Exact solution of the(2+1)-dimensional hyperbolic nonlinear Schrödinger equation by Adomian decomposition method

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1 Malaa J Ma (( Exac soluion of he(+1-dimensional hperbolic nonlinear Schrödinger equaion b Adomian decomposiion mehod Ifikhar Ahmed, a, Chunlai Mu b and Pan Zheng c a,b,c College of Mahemaics and Saisics, Chongqing Universi, Chongqing , PR China Absrac This paper sudies he exac soluion of he he(+1-dimensional hperbolic nonlinear Schrödinger equaion b he aid of Adomian decomposiion mehod Kewords: Exac soluion, Hperbolic Schrödinger equaion, Adomian decomposiion mehod 010 MSC: 83C15,35L70, 49M7 c 01 MJM All righs reserved 1 Inroducion Nonlinear equaions describe fundamenal phsical phenomena in naure ranging from chaoic behaviour in biological ssems, plasma conainmen in okamaks and sellaraors for energ generaion, o solionic fibre opical communicaion devices The consrucion of he exac soluions of nonlinear parial differenial equaions (PDEs is one of he mos imporan and essenial asks which help us for beer undersanding of nonlinear complex phsical phenomena In he pas couple of decades, here are various mahemaical echniques have been developed o carr ou he inegraion of hese equaions Some of hese commonl sudied echniques are Inverse Scaering Transform [5], bilinear ransformaion[4], he anh-sech mehod[6, 7], adomian decomposiion mehod [3], he anh-coh mehod[8], homogeneous balance mehod[9], Exp-funcion mehod [10], and man ohers The Adomian decomposiion mehod was inroduced and developed b George Adomian in [11, 1] and is well addressed in he lieraure A reliable modificaion of he Adomian decomposiion mehod developed b Wazwaz and presened in [3] A considerable amoun of research work has been invesed recenl in appling his mehod o a wide class of linear and nonlinear equaions for deail see [13, 14, 15, 16, 17, 18] and he references herein In his paper he Adomian decomposiion mehod will deermine exac soluion o (+1-dimensional hperbolic nonlinear Schrödinger equaion In Secion, we described his mehod for finding exac soluions for nonlinear PDEs In Secion 3, we illusraed his mehod in deail wih he hperbolic Schrödinger equaion In Secion 4, we gave some conclusions Adomian decomposiion mehod for nonlinear PDEs We firs consider he nonlinear parial differenial equaion given in an operaor form Corresponding auhor addresses: ifi cqu@homailcom (Ifikhar Ahmed L x u(x, + L u(x, + R(u(x, + F(u(x, = g(x,, (1

2 Ifikhar Ahmed e al / Exac soluion 161 where L x is he highes order differenial in x, L is he highes order differenial in, R conains he remaining linear erms of lower derivaives, F(u(x, is an analic nonlinear erm, and g(x, is an inhomogeneous or forcing erm he decision as o which operaor L x or L should be used o solve he problem depends mainl on wo bases: (i The operaor of lowes order should be seleced o minimize he size of compuaional work (ii The seleced operaor of lowes order should be of bes known condiions o accelerae he evaluaion of he componens of he soluionfor more deail see[3] Assume ha L mee hese wo condiions, herefore we se Appling where L u(x, = g(x, L x u(x, R(u(x, F(u(x, ( o boh sides of ( gives u(x, = Φ 0 Φ 0 = Take he soluion u(x, in a series form and he nonlinear erm F(u(x, b g(x, L x u(x, R(u(x, F(u(x,, (3 u(x, 0 u(x, 0 + u (x, 0 u(x, 0 + u (x, 0 + 1! u (x, 0 u(x, 0 + u (x, 0 + 1! u (x, ! 3 u (x, 0 u(x, = F(u(x, = L =, L =, L = 3 3, L = 4 4, u n (x,, (4 A n, (5 where A n are Adomian polnomials ha can be generaed for all forms of nonlineari and can be evaluaed b using he following expression [ ( A n = 1 d n n ] n! dλ n F λ i u i, n = 0, 1, (6 i=0 λ=0 Based on hese assumpions, Eq (3 become u n (x, = Φ 0 ( g(x, L x ( ( R u n (x, u n (x, A n The componens u n (x,, n 0 of he soluion u(x, can be recursivel deermined b using he relaion (7 u 0 (x, = Φ 0 u k+1 (x, = g(x,, L x u k R(u k (A k, k 0 (8 Nex find he componens of u n (x, b u 0 (x, = Φ 0 u 1 (x, = u (x, = u 3 (x, = u 4 (x, = g(x,, L x u 0 (x, L x u 1 (x, L x u (x, L x u 3 (x, R (u 0 (x, A 0, R (u 1 (x, A 1, R (u (x, A, R (u 3 (x, A 3, where each componen can be deermined b using he preceding componen Having he calculaed he componens u n (x,, n 0, he soluion in a series form is readil obained

