Research Article On Compact and Noncompact Structures for the Improved Boussinesq Water Equations

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1 Mthemticl Problems in Engineering Volume 2013 Article ID pges Reserch Article On Compct nd Noncompct Structures for the Improved Boussinesq Wter Equtions Jun Yin 12 nd Hongyi Chen 3 1 Institute for Fiscl Science Reserch Ministry of Finnce of Chin Beijing Chin 2 Pnzhihu Municipl Finnce Bureu of Chin Pnzhihu Sichun Chin 3 Business School Imperil College London South Kensington Cmpus Exhibition Rod London SW7 2AZ UK Correspondence should be ddressed to Jun Yin; yinjun520777@sohu.com Received 3 November 2013; Accepted 15 November 2013 Acdemic Editor: Shoyong Li Copyright 2013 J. Yin nd H. Chen. This is n open ccess rticle distributed under the Cretive Commons Attribution License which permits unrestricted use distribution nd reproduction in ny medium provided the originl work is properly cited. The nonliner vrints of the generlized Boussinesq wter equtions with positive nd negtive exponents re studied in this pper. The nlytic expressions of the compctons solitons solitry ptterns nd periodic solutions for the equtions re obtined by using technique bsed on the reduction of order of differentil equtions. It is shown tht the nonliner vrints or nonliner vrints together with the wve numbers directly led to the qulittive chnge in the physicl structures of the solutions. 1. Introduction One of the clssicl Boussinesq wter equtions cn be written by the form u tt = αu xxxx +u xx +β(u 2 ) xx (1) where u(x t) is the elevtion of the free surfce of fluid the subscripts denote prtil derivtives nd the constnt coefficients α nd β depend on the depth of fluid nd the chrcteristic speed of the long wves. It is well known tht Boussinesq eqution (1) ws originlly introduced s model for one-dimensionl wekly nonliner dispersive wves in shllow wter nd ws subsequently pplied to explin problems in the percoltion of wter in porous subsurfce strt. It lso crops up in the nlysis of mny other physicl processes. Vrious generliztions of the clssicl Boussinesq wter eqution hve been proposed nd studied to probe the dynmic properties of their solutions. The generlized Boussinesq s equtions hve been successfully pplied in costl engineering for simulting wve propgtion from the deep se to shllow-wter region nd in nvl rchitecture for computing ship wves in shllow wter (See 1 2). Ky 3 obtined the exct nd numericl solitry-wve solutions for generlized modified Boussinesq eqution. Elgryhi nd Elhnbly 1 discussedndnlyzednew exct trveling wve solutions for the two-dimensionl KdV- Burgers nd Boussinesq equtions. Li nd Wu 4 nd Li et l. 5 proposed n pproch for constructing symptotic solution of the Boussinesq equtions. Rosenu 6 proved tht the vibrtion of the nhrmonic mss-spring chin leds to new Boussinesq eqution dmitting compctons nd compct brethers. It ws formlly shown tht the collision of two compctons results in the cretion of low-mplitude compcton nd nticompcton