Nonlocal symmetries of Frobenius sinh-gordon systems

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1 Zhou et al. Advances n Dfference Equatons ( :71 R E S E A R C H Open Access Nonlocal symmetres of Frobenus -Gordon systems Hujuan Zhou 1, Chuanzhong L 1*,XnyueL and Fushan L 3 * Correspondence: lchuanzhong@nbu.edu.cn 1 Department of Mathematcs, Nngbo Unversty, Nngbo, Chna Full lst of author nformaton s avalable at the end of the artcle Abstract In ths paper, we consder a weakly coupled -Gordon equaton whch takes values n a commutatve Frobenus subalgebra of gl(,c. Then we construct some nonlocal symmetres of the Frobenus -Gordon system usng ts Bäcklund transformaton and nfntesmal transformatons. Based on the nonlocal symmetres, we show some conserved denstes of the Frobenus -Gordon system. Usng these symmetres, we also construct some new coupled ntegro-dfferental systems. Keywords: Bäcklund transformatons; Frobenus -Gordon equaton; Nonlocal symmetry 1 Introducton The -Gordon equaton and sne-gordon equaton are mportant ntegrable equatons and they descrbe many nterestng phenomena ncludng dynamcs of coupled pendulums, Josephson juncton arrays [1], nonlnear exctatons n complex systems n physcs, and lvng cellular structures []. These two models have a transformaton whch lnks them together. In [3], Grauel studed the Panlevé property and Bäcklund transformaton of -Gordon equaton. As we know, Le symmetres are very mportant n fndng solutons of ntegrable equatons [4 13], partcularly the resdual symmetres and nonlocal symmetres [14 16]. In [17], nonlocal symmetres of the (1 + 1-dmensonal -Gordon equaton are obtaned. Makng advantages of the consstent condtons ntroduced when solvng the nonlocal symmetres, some new nonlocal conservaton laws of the -Gordon equaton related to the nonlocal symmetres are obtaned. Some new fnte and nfnte dmensonal nonlnear systems are constructed by takng the nonlocal symmetres as symmetry constrant condtons mposed on the Bäcklund transformatons. Nonlocal symmetres were frst studed rgorously early n 1980 [18]n whch a satsfactory geometrc formulaton was developed, and later a seres of works [19, 0] appeared. A constructve method for dervng nonlocal symmetres of dfferental equatons based on the Le Bäcklund theory of groups was developed n [1]. Systematc procedures were presented for fndng nonlocally related partal dfferental equatons and ther many local and nonlocal conservaton laws and nonlocal symmetres n []. Nonlocal symmetres are of nterest because they are assocated wth the exstence of lnearzng transformatons, Bäcklund transformatons, and Darboux transformatons. Applyng the nfntesmal transformaton on the nonlnear system and ts lax par smultaneously, some useful The Author(s 018. Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made.

2 Zhou et al. Advances n Dfference Equatons ( :71 Page of 7 nonlocal symmetres nvolvng the egenfuncton can be obtaned. These nonlocal symmetres are also known as the egenfuncton symmetres [3, 4],and they have been recently studed to construct explct solutons [5]. In [6], from the algebrac reductons from the Le algebra gl(n, C totscommutatve subalgebra Z n, we construct the general Z n -sne-gordon and Z n --Gordon systems whch contan many mult-component sne-gordon type and -Gordon type equatons. Meanwhle, we gve the Bäcklund transformatons of the Z n -sne-gordon and Z n - -Gordon equatons whch can generate new solutons from seed solutons. A natural queston s what s the nonlocal symmetry of them, partcularly the Z --Gordon equaton (also named as Frobenus -Gordon equaton n ths paper. In ths paper, we wll answer ths queston n detal. Ths paper s arranged as follows. In Sect., werecall some basc facts about the Frobenus -Gordon equatons and ther Bäcklund transformatons. In Sect. 3, we construct some coupled ntegro-dfferental systems usng the nonlocal symmetres. The Frobenus -Gordon equaton and ts Bäcklund transformaton In ths secton, we recall the Frobenus -Gordon equaton whch was constructed frstly n our recent paper [6]. The Frobenus -Gordon equaton was constructed n the commutatve algebra Z = C[Ɣ]/(Ɣ andɣ =(δ,j+1 j gl(, C. In ths secton, we wll use a smlar method n the last secton to consder the Bäcklund transformaton of the Frobenus -Gordon equaton. Based on the well-known -Gordon equaton u xt = u, (1 the followng equaton n Z s the Frobenus -Gordon equaton: u xt = u, v xt = v u. ( The Frobenus -Gordon equaton has the followng Bäcklund transformaton [6]: +u +v u x = a u u x = v v, u u, t = 1 a u +u v t = 1 a, v +v u +u. (3 3 Nonlocal symmetres of the Frobenus -Gordon equaton Suppose that the above Frobenus -Gordon equaton ( and the Bäcklund transformaton (3 are nvarant up to an nfntesmal transformaton u u + ɛτ u, v v + ɛτ v, (4 u u + ɛτ u, v v + ɛτ v, (5 a a ɛδ, (6