3 16 Ifikhar Ahmed e al / Exac soluion 3 Exac soluions for(+1-dimensional hperbolic Schrödinger equaion In his secion we obain exac soluion of (+1-dimensional hperbolic nonlinear Schrödinger equaion b using he decomposiion mehod The hperbolic nonlinear Schrödinger equaion given b[1] is iu + 1 u xx 1 u + u u = 0 (31 where u is a complex valued funcion, while x, and are he independen variables In order o seek exac soluion, we assume ha u(x,, 0 = e i(mx+n Mulipling Eq(31 b i, we ma express his equaion in an operaor form as follows where L is defined b L = Appling L u(x,, = i L xxu(x,, i L u(x,, + i u(x,, u(x,, (3 and he inverse operaor L 1 ( = 0 is idenified b ( d o boh sides of (3 and using he iniial condiion we obain u(x,, = e i(mx+n + i L 1 where u(x,, u(x,, is nonlinear erm Subsiuing and nonlinear erm (u(x,, xx i L 1 (u(x,, u(x,, u(x,,, (33 u(x,, = u(x,, u(x,, = ino (33 gives ( u n (x,, = e i(mx+n + i L 1 u n (x,, Adomian s analsis inroduces he recursive relaion u n (x,, (34 xx A n (35 i L 1 ( u n (x,, ( A n (36 u 0 (x,, = e i(mx+n, u k+1 (x,, = i L 1 since u is a complex funcion so we can wrie (u k xx i L 1 (u k (A k, k 0 (37 where ū is he conjugae of u his means ha (35 can be wrien as u = uū (38 u ū = A n (39 B using formal echnique o find adomian polnomial used in [3] we find ha (39has he following polnomial represenaion A 0 = u 0ū0, A 1 = u 0 u 1 ū 0 + u 0ū1, A = u 0 u ū 0 + u 1ū0 + u 0 u 1 ū 1 + u 0ū, A 3 = u 0 u 3 ū 0 + u 1 u ū 0 + u 0 u ū 1 + u 1ū1 + u 0 u 1 ū + u 0ū3 (310

4 Ifikhar Ahmed e al / Exac soluion 163 ha in urn gives he firs few componens b u 0 (x,, = e i(mx+n, u 1 (x,, = i L 1 (u 0xx i L 1 (u 0 u (x,, = i ( L 1 u1xx i (u 1 u 3 (x,, = i L 1 (u xx i L 1 (A 0, (A 1, (u (A, (311 we obain u 0 (x,, = e i(mx+n (, A 0 = u 0ū0 = e i(mx+n, u 1 (x,, = i L 1 m e i(mx+n ( i L 1 n e i(mx+n u (x,, = i ( ( L 1 u1xx i (u 1 (A 1 = (i n e! m + 1 i(mx+n ( u 3 (x,, = i L 1 (u xx i L 1 (u (A = (i3 n 3e 3! m + 1 i(mx+n Accordingl, he series soluion is given b u(x,, = u n (x,, = u 1 + u + u 3 + (e i(mx+n = i( n m + 1ei(mx+n, (31 u(x,, = e i(mx+n [ ha gives exac soluion of (31 in closed form 1 + i ( n 1! m (i! ( n ( ( u(x,, = e i mx+n+ n m +1 m ] (313 (314 4 Conclusion The Adomian decomposiion mehod is successfull used o esablish new exac soluion The performance of his mehod is found o be reliable and effecive and can give more soluions, which ma be imporan for he explanaion of some nonlinear complex phsical phenomena References [1] SP Gorz, PK Ockaer, P Empli, and M Haelerman, Oscillaor neck insabili of spaial brigh solions in hperbolic ssems, Phsical Review Leers, 10(13(009, [] A Biswas, S Konar, Inroducion o Non-Kerr Law Opical Solions, CRC Press, Boca Raon, FL, USA, 006 [3] A M Wazwaz, Parial Differenial Equaions and Soliar Waves Theor, Springer, 009 [4] R Hiroa, Direc Mehod of Finding Exac Soluions of Nonlinear Evoluion Equaions,in: R Bullough, P Caudre (Eds, Backlund Transformaions, Springer, Berlin, 1980, p 1157 [5] VO Vakhnenko, EJ Parkes, AJ Morrison, A Backlund ransformaion and he inverse scaering ransform mehod for he generalised Vakhnenko equaion, Chaos Solion Frac, 17(4(003, [6] W Malflie, W Hereman, The anh mehod I Exac soluions of nonlinear evoluion and wave equaions, Phs Scripa 54 (1996, [7] AM Wazwaz, The anh mehod for ravelling wave oluions of nonlinear equaions, Appl Mah Compu, 154 (3, (004,

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