pirs. One cn find tht mny meningful chievements tht outline the structure of compctons nd the collision of two compctons were reported in 6 8.For more detils bout the compctons nd soliton solutions for nonliner evolution equtions the reder is referred to in which mny nlyticl nd numericl methods such s the pseudospectrl methods the Glerkin method the finite differences method the sinecosine nstz nd the tnh method hve been presented. The following (2+1)-dimensionl Boussinesq wter system u tt u xx u yy (u 2 ) xx u xxxx =0 (2) ws studied in 11. The uthor discussed the stbility of solitry wve solution of (2). It is found tht pulse-like

2 2 Mthemticl Problems in Engineering solutions to the Boussinesq wter eqution re stble to liner perturbtions. Wzwz 12 studied the vrints of the following improved Boussinesq wter equtions. (I) Vrint of the (1+1)-dimensionl improved Boussinesq eqution with positive exponents is given by u tt u xx (u 2n ) xx bu n (u n ) xx =0 n>1 (3) nd with negtive exponents is given by u tt u xx (u 2n ) xx bu n (u n ) xx =0 n>1 (4) where =0nd b =0re constnts. (II) Vrint of the (2+1) dimensionl improved Boussinesq eqution with positive exponents (some typing s errors occurred for (5) nd(6) in12. This cn be found through the reding of the whole pper) is given by u tt u xx u yy 1 (u 2n ) xx 2 (u 2n ) yy b 1 u n (u n ) xx b 2 u n (u n ) yy =0 n>1 nd with negtive exponents is in the form u tt u xx u yy 1 (u 2n ) xx 2 (u 2n ) yy b 1 u n (u n ) xx b 2 u n (u n ) yy =0 n>1 where =0nd b 1 +b 2 =0. Wzwz 12 ssumed tht the solutions of (3) (4) (5) nd (6) tke the forms in terms of sine or cosine; nmely or the nstz u (x t) ={λcos β (μξ 1 ) u (x t) ={λsin β (μξ 1 ) (5) (6) ξ 1 π 2μ (7) ξ 1 π μ (8) where ξ 1 = x ct or ξ 1 = x + y ct nd μ nd c re the wve number nd the wve speed respectively. Using the blncing rule Wzwz 12 clcultedndfoundoutthe concrete vlues of β μ λ. In this pper by using mthemticl technique different from those in 9 12 we derive nlyticl expressions of the trvelling solutions with wve vrible ξ = μ(x ct) for (3) nd(4) or ξ = μx+ηy ctfor (5) nd(6) where μ η ndc re rbitrry nonzero constnts. The technique used in this pper is proved very useful in constructing the exct trvelling wve solutions for (3) (6). It is shown tht the nonliner vrints or nonliner vrints together with the wve numbers directly determine the physicl structures of the solutions such s compctons solitons solitry ptterns nd periodic solutions. Moreover the results presented in this pper include those crried out in Wzwz Vrint of the (1+1)-Dimensionl Improved Boussinesq Eqution with Positive Exponents Firstly we consider solutions of the following eqution: ( dw 2 dz ) = 0 W 2 (9) where 0 =0nd =0re constnts. When >0(9)dmits two solutions: W 1 =± 0 sin (z+a) W 2 =± 0 cos (z+a) (10) where A is n rbitrry constnt. When <0 noticing tht cosh 2 z sinh 2 z=1 we derive tht (9) hs two solutions of the form W 3 =± 0 sinh (z+a) W 4 =±i 0 cosh (z+a) (11) where i= 1. Nowwewriteoutthefollowingformulswhichwillbe