An Invariance for (2+1)-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation

Transcription

1 Commun Theor Phys Beijing, China pp c Inernaional Academic Publishers Vol 43, No 4, April 15, 2005 An Invariance for 2+1-Eension of Burgers Equaion Formulae o Obain Soluions of KP Equaion TIAN Yong-Bo, 1 TIAN Chou, 1 SHAO Nan 2 1 Deparmen of Mahemaics, Universiy of Science Technology of China, Hefei , China 2 Deparmen of Chemisry, Universiy of Science Technology of China, Hefei , China Received Augus 23, 2004 Absrac In his aricle, we sudy he 2+1-eension of Burgers equaion he KP equaion A firs, based on a known Bäcklund ransformaion corresponding La pair, an invariance which depends on wo arbirary funcions for 2+1-eension of Burgers equaion is worked ou Given a known soluion using he invariance, we can find soluions of he 2+1-eension of Burgers equaion repeaedly Secondly, we pu forward an invariance of Burgers equaion which canno be direcly obained by consraining he invariance of he 2+1-eension of Burgers equaion Furhermore, we reveal ha he invariance for finding he soluions of Burgers equaion can help us find he soluions of KP equaion A las, based on he invariance of Burgers equaion, he corresponding recursion formulae for finding soluions of KP equaion are digged ou As he applicaion of our heory, some eamples have been pu forward in his aricle some soluions of he 2+1-eension of Burgers equaion, Burgers equaion KP equaion are obained PACS numbers: 0230Jr Key words: invariance, 2+1-eension of Burgers equaion, formulae for obaining soluions of KP equaion 1 Inroducion As we know, here are many radiional ways see Refs [1] [3] for finding eac soluions of he nonlinear evoluion equaions In recen years, we have developed a echnology see Refs [4] [5] o consruc formulae for finding eac soluions of he inegrable nonlinear evoluion equaions Using he echnology o solve a nonlinear inegrable equaion is based on wo basic poins: i New characerisic funcion ϕ of he La pair of he inegrable nonlinear evoluion equaion can be worked ou when he poenial funcion u is fied ii Using he renewed characerisic funcion ϕ he corresponding Bäcklund ransformaion, new poenial funcion u will be obained The repea of he procedure will lead o recursion formulae esablished In his aricle, based on he echnology making a series corresponding analyses, we will consruc formulae for finding eac soluions of he 2+1-eension of Burgers equaion [6,7] he KP equaion: u + u y 2 u u y = 0, u + 6uu + u + u yy = 0 We pu he wo equaions ogeher no only for he reason ha hey are all 2+1-dimensional equaions bu also we will reveal ha he formulae for finding he soluions of Burgers equaion can help us find he soluions of KP equaion The main resuls can be concluded as he following respecs: i In Sec 2, by a reconsrucion of a Bäcklund ransformaion of he 2+1-eension of Burgers equaion, an invariance depending on wo arbirary funcions of his equaion is worked