NEW PERIODIC SOLITARY-WAVE SOLUTIONS TO THE (3+1)- DIMENSIONAL KADOMTSEV-PETVIASHVILI EQUATION

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1 Mathematical and Comptational Applications Vol 5 No 5 pp Association for Scientific Research NEW ERIODIC SOLITARY-WAVE SOLUTIONS TO THE (+- DIMENSIONAL KADOMTSEV-ETVIASHVILI EQUATION Zitian Li Zhengde Dai Jn Li College of Mathematics and Information Science Qjing Normal University Qjing Ynnan 6550 R China School of Mathematics and Statistics Ynnan University KnmingYnnan R China lizitian88@6com Astract- By the extended homoclinic test technie explicit soltions of the (+-dimensional Kadomtsev-etviashvili(K eation are otained These soltions inclde doly periodic wave soltions doly soliton soltions and periodic solitary-wave soltions It is shown that the extended homoclinic test technie is a straightforward and powerfl mathematical tool for solving nonlinear evoltion eation Keywords- Extended homoclinic test Doly periodic Soliton eriodic solitary wave INTRODUCTION Recently many effective and powerfl methods have een proposed to solve nonlinear evoltion eations sch as the inverse scattering transform [] the tanh fnction method [] the homogeneos alance method [] the axiliary fnction method [4] the Exp-fnction method [5-8] and so on Very recently a new technie called "extended homoclinic test technie"was proposed [9] and has een applied to see periodic solitary wave soltions of integrale eations [0] In this wor we apply the technie to the (+-dimensional K eation New exact soltions inclding doly periodic wave soltions doly soliton soltions and periodic solitary-wave soltions are otained ROCEDURES FOR SOLVING THE (+D K EQUATION The (+-dimensional Kadomtsev-etviashvili(K eation reads as []: By sing ainlevé analysis we sppose ( xt where F (x y z t is an nnown real fnction Sstitting ( into ( we have a Hirota ilinear eation: x xx xxxx xx yy zz (lnf ( ( z 4 D x D t D x D y D F F 0 (

2 878 Z Li Z Dai and J Li Then y introdcing different ansätz test fnction F(x y z t to E ( we can otain a series of exact soltions to the (+D K eation ( ( Sppose that the test fnction F(x y z t has the following ansätz: F(x y z t (4 y+ lz ωt y+ lz ωt cos[(x + y + lz + ωt] e where pl and ω are constants to e determined later On sstitting (4 into ( eating all the coefficients of different powers of e iy+lz-ωt (i-0 to zero yields a set of algeraic eations Solving the reslting eations simltaneosly we get + l ( p ω p + (5 4p Case (I When p i we have ± + l 0 ω Therefore we get a doly periodic wave soltion where (x yz t ( 8 + cos[(x + y + lz + t] + 9cos[(x + y + lz + 9 t] ( cos[(x + y + lz t] + cos[(x + y + lz + t] l are aritrary real constants Case (II When p we have ± + l 6 ω 5 Therefore we otain a periodic solitary-wave soltion (x yz t 4 [ 4 ± 4cosh( ξsin( sinh( ξcos( ] (6 [sinh( ξ ± cos( ] where ξ (x + y + lz 5 t η (x + y + lz 5 ( Let the test fnction F(x y z t e t and + l 6 F y+ lz ωt y+ lz ωt cos[(x + y + lz] e where pl and ω are constants to e determined By sing a similar procedres to derive (5 we otain + l p + ω (p + 4p (i Choosing and p i we get + l ω 0 ± Ths a triangle fnction soltion to eation ( is given as sec [(x + y lz] 4 + (ii Choosing and p i we have + l 7 ± ω 6 Therefore we get a doly periodic wave soltion