used in this work: sinh x= 1 sin ix i sin cosh x=cosix x=1 sinh ix i cosx =cosh ix. (12) Remrk 1. Here we point out tht we will solve (9) inthe complex vlue district. By the theory of solution structure for the first-order ordinry differentil eqution we know tht the formuls (10) nd(11) include ll the nlyticl solutions of (9). In the following discussions we let the phse number A=0in formuls (10) (11) nd the forthcoming nlyticl expressions of trvelling wve solutions. In this section we seek forml trvelling wve solutions for (3); nmely u (x t) =u(ξ) (13) where the wve vrible ξ tkes the form ξ = μ(x ct) with constnts μ =0nd c =0. The wve vrible ξ = μ(x ct) turns (3)intothefollowing ODE: (c 2 1)u ξξ (u 2n ) ξξ bc 2 μ 2 u n (u n ) ξξ ξξ =0. (14) Integrting (14) twice ndsetting theconstntsof integrtion to be zero we hve (c 2 1)u u 2n bc 2 μ 2 u n (u n ) ξξ =0. (15)

3 Mthemticl Problems in Engineering 3 Since c 2 ppers in (15) we cn ssume tht c>0lwys holds. Cse 1. When c 2 =1if/b < 0 the solution of (15)tkesthe form u n =k 1 exp ( b (x ct)) +k 2 exp ( b (x ct)) (16) where k 1 nd k 2 re rbitrry constnts. If /b > 0wesolve(15)by u n =k 3 sin ( b (x ct))+k 4 cos ( (x ct)). (17) b Here k 3 nd k 4 re rbitrry constnts. Cse 2. When c 2 =1lettingdu n /dξ = Zwehved 2 u n /dξ 2 = Z(dZ/du n ) nd where A isfreeconstnt.choosingthephsenumbera=0 in formuls (21) nd(22) nd limiting the domin of ξ we hve the following compcton solutions: u={ 2n (c2 1) sin 2 () 2nc b (x ct) u=0 otherwise (x ct) < 2ncπ () /b u={ 2n (c2 1) cos 2 () 2nc b (x ct) (x ct) < ncπ () /b (23) (c 2 1)u u 2n bc 2 μ 2 u n Z dz =0. (18) dun u=0 otherwise. (24) Using du n =nu n 1 duwehve (c 2 1) u 2n 1 bc2 μ 2 Further clcultion of (19)gives rise to n ZdZ du =0. (19) ( du(2n 1)/2 2 ()2 ) = dξ bc 2 μ 2 (2n) 2 2n (c2 1) (u (2n 1)/2 ) 2. (20) or If /b > 0itfollowsfrom(10)nd(20)tht u (2n 1)/2 =± 2n (c2 1) u (2n 1)/2 () sin { 2ncμ μ (x ct) +A b (21) However for /b < 0 from(11) nd(20) we obtin the solitry pttern solutions: u={ 2n (c2 1) sinh 2 () 2nc b (x ct) u={ 2n (c2 1) cosh 2 (). 2nc b (x ct) (25) Remrk 2. From the formuls (21)-(22) we conclude tht the compctons nd solitry pttern solutions re independent of wve number μ when we choose the phse number equls zero. These results extend those obtined by Wzwz 12 in which the wve number μ wschosensspecilconstnt. 3. Vrint of the (1+1)-Dimensionl Improved Boussinesq Eqution with Negtive Exponents Now we consider the (1+1)-dimensionl improved Boussinesq eqution with negtive exponents; nmely =± 2n (c2 1) () cos { 2ncμ μ (x ct) +A b (22) u tt u xx (u 2n ) xx bu n (u n ) xx =0 n>1 =0 b=0. (26)