3 Zhou et al. Advances n Dfference Equatons ( :71 Page 3 of 7 we can derve the followng denttes: τ u xt n (uτ u =0, (7 τxt u τ u =0, (8 ( u τx u τ x u + u a ( u (τ u + τ u =0, (9 τt u τt u 8 ( u a δn u + a ( u (τ u τ u = 0, (10 τxt v n (uτ v + n v (uτ u = 0, (11 τxt v n τ v + n v τ u = 0, (1 ( ( v τx v τ x v δn + v u + u + a ( ( v n + v u + u (τ u + τ u a ( u n + u (τ v + τ v = 0, (13 τt v τ t v 8 a δn v ( v a n v sn u u + a ( u n u (τ v τ v (τ u τ u = 0. (14 Smlar to [17], we can derve the followng three symmetres. I: If δ =0, τ u = τ v =0, then the Frobenus -Gordon equaton has a nonlocal symmetry wth τ u = e ap, τ v = aqe ap, (15 where p x =, u p t = 1a + u ( q x = v u, u q t = 1a + v, (16 ( u + u. (17 II: Ifδ = 1 n, τ u = τ v =0, then the Frobenus -Gordon equaton has a nonlocal symmetry wth τ u = re ap, τ v = se ap + raqe ap, (18 where r x = e ap u s x = e ap v s t = 1 a e ap + v, r t = 1 a e ap u aqe ap + u + q a e ap + u u + u, (19, (0. (1

4 Zhou et al. Advances n Dfference Equatons ( :71 Page 4 of 7 III: If δ =0, τ u = u x, τ v = v x, then the Frobenus -Gordon equaton has a nonlocal symmetry wth where τ u = u x nafe a np, τ v = v x nage a np + n a qfe a np, ( f x = u x cos + u [ g x = v x u x + v e a np, f t = a nu xt e a np, (3 ] cos + u e a np + a nqu x cos + u e a np, (4 g t = a nv xt e a np + a 4 n qu xt e a np. (5 It s evdent that symmetres of the Frobenus -Gordon equaton ( obtanedabove are really nonlocal as they depend on the functon u, v, whch s related to the functon u, v through the Bäcklund transformaton (3. Integratng wth respect to x and t wll leadto τ u = 4e ap rδ ae ap p x τ u e ap dx τ u + e ap G 0 (t, (6 τ v = 4aqe ap rδ 4e ap sδ a qe ap p x τ u e ap dx ae ap q x τ u e ap dx ae ap p x τ v e ap dx +a e ap p x τ u qe ap dx τ v + aqe ap G 0 (t+e ap G 1 (t, (7 and τ u = 4w a e h a δ + a e h a h x τ u e h a dx τ u + e h a G 3 (x, (8 τ v = 4 w a e h a δ + 4w h a a e h a δ + a he h a h x τ u e h a dx + a e h a h x τ u e h a dx + a e h a h x τ v e h a dx a e h a h x τ u he h a dx τ v + h a e h a G 3 (t+e h a G 4 (x, (9 where p, q, r, s, h, w, h, w satsfy p x = r x = u u h t = + u w t = + u, q x = e ap, s x =, h t = e h a, w t = v v u u + u + u + v + v, (30 ( u e ap aq u e ap,, (31 e h a + h ( u a + u e h a.