ou By using he invariance, one can find he soluions of he 2+1-eension of Burgers equaion repeaedly jus from a known soluion while he resuls in Refs [4] [5] are he recursion formulae for La pair of he corresponding equaions ii In Sec 3, we pu forward he corresponding invariance of Burgers equaion which canno be direcly obained by consraining he invariance of 2+1-eension of Burgers equaion Furhermore, we will reveal ha he invariance for finding he soluions of Burgers equaion can help us find he soluions of he KP equaion jus from zero soluion iii In Sec 4, by using he resuls in Sec 3, he recursion formulae by which we can find soluions of KP equaion are digged ou In his aricle, f d or f d indicaes any primiive funcion of f which is aken definiely 2 An Invariance Depending on Two Arbirary Funcions for he 2+1-Eension of Burgers Equaion In his secion, we consider he 2+1-dimensional sysem: 2+1-eension of Burgers equaion: u + u y 2 u u y = 0, 1 The projec suppored by Naional Naural Science Foundaion of China under Gran No

2 592 TIAN Yong-Bo, TIAN Chou, SHAO Nan Vol 43 whose La pair is [7] ϕ = λ 2 u ϕ, 2a ϕ = λϕ y 1 2 u yϕ 2b The La pair has he following Bäcklund ransformaion: ū = u 2 ln ϕ 3 ϕ I is self-eviden ha if ϕ in Eq 3 can be epressed by u only, hen he Bäcklund ransformaion 3 can be reconsruced o an invariance for Eq 1 Ne we will show ha he reconsrucion can be realized In fac, by Eq 2a, we have ln ϕ = λ 2 u, which means ϕ = C 1 e λ+u/2 d + C 2, 4 where C 1 C 2 are undeermined funcions of y In order o deermine C 1 C 2, we should use he condiion ha ϕ saisfies he second par of he La pair 2 A firs, by differeniaing Eq 2b for, we obain ϕ = λϕ y 1 2 u yϕ 1 2 u y λ 2 u ϕ 5 Subsiuing Eq 4 ino Eq 5, one finds where C 1 = λc 1y + HC 1, 6 H = 1 2 u y 1 4 u yu 1 2 u Furhermore H = 1 [u + u y ] 2 2 u u y = 0, which means H is he funcion of y only, ie, H = Hy, solving Eq 6 by using he firs inegral heory, [8] C 1 is obained, C 1 = fθ e Hθ+λ,d, where θ = y λ f is an arbirary differeniable funcion by subsiuing Eq 4 ino Eq 2b, he equaion for C 2 is obained as where W = C 1 λ C 1 C 2 = λc 2y + W, 7 e λ+u/2 d e λ+u/2 d y 1 2 u yc 1 e λ+u/2d Similarly, we can check ha W is he funcion of y only, ie, W = W y, Finally by solving Eq 7, C 2 is obained: C 2 = W θ + λ, d + gθ, where g is an arbirary differeniable funcion The conclusion in his secion By he above inducion, we can know he characerisic funcion ϕ in La pair 2 can be epressed by u Thus, he Bäcklund ransformaion 3 has been reconsruced o an invariance depending on wo arbirary funcions for he 2+1-eension of Burgers equaion Therefore, according o he above discussions, we can conclude he following heorem: Theorem 21 If u n is a soluion of 2+1-eension of Burgers equaion 1, hen u n+1 = u n 2 ln ϕ n ϕ n is he soluion of 1 as well Here ϕ n = C 1n y, e λ+un/2 d + C 2n y,, H C 1n = f n θ e nθ+λ,d, C 2n = W n θ + λ, d + g n θ, θ = y λ, while f n g n are arbirary