ (43) i C( ) ψ = ψ + ψ (44) 2 C ( ) = c 2 c N ohewise c being an abiay funcion of c is an abiay consan and being a posiive inege and ψ saisfy he following equaions: 2 2 i φ n αφ n i φ2 n αφ 2 n = ( ) = ( ) (45) (46) 483

7 i ψ k αψ 2 k i ψ2 k αψ 2 2 k = ( ) = ( ) Fom now on he deeminan (42) will fo simpliciy be denoed as. (47) m ; 2N m. (48) aking ino accoun Equaions (42)-(48) we can sae he following Poposiion: Poposiion he soluions o Equaions (7) (2) (25) and (28)-(3) fo = 2 can be expessed as he following double-casoai ype deeminans: whee he pfaffian elemens ae defined by F = m ; 2N m (49) G = m ; ( 2N m 2 ) (5) H = m 2 ; ( 2N m ) (5) 2 ( ˆ ) P = i c d d d d N (52) m N m ( α ) Q = α c d 2 d d d2 N (53) m N m ( ˆ ) J = c d d d d N (54) m N m ( α ) L= iα2 c d dm d d2n m 2N (55) ( d j) φ ( n l) ( d j) ψ ( k s) = + = + (56) l j s j * ( i j) ( dl ds ) ( dl ds) ( dl ds ) = = = = (57) ( d α ) φ ( n l) ( d α ) φ ( k s) = + = + (58) l 2 s 2 in which ls ae ineges i j = 2 2N = 2 and ˆ in he pfaffians indicaes deleion of he lee unde i. Poof: he double Casoai deeminans in ()-(3) can be expessed by pfaffians [28] in he following way: ( 2 m 2Nm ) ( 2 m 2Nm2 ) F = d d d d N (59) G = d d d d N (6) H = ( d dm2 d d2nm 2N ) (6) 2 whee he pfaffian elemens ae given in (56)-(58). We fis show ha funcions (49)-(55) saisfy Equaions (2) and (25). Using Equaions (43)-(47) we can calculae he following diffeenial and diffeence fomula fo F G H : ( 2 ) α ( 2 ) if = α d d d d d N + d d d d d N m 2N m 2 m 2N m ( ˆ 2 α ) + c d d d d 2N m N m = = ( ) (62) F 2 n+ d dmd d2nm N (63) F 2 k = d dm d d d2nm2 N (64) 484

8 ( 2 ) α ( 2 ) ig= α d d dm d d2n m 2 N + 2 d dm d d d2n m 2 N 2 + c d dm d d N m N = ( ˆ 2 2 α ) G 2 n = d d dm d d2nm2 N (66) (65) G 2 k+ = d dm d d2nm N (67) ih = α d d d d d N + d d d d d N ( 2 ) α ( 2 ) m 2 2N m 2 m 2 2N m 2 + c d dm d d N m N = ( ˆ α ) = ( ) (68) F 2 n = d d dm2 d d2nm N (69) Fk+ d dm d d2nm2n (7) H = d d d d N (7) ( 2 ) n+ m 2Nm 2 H = d d d d d N. (72) ( 2 ) k m2 2Nm 2 Subsiuion of Equaions (52)-(55) and (62)-(72) ino Equaions (2) and (25) yields he following deeminan ideniies especively: and ( d d ˆ m d d2nm2 2N α)( d dm d d2nm 2N ) ( d d )( ˆ m d d Nm N d dm d d2nm 2N α) ( d ˆ dm d d2nm2 2N ) ( d dm d d2nm N α ) 2 = ( d d ˆ m d d2nm 2N α)( d dm2 d d2nm 2N ) ( d d 2 2 )( ˆ m d d Nm N d dm2 d d2nm 2N α) ( d ˆ dm2 d d2nm 2N ) ( d dm d d2nm N α ) 2 =. I is easy o show ha (49)-(5) saisfy Equaion (7). Now we pove ha funcions (49)-(55) saisfy Equaions (28)-(3). Fom Equaions (52)-(58) we can deive he diffeence fomula fo pfaffians P Q J L = as follows: ( + ) = ( ˆ ) P n i c d d d d N j (75) m N m ( + ) = ( ˆ 2 2 ) J n c d d d d N j (76) m N m ( + ) = α2 ( 2 2 β ) L n i c d d d d N (77) m N m ( ) = α ( β ) Q k c d d d d d N (78) m N m ( ) = α2 ( 2 2 β ) L k i c d d d d d N (79) m N m ( ) = ( ˆ ) J k c d d d d d N j. (8) m N m (73) (74) 485