3 New eriodic Solitary-Wave Soltions [ 8 + cos[(x + y + lz t] + 9cos[(x + y + lz + 6 t] ] [ cos[(x + y + lz 6 t] + cos[(x + y + lz] ] (iiiwhen and p we have + l ± ω 0 Then we otain periodic solitary-wave soltions 4 [ 4 ± 4cosh( ξsin( sinh( ξcos( (x yz t [sinh( ξ ± cos( ] 67 where ξ (x + y + lz 0 t η (x + y + lz and is an aritrary constant ( We now sppose that the test fnction F(x y z t has the following ansätz: F y ωt y ωt cos[(x + lz + λt] e where pl ω and λ are constants to e determined By sing a similar procedres to derive (5 we get p + l p + l ω + p λ + p (7 p + p + [( + l p (p 4p [( + l + (p + ] + ] By choosing we have three types of soltions to E ( 4 Type (i Choosing p i Q satisfies Q > we have ( Q (Q Q 4 + l ξ (x + y ω t η Q(x + lz + λt (8 Conseently we otain a doly periodic wave soltion 8 (x yz t 8 + Q + 4( + Q cos( ξcos( + 8Q sin( ξsin( (9 [ cos( ξ + cos( ] where ηξ are given y (8 and ωλ are given y (7 Soltion expressed y (9 is a doly periodic wave soltion with different period aot different spatiotemporal variale ( x y t and (x z t respectively Type (ii Choosing p iq satisfies > / 4Q Then from soltion (9 we have a doly soliton soltion

4 880 Z Li Z Dai and J Li 9 (x yz t 8 + Q + 4( + Q cosh( ξcosh( 8Q sinh( ξsinh( (0 [ cosh( ξ + cosh( ] ( Q (4 Q where ξη are given y (8 and + l ; Q are real constants Q Type (iii Choosing solitary-wave soltion p i iq then from soltion (9 we have a periodic 0 (x yz t 8 + Q 4( Q cos( ξcosh( + 8Q sin( ξsinh( ( [ cos( ξ + cosh( ] ( + Q (4 + Q where ξ η are given y (8 and + l ; Q are real constants Q (4 Finally we sppose that the test fnction F(x y z t has the following ansätz: F y ωt y ωt cos[(x + y + lz + ω t] e where pl and ω are constants to e determined By sing a similar procedres to derive (5 we get l (p ( p ω p + ( p + 4p [ l + (p + ] [p l + ] Choosing ( is an aritrary constant then ( is redced to (p + (4p + (p + (p + ( p l ω p p Therefore we have p i QQ > 4 and ( + Q (4 Q Q ( + Q l ω p i iqq > ( Q ( Q ( + Q (4 Q + Q + Q ( + Q ( Q l ω Conseently we otain the following soltions (x yz t 8 + Q + 4( ( + Q cos( ξcos( + 8Q sin( ξsin( [ cos( ξ + cos( ]

5 New eriodic Solitary-Wave Soltions 88 where (x ξ η ς yz t 8 + Q 4( Q cos( ςcosh( τ + 8Q sin( ςsinh( τ [ cos( ς + cosh( τ] ( Q ( Q [x + ( + Q y t] ( + Q (4 Q ( Q ( [x + ( + Q y + z + Q Q ( + Q ( + [x + ( Q y Q t] Q t] and τ ( + Q (4 + Q ( + Q ( + [x + ( Q y + z + Q Q Q are aritrary real constants Q t] CONCLUSION In this paper the extended homoclinic test technie is applied to solve the (+-dimensional Kadomtsev-etviashvili(K eation As a reslt explicit soltions inclding doly periodic wave soltions doly soliton wave soltions and periodic solitary-wave soltions are otained these soltions enrich the strctres of exact soltions 4 ACKNOWLEDGEMENTS This wor is spported y the National Natral Science Fondation of China nder Grant No 0608 the Natral Science Fondation of Ynnan rovince nder Grant No 00CD080 and the Science Research Fondation of Ynnan rovince Edcational Department nder Grant No 00Y079 5 REFERENCES M J Alowitz A Clarson Solitons Nonlinear Evoltion Eations and Inverse Scattering Transform Camridge Univ ress 99 E J ares B R Dffy An atomated tanh-fnction method for finding solitary wave soltions to non-linear evoltion eations Compt hys Commn Z S Feng Comment on "On the extended applications of homogeneos alance method" Appl Math Compt S Zhang T C Xia A generalized new axiliary eation method and its applications to nonlinear partial differential eations hys Lett A X H(Benn W J H He Solitary soltions periodic soltions and compacton-lie soltions sing the Exp-fnctionmethod Compters and Mathematics with Applications

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