differeniable funcions H n = H n y,, W n = W n y, which can be epressed as H n = 1 2 u ny 1 4 u nyu n 1 2 u n, W n = C 1n e λ+un/2 d λ C 1n e λ+un/2 d 1 y 2 u nyc 1n e λ+un/2 Eample 21 We ake he soluion of equaion 1: u 0 = 2 ln + 2 ln, by heorem 21 we have hen H 0 = 1, W 0 = 0, C 10 = 1 f 0θ, C 20 = g 0 θ, θ = y λ, λ 0, where f 0 g 0 are arbirary differeniable funcions Correspondingly ϕ 0 = f 0 θ 1 λ e λ 1 λ 2 e λ + g 0 θ, u 1 = 2 ln + 2 ln 2 ln λ 2 f 0 θ e λ λ 2 g 0 θ f 0 θ e λ + λ e λ Eample 22 We ake he soluion of Eq 1: u 0 = 0 by Theorem 21, we have hen H 0 = 0, W 0 = 0, C 10 = f 0 θ, C 20 = g 0 θ,

3 No 4 An Invariance for 2+1-Eension of Burgers Equaion Formulae o Obain Soluions of KP Equaion 593 θ = y λ, λ 0, where f 0 g 0 are arbirary differeniable funcions Correspondingly ϕ 0 = 1 λ f 0θ e λ + g 0 θ, λf 0 θ e λ u 1 = 2 ln λg 0 θ f 0 θ e λ Saring from u 1 using Theorem 21 again, we have ϕ 1 = H 0 = 0, W 0 = C 11 = f 1 θ, C 21 = g0 g0 θ f 0 θ λ 2 e λ f 1 θ + θ, f 0 g0 θ + g 1 θ f 0 θ g0 θ + g 1 θ, f 0 θ where f 1, g 1 are arbirary differeniable funcions Finally, we obain u 2 = 2 ln λf 0 θ e λ λg 0 θ f 0 θ e λ [ 2 ln g 0 θ/f 0 θ e λ /λf 1 θ g 0 θ/f 0 θ /λ 2 e λ f 1 θ + g 0 θ/f 0 θ + g 1 θ ] 3 The Corresponding Invariance of Burgers Equaion a Discussion for Is Furher Applicaion By he consrain: y =, u = 2q, 8 he 2+1-eension of Burgers equaion 1 can be consrained o he Burgers equaion: [9] q + q + 2qq = 0, 9 he La pair 2 can be consrained o he La pair of Burgers equaion: ϕ = λ + qϕ, ϕ = λ qϕ 10a 10b According o consrain 8 by using Eq 10a, he Bäcklund ransformaion 3 can be specialized o: ū = λ + ln ϕ As a maer of fac, we should poin ou ha he invariance of he 2+1-eension of Burgers equaion shown in Theorem 21 canno be direcly consrained o an invariance of he Burgers equaion 9 Of course, we can use similar procedure shown in Sec 2 o work ou he corresponding invariance of Burgers equaion Here we jus announce he resuls: Theorem 31 If q n is a soluion of Burgers equaion 9, hen q n+1 = λ + ln ϕ n is he soluion of Eq 9 as well Here λ is an arbirary consan λ+q ϕ n = K 1n e nd d + K 2n, while P K 1n = m 1n e n d, K 2n = Q n d + m 2n, m 1n, m 2n are arbirary consans P n = λ + q n d λ + q n λ + q n q n, λ+q Q n = K 1n e nd d λ+q λ + q n K 1n e nd Furher Discussion Ne, we will show ha Theorem 31 will help us esablish he formulae for obaining he soluions of KP equaion In fac, he KP equaion u 6uu u = 3u yy 11 has he Ricai form La pair: [3] q y = q 2qq u, q = 4q + 3q 2 + 3qq + 3q 2 q 6u q 6uq 3u u y, he corresponding Bäcklund ransformaion: 12a 12b ū = u + 2q 13 Especially, when we ake he poenial funcion u = 0 in La pair 12, we have q y = q 2qq, q = 4q + 3q 2 + 3qq + 3q 2 q 14a 14b The Bäcklund ransformaion 13 will be simplified as: ū = 2q Therefore, we can conclude as follows Theorem 32 If q saisfies sysem 14, hen u = 2q is a soluion of KP equaion 11 According o Theorem 32, we know ha we can obain he soluions of KP equaion by solving sysem 14 By a minor observaion, we find ha equaion 14a is in a form of Burgers equaion hough i is in fac a 2+1- dimensional equaion Therefore, i is a very naural idea ha we can esablish formulae o solve sysem 14 by using he invariance of Burgers equaion, which has been