5 Zhou et al. Advances n Dfference Equatons ( :71 Page 5 of 7 G 0 (t, G 0 (t, G 3 (x, G 4 (x are arbtrary ntegraton functons. The followng condtons should be satsfed: h = a p, w = a r, h = a q, w = a s, (3 p x τ u e ap dx + p x τ u e ap dt + 1 a τ u e ap = 0, (33 q x τ u e ap dx + p x τ u e ap dx + p x τ v e ap dx a p x qτ v e ap dx + q x τ u e ap dt + p x τ v e ap dt a p x qτ u e ap dt + 1 a τ v e ap qτ u e ap = 0. (34 Then we can get the followng correspondng conserved densty and flux: ρ 1 = a u ρ 1 = a v ( ρ = a u eap ρ = a e ap J = 1 τ u ρ 3 = a, J 1 = + u ( u v, J 1 = + v u, J = 1 eap + u ( ( v u u a3 q u eap u ( ( v v u u aq eap u τ u e, J ap 3 = τ u eap + u, (35 ( u + u, (36, (37, (38, (39, (40 ρ 3 = a τ v e + a aqτ u, (41 ap eap [ aqτ u ( u τ v J 3 = eap e ap + v τ ] ( v u e ap + u. (4 These conservaton laws of the Frobenus -Gordon equaton satsfy the dentty t ρ = x J, t ρ = x J. (43 4 Coupled ntegro-dfferental systems From the nonlocal symmetry (16 and(17, we can construct the coupled ntegrodfferental ntegrable systems wth respect to the varable x: u x = =1 ( b exp a u x + u x =a u v x = =1 [ b a v u dx, (44, =1,,...,m, (45 u ] ( dx exp a u dx, (46

6 Zhou et al. Advances n Dfference Equatons ( :71 Page 6 of 7 v x + v x =a v ( u u, =1,,...,m. (47 Smlarly, from the nonlocal symmetry, we can construct the coupled ntegro-dfferental ntegrable systems wth respect to the varable t: ( ( 1 u u t = c exp a + u =1 u t u t = a + u [ ( 1 v v t = c a + v ( u + u =1 v t v t = a + v ( u + u dx, (48, =1,,...,m, (49 ] ( 1 dx exp a + u dx, (50, =1,,...,m. (51 Of course, these coupled ntegro-dfferental systems are ntegrable systems whch mght be taken nto our detaled study n the future. Acknowledgements The authors would lke to thank the referees for the careful readng of the manuscrpt and valuable suggestons. Fundng CL s supported by the Natonal Natural Scence Foundaton of Chna under Grant No and K. C. Wong Magna Fund n Nngbo Unversty. XL s supported by the Nature Scence Foundaton of Chna (No and the Scence and Technology Plan Project of the Educatonal Department of Shandong Provnce of Chna (No. J16LI1. Competng nterests The authors declare that they have no competng nterests. Authors contrbutons CL contrbuted to the dea. Other authors contrbuted to the calculaton of ths paper. The authors read and approved the fnal manuscrpt. Author detals 1 Department of Mathematcs, Nngbo Unversty, Nngbo, Chna. College of Mathematcs and Systems Scence, Shandong Unversty of Scence and Technology, Qngdao, Chna. 3 School of Mathematcal Scences, Qufu Normal Unversty, Qufu, Chna. Publsher s Note Sprnger Nature remans neutral wth regard to jursdctonal clams n publshed maps and nsttutonal afflatons. Receved: Aprl 018 Accepted: 31 July 018 References 1. Goldobn, E., Sterck, A., Gaber, T., Koelle, D., Klener, R.: Dynamcs of semfluxons n Nb long Josephson 0-φ junctons. Phys. Rev. Lett. 9, Artcle ID (004. Ivancevc, V.G., Ivancevc, T.T.: Sne-Gordon soltons, knks and breathers as physcal models of nonlnear exctatons n lvng cellular structures. J. Geom. Symmetry Phys. 31, 1 56 ( Grauel, A.: Snh-Gordon equaton, Panlevé property and Bäcklund transformaton. Physca A 13, ( Baleanu, D., Inc, M., Yusuf, A., Alyu, A.I.: Le symmetry analyss and conservaton laws for the tme fractonal smplfed modfed Kawahara equaton. Open Phys. 16, ( Inc, M., Yusuf, A., Alyu, A.I., Baleanu, D.: Le symmetry analyss, explct solutons and conservaton laws for the space-tme fractonal nonlnear evoluton equatons. Physca A 496, ( Baleanu, D., Inc, M., Yusuf, A., Alyu, A.I.: Travelng wave solutons and conservaton laws for nonlnear evoluton equaton. J. Math. Phys. 59, Artcle ID ( Inc, M., Yusuf, A., Alyu, A.I., Hashem, M.S.: Solton solutons, stablty analyss and conservaton laws for the Brusselator reacton dffuson model wth tme- and constant-dependent coeffcents. Eur. Phys. J. Plus 133, Artcle ID168 ( Akgül, A., Inc, M., Klcman, A., Baleanu, D.: A new approach for one-dmensonal sne-gordon equaton. Adv. Dffer. Equ. 016, Artcle ID 8 ( Ma, W.X.: Conservaton laws by symmetres and adjont symmetres. Dscrete Contn. Dyn. Syst., Ser. S 11, (018

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