Darboux transformation of lax pair for an integrable coupling of the integrable differential-difference equation

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1 App Copo M ; 5: -6 P o Oo p://popo// o: 68/58 ISS: P; ISS: 8-56 O Dox oo o x p o op o - o X-X X Co o M Sy S So vy o S ooy Qo 6659 C E : o : X-X X Dox oo o Lx P o I Cop o I D-D Eo App Copo M Vo o 5 pp -6 o: 68/58 A: A op o ko - o Lx p p B o oo opo o-y- o x p po Dox oo o Lx p o op A ppo o o Dox oo xp oo v Kyo: I D-D Eo I Cop Dox oo Exp Soo Ioo S o ok o F P 96 - o pp y o Py My - o v pop v o po o v S v oo y y Ho -Ho -6 o opxo oo y Co 7 yy o 8 o o Ry vo o op y o oo o o op o o ok o po o oo o voo o 9 po A ppo o o op y o voo o popo Fo xp po o - p po - o L -7 o o Fo v - o y K y I y y vo-v o ov Z R W y o v - o oo: y z K y ξ y z H z vo-v o ov Z R vo-v o ξ y z o ξ y z y o-vy oo y y y y y o - o y vov p v o o o E op y o E I o Dox oo po oo o o xp oo o - o 8- x p po o - o py ky o oy o Dox oo Fo o x p po y o Dox oo o - o o xp oo o - o I R - o y v o x p

2 App Copo M ; 5: -6 po Ho o y y y I Dox oo I o oo o y y K y -o po y o o y p I y z z K y ' K y K y z ' K y z o Gx vv o K y p oy o z I y o o ov -o po y o o o 5 o opo E v o E y E E Z By o o op 5 oo Lx p E χ 6 Vϕ ϕ B o o E5 o E E5 op o - o H v 7 y opy oo o E 6 E 7 EV V 8 E 8 y E 5 pp oz oo I o y o oo o Lx p o Dox oo o Lx p o E5o o o ko Dox oo o Lx p o op o I So ppo o Dox oo xp oo o E 5 So oo k v So Dox oo I ko Dox oo pov py po o o xp oo o o - o o y A x p po o - o py ky o 9- ko oo o x p po Dox oo o p po o o p po o yp I oo o o Dox oo o - o 5I oy o Dox oo ky po oo x popy p W o oo H ϕ ϕ 9 o o

3 X-X X: Dox oo o Lx P o I Cop o I D-D Eo o o v E 9 o o p po 7 8 o E ɶ ɶ ɶ ɶ V ϕ ϕ V V I oo V E v o V E 6 E 7 pvy L o p oo o E 7 E 8 o oo y o I 8 y o o o o 8 E E ozo o y Fo 7 E E y o -o poyo o oo v 9 Popoo x y E o o po pp o po o o Poo: L

4 App Copo M ; 5: -6 I y o o * W o poyo o poyo Fo o E 6 E 7 ω σ µ W ω σ µ Ao o E E xpo o Mk o 8 E E o o o o k k o v * M M W p H M E o o E v poo op Popoo x V y o V E 7 oo V 5 Poo: L

5 X-X X: Dox oo o Lx P o I Cop o I D-D Eo V * Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ W y Σ Σ Σ Σ Σ o poyo Σ o poyo D o E E8 y y Σ k k oo o * V Q 7 H Q p o Fo E 7 o V Q 8 Cop o o E 8 v poo op H o oo 9 Lx p 6 7 o o Lx p o A oo 9 : ϕ ; ϕ ; 9 Dox oo o E 6 E 7 E o- Bɺ ɺ k oo B oo ϕ ; o oo ϕ ; I oo o o popoo v oo o E ooϕ o E 5 pp o oo ; ϕ ; oo v y E E7 Rk B o poyo o x o poyo Dox oo 9 o -o Dox oo 9 o ko x o -o Dox oo Appo o Dox oo I o ppy Dox oo 9 o xp oo o E 5 F o oo p p oo ϕ ; o E 5 o -o Dox oo o E 6 E 7 y S oo o E 6 E 7 o oo Lx p

6 App Copo M ; 5: -6 5 E Sov ov o o v xp xp xp xp xp xp xp xp xp xp xp xp xp xp xp xp xp xp xp xp xp xp Fy y o oo o -o oo o E 5: S o o xp oo ppy Dox oo 9 o o oo o E 5 o po o oy o o o o xp oo o - o 5 Coo Rk I pp v o op o ko - o p Lx p W p o oo o opo Lx p -o Dox oo o Lx o op A ppo xp oo v y o Dox oo Fo y o o y po o o op o v oo o opxo oo y Co o o I o o o oo o o Dox oo o op o oo y o o So opo Dox oo o op o oo y o ok R EF JP S Co Pp o Eo F IIvy o Co P Co 965 MAoz JLk o - o JMPyVo6 975 pp598-6 WOv HZ BF My -Ho oo o o y Poo PyVo8 989 pp9-8 GZ A y ppo o oy o y JpyA:MGVo 99 pp9-9 5 WXM A vo y o - o L JpyA: MoVo 7 pp XXX A op y o o y -Ho oop y JMPyVo5 pp5-8

7 6 X-X X: Dox oo o Lx P o I Cop o I D-D Eo 7 W X M KMo Copxo oo o o o Py AVo pp9-7 8 WXM XGG Bɺ ɺ k oo o oo y o yy o CRM Po L ovo9 pp- 9 KMz MLk Cop y o Ko- V o po o oo: L-Bk yy ppo J PyAMGVo6 98 pp77-8 BF I Appo o Ay Go Mo o o D Eo y P ACko K Do 99 WXM BF I oy o po o Co Soo &FVo pp7-5 WXM I op o oo o y po I A oy ppo o KV y MAppAVo7 pp-56 WXM E p po o o op o oo o PyLAVo6 pp WXM XXX YFZ S- o L oo op PyLAVo 5 6 pp5-6 WXM XXX YFZ S o L op JMPyVo 7 6 pp XXX A y o Lov Ho o Py L A Vo7 8 pp VMvv MS Dox o Soo B: Sp 99 9 G RM G-oo oo o AKS o y ko PyLAVo 98 pp67-7 CHG HSH ZXZo Dox o Soo oy I Go Appo S S & P 999HYD XXX XDZo A y o oo o Dox oo CPyVo pp5- YW XGG A y o - o Dox oo JPyAMGVo 998 ppl677-l68 XXX HXY YPS Dox oo o o o o MoPyLBVo 6 pp6-68 XXX Dox oo o op oo o PyLA Vo6 7 pp5-

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