4 594 TIAN Yong-Bo, TIAN Chou, SHAO Nan Vol 43 shown in Theorem 31 In he ne secion we will give a deailed deducion finish he work 4 Formulae o Obain Soluions of KP Equaion Some Eamples According o Theorem 31 afer making some needed revision, he following lemma can be naurally obained Lemma 41 If q n is a soluion of Eq 14a, hen q n+1 = λ + ln ϕ n is he soluion of Eq 14a as well Here λ is an arbirary consan λ+q ϕ n = G 1n y, e nd d + G 2n y,, 15 while δ G 1n = M 1 e n dy, G 2n = n dy + M 2, 16 M 1, M 2 are arbirary differeniable funcions of δ n = λ + q n d λ + q n λ + q n q n, 17 λ+q n = G 1n e nd d Γ n = δ n dy λ + q n d Ω n = G 1n y y λ+q λ + q n G 1n e nd 18 The funcion M 1 M 2 in he above lemma should be deermined so ha he invariance for sysem 14 can be esablished Therefore, subsiuing he ransformaion q n+1 = λ + lnϕ n ino 14b using he relaionship ϕ n = λ + q n ϕ n, we find ha he coefficiens of ϕ n 4 ϕ n 3 can be well balanced he coefficiens of ϕ n 1, ϕ n 2 yield ϕ n 1 : ϕ n = 4q n + 3q n q n + q 3 n λ 3 ϕ n, 19a ϕ n 2 : ϕ n = 4q n + q 2 n + λq n + λ 2 ϕ n 19b Finally, by subsiuing Eqs ino he sysem 19, M 1, M 2 can be deermined: Γ M 1n = d 1n e n d, M 2n = Ω n d + d 2n, where d 1n, d 2n are arbirary consans 4q n + 3q n q n + qn 3 λ 3, 20 λ+q e d 1nq n + q 2 n + λ 2 λ+q + λq n e n dy 21 5 Conclusions To sum up, we pu forward he following heorem: Theorem 51 If q n is a soluion of sysem 14, hen q n+1 = λ + ln ϕ n is he soluion of Eq 14 as well Here λ is an arbirary consan λ+q ϕ n = G 1n y, e nd d + G 2n y,, δ G 1n = d 1n e n dy Γ e n d, G 2n = n dy + Ω n d + d 2n, while d 1n, d 2n are arbirary consans δ n, n, Γ n, Ω n are deermined by Eqs 17, 18, 20, 21 respecively The Eamples for Burgers Equaion KP Equaion By using Theorem 32 Theorem 51, we can consruc soluions of KP equaion In view of ha he formulae for obaining soluions of Burgers equaion 10 have inheren relaionship wih hose of KP equaion 12, we have he following eamples: Eample 51 10: We ake he soluion of Burgers equaion q 0 = ω, where ω is an arbirary consan, by Theorem 31 aking λ = 0, we have P 0 = ω 2, Q 0 = 0, K 10 = m 10 e ω2, K 20 = m 20, where m 10, m 20 are arbirary consans Correspondingly q 1 = ϕ 0 = m 10 ω 2 e ω +ω + m 20, [ m10 ] ln 2 ω e ω +ω + m 20 Saring from q 1 using Theorem 31 again, we have P 1 = 0, Q 1 = 0, K 11 = m 11, K 21 = m 21 ϕ 1 = m 11 m10 ω 2 Finally, we obain [ m10 q 2 = ln m 11 ω 2 2 e ω +ω + m 20 + m 21 ] 2 e ω +ω + m 20 + m 21, where m 11, m 21 are arbirary consans