9 Subsiuing Equaions (59)-(6) (63)-(64) (7)-(72) and (75)-(8) ino Equaions (28)-(3) we obain he following deeminan ideniies especively: ( d d ˆ m d d2nm2 2N α)( d dm d d2nm 2N ) ( d d )( ˆ m d d Nm N d dm d d2nm 2N α) ( d ˆ dm d d2nm2 2N ) ( d dm d d2nm N α ) ( d d ˆ m d d2nm2 2N α)( d dm d d2nm 2N ) ( d d )( ˆ m d d Nm N d dm d d2nm 2N α) ( d ˆ dm d d2nm2 2N ) ( d dm d d2nm N α ) ( d d ˆ m d d2nm2 2N α)( d dm d d2nm 2N ) ( d d )( ˆ m d d Nm N d dm d d2nm 2N α) ( d ˆ dm d d2nm2 2N ) ( d dm d d2nm N α ) ( d d ˆ m d d2nm2 2N α)( d dm d d2nm 2N ) ( d d )( ˆ m d d Nm N d dm d d2nm 2N α) ( d ˆ dm d d2nm2 2N ) ( d dm d d2nm N α ) 4. Conclusions 2 = 2 = 2 = 2 =. In his pape we apply he souce geneaion pocedue o he diffeenial-diffeence Davey-Sewason sysem ()-(3) o geneae a diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces (35)-(4) and claify he algebaic sucues of he esuling coupled discee sysem by expessing he soluions in ems of wo ypes of deeminans Casoai-ype deeminan and Gam-ype deeminan. In [29] a Davey-Sewason equaion wih self-consisen souces is consuced. I would be of inees o find he pope educion and ceain coninuous limis which give he Davey-Sewason equaion wih selfconsisen souces invesigaed in [29] fom he diffeenial-diffeence Davey-Sewason sysem wih self-consisen souces (35)-(4). Acknowledgemens he auho would like o expess he sincee hanks o Pof. Xing-Biao Hu fo his helpful discussions and encouagemen. his wok was suppoed by he pogam of highe-level alens of Inne Mongolia Univesiy (253) and he Naional Naual Science Foundaion of China (Gan No. 222). Refeences [] Hioa R. (977) Nonlinea Paial Diffeence Equaions: I. A Diffeence Analogue of he oeweg-de Vies Equaion. Jounal of he Physical Sociey of Japan hp://dx.doi.og/.43/jpsj [2] Hioa R. (977) Nonlinea Paial Diffeence Equaions: II. Discee-ime oda Equaion. Jounal of he Physical Sociey of Japan hp://dx.doi.og/.43/jpsj [3] Hioa R. (977) Nonlinea Paial Diffeence Equaions: III. Discee Sine-Godon Equaion. Jounal of he Physical Sociey of Japan hp://dx.doi.og/.43/jpsj [4] Feng B.F. Inoguchi J. ajiwaa. Mauno. and Oha Y. (23) Inegable Disceiaions of he Dym Equaion. Fonies of Mahemaics in China hp://dx.doi.og/.7/s y [5] Hu X.B. and Yu G.F. (27) Inegable Disceiaions of he (2+)-Dimensional Sinh-Godon Equaion. Jounal of Physics A: Mahemaical and heoeical hp://dx.doi.og/.88/75-83/4/42/s (8) (82) (83) (84) 486

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