5 No 4 An Invariance for 2+1-Eension of Burgers Equaion Formulae o Obain Soluions of KP Equaion 595 Eample 52 We ake he soluion of equaion Burgers equaion 9: q 0 = co, by Theorem 31, we have P 0 = λ 2, Q 0 = 0, K 10 = m 10 e λ2 +1, K 20 = m 20, where m 10, m 20 are arbirary consans Correspondingly, Saring from q 1 using heorem 31 again, we have ϕ 0 = m 10 e λ2 +1 λ λ 2 λ sin cos + m 20, [ m10 e λ2 +1 λ ] q 1 = λ + ln λ 2 λ sin cos + m 20 P 1 = 0, Q 1 = 2λm 20, K 11 = m 11, K 21 = 2λm 20 + m 21 m10 e λ2 +1 λ ϕ 1 = m 11 λ 2 λ 2 sin + 2λ cos sin + m 20 2m 20 + m 21, here m 11 m 21 are arbirary consans Finally, we obain [ m10 e q 2 = λ + ln m λ2 +1 λ ] 11 λ 2 λ 2 sin + 2λ cos sin + m 20 2m 20 + m 21 Eample 53 We ake he soluion of sysem 14: q 0 = ω, where ω is an arbirary consan, by Theorem 51 aking λ = 0, we have δ 0 = ω 2, Γ 0 = 4ω 3, 0 = 0, Ω 0 = 0, G 10 = d 10 e ω2 y 4ω 3, G 20 = d 20, where d 10, d 20 are arbirary consans Correspondingly q 1 = ϕ 0 = m 10 ω 2 e ω y 4ω 3 +ω + m 20, [ d10 ] ln 2 ω e ω y 4ω 3 +ω + d 20 according o Theorem 32, he soluion of KP equaion 11 is obained: [ u 1 = ln m ] 10 2 ω e ω y 4ω 3 +ω + m 20 Saring from q 1 using Theorem 51 again, we have δ 1 = 0, Γ 1 = 0, 1 = 0, Ω 1 = 0, K 10 = d 11, K 20 = d 21, q 2 = d10 ϕ 1 = d 2 11 ω 2 e ω y 4ω 3 +ω + d 20 + d 21 [ d10 ] ln d 2 11 ω 2 e ω y ω 3 +ω + d 20 + d 21, where d 11, d 21 are arbirary consans Finally, anoher soluion of KP equaion 11 is obained: [ d10 ] u 2 = ln d 2 11 ω 2 e ω y ω 3 +ω + d 20 + d 21 Eample 54 We ake he soluion of sysem 14:, by Theorem 51, we have q 0 = co + 4 δ 0 = λ 2, Γ 0 = 4λ 3, 0 = 0, Ω 0 = 0, G 10 = d 10 e λ2 +1y+4λ 3, G 20 = d 20,

6 596 TIAN Yong-Bo, TIAN Chou, SHAO Nan Vol 43 where d 10, d 20 are arbirary consans Correspondingly ϕ 0 = d 10 e λ2 +1y+4λ 3 λ λ 2 λ sin + 4 cos d 20, [ d10 e λ2 +1y+4λ 3 λ ] q 1 = λ + ln λ 2 λ sin + 4 cos d 20 according o Theorem 32, one soluion of KP equaion 11 is obained: [ d10 e λ2 +1y+4λ 3 λ ] u 1 = 2 ln λ 2 λ sin + 4 cos d 20 Saring from q 1 using Theorem 51 again, we have δ 1 = 0, Γ 1 = 0, 1 = 2λd 20, Ω 1 = 3λ 2 d 20, G 10 = d 11, G 20 = 2λd 20 y 3λ 2 d 20 + d 21, where d 11, d 21 are arbirary consans [ d10 e λ2 +1y+4λ 3 λ ] ϕ 1 = d 11 λ 2 2 λ 2 sin λ cos + 4 sin d 20 Thus, anoher soluion of KP equaion 11 is digged ou: u 2 = 2ln ϕ 1 2d 20 y 3λ 2 d 20 + m 21 References [1] Gu Chao-Hao, e al, Solion Theory Is Applicaion, Zhejiang Publishing House of Science Techonology, Hangzhou 1990 [2] Li Yi-Sheng, e al, Nonlinear Science Seleced, Publishing House of Universiy Science Technology of China, Heifei 1994 [3] VB Maveev MA Salle, Darbou Transformaion Soluions, Springer-Verlag, Berlin 1991 [4] Tian Yong-Bo, Commun Theor Phys Beijing, China [5] Tian Yong-Bo, Cheng, Yi Tian Chou, Commun Theor Phys Beijing, China [6] A Pickering, J Mah Phys [7] Zhou Kou-Hua, Tian Chou, Tian Yong-Bo, Commun Theor Phys Beijing, China [8] Wang Rou-Huai Wu Zhou-Qun, Ordinary Differenial Equaions, People s Educaion Publishing House, Beijing 1967 [9] Wang Ming-Liang, Nonlinear Evoluion Equaions Solions, Science Press, Beijing 1990