4 4 Mthemticl Problems in Engineering Since (3) nd(26) resymmetricboutn nd ( n) wecn redily obtin the following periodic solutions for (26): (2n + 1) u={ 2n (c 2 1) csc2 2nc b (x ct) >0 c b =±1 (2n + 1) u={ 2n (c 2 1) sec2 2nc b (x ct) b >0c=±1. (27) For the cse where /b < 0(26) hs solitons of the form (2n + 1) u={ 2n (c 2 1) csch2 2nc b (x ct) c =±1 (2n + 1) u={ 2n (c 2 1) sech2 2nc b (x ct) 4. Vrint of the (2+1)-Dimensionl Improved Boussinesq Eqution with Positive Exponents c =±1. (28) In this section we consider the solution of (5). The use of wve vrible ξ=μx+ηy ctcrriestheequtionintothe following ordinry differentil eqution (ODE): (c 2 μ 2 η 2 )u ξξ ( 1 μ η 2 )(u 2n ) ξξ (b 1 μ 2 +b 2 η 2 )c 2 (u n (u n ) ξξ ) ξξ =0. (29) Integrting (29) twice nd setting the constntsof integrtion to be zero we get (c 2 μ 2 η 2 )u ( 1 μ η 2 )u 2n (b 1 μ 2 +b 2 η 2 )c 2 u n (u n ) ξξ =0. (30) Using the trnsformtion du n /dξ = Z wehved 2 u n /dξ 2 = Z(dZ/du n ) nd Furthermore (c 2 μ 2 η 2 )u ( 1 μ η 2 )(u 2n ) (b 1 μ 2 +b 2 η 2 )c 2 u n Z dz (31) du n =0. (c 2 μ 2 η 2 ) ( 1 μ η 2 )(u 2n 1 ) (b 1 μ 2 +b 2 η 2 )c 2 Z dz (32) ndu =0. In the following discussions we ssume tht (c 2 μ 2 η 2 ) =0 1 μ η 2 =0ndb 1 μ 2 +b 2 η 2 =0. Integrting (32) yields (b 1 μ 2 +b 2 η 2 )c 2 2n Z 2 =(c 2 μ 2 η 2 )u ( 1 μ η 2 ) 1 2n (u2n ). It follows from (33)tht (b 1 μ 2 +b 2 η 2 )c 2 ( 2ndu(2n 1)/2 2 2n () dξ ) =(c 2 μ 2 η 2 ) ( 1 μ η 2 ) 1 2n (u2n 1 ). Setting W=u (2n 1)/2 wehve ( dw 2 dξ ) () 2 = 2nc 2 (b 1 μ 2 +b 2 η 2 ) (33) (34) (c 2 μ 2 η 2 ) ( 1 μ η 2 ) 1 2n W2. (35) (1) If ( 1 μ η 2 )/(b 1 μ 2 +b 2 η 2 ) > 0 nd n > 1/2it follows from (10)nd(35)tht W= 2n (c2 μ 2 η 2 ) 1 μ η 2 sin b 1 μ 2 +b 2 η 2 (ξ+a) W= 2n (c2 μ 2 η 2 ) 1 μ η 2 cos b 1 μ 2 +b 2 η 2 (ξ+a). (36) Letting A = 0 in formuls (36) we hve the following compcton solutions: u= { { 1 μ η 2 { u=0 sin 2 b 1 μ 2 +b 2 η 2 (μx + ηy ct) otherwise (μx+ηy ct) < 2ncπ () b 1μ 2 +b 2 η 2 1 μ η 2 (37)

5 Mthemticl Problems in Engineering 5 u= { { 1 μ η 2 { u=0 cos 2 b 1 μ 2 +b 2 η 2 (μx+ηy ct) otherwise. (μx+ηy ct) < ncπ () b 1μ 2 +b 2 η 2 1 μ η 2 (38) (2) For 1 μ η 2 /b 1 μ 2 +b 2 η 2 <0nd n>1/2weobtin the solitry pttern solutions u= { { 2n (c2 μ 2 η 2 ) 1 μ η 2 { sinh 2 2nc 1μ η 2 b 1 μ 2 +b 2 η 2 u={ 2n (c2 μ 2 η 2 ) 1 μ η 2 (μx+ηy ct) cosh 2 2nc 1μ η 2 b 1 μ 2 +b 2 η 2 (μx + ηy ct). (39) (40) Letting μ=ηnd c=μc 1 in formuls (37)to(40) we obtin the nlyticl expressions of compctons nd solitry pttern solutions for (5) which ws estblished in Wzwz s pper Vrint of the (2+1)-Dimensionl Improved Boussinesq Eqution with Negtive Exponents We firstly ssume tht (c 2 μ 2 η 2 ) =0 1 μ η 2 =0nd b 1 μ 2 +b 2 η 2 =0.Tkingthesymmetricpropertyof( n) nd n into ccount in (5) nd(6) we find the periodic solutions in the cse where ( 1 μ η 2 )/(b 1 μ 2 +b 2 η 2 )>0for (6) s follows: u={ 1μ η 2 csc 2 2n+1 b 1 μ 2 (μx + ηy ct) +b 2 η2 u={ 1μ η 2 cos 2 2n+1 b 1 μ 2 +b 2 η 2 (μx + ηy ct) (41) where n> 1/2. If ( 1 μ η 2 )/(b 1 μ 2 +b 2 η 2 )<0weobtinthesolitons u={ 1μ η 2 csch 2 2n+1 2nc 1μ η 2 b 1 μ 2 (μx + ηy ct) +b 2 η2 u={ 1μ η 2 sech 2 2n+1 2nc 1μ η 2 b 1 μ 2 (μx+ ηy ct) +b 2 η2 (42) where n> 1/2. Also choosing μ=ηnd c=μc 1 in (42) producesthe nlyticl expressions of periodic nd soliton solutions for (6) which ws obtined in Wzwz 12. Remrk 3. From the formule (37) to(46)wecnconclude tht the exponent n thewvenumbersμ nd η ndthe coefficients 1 2 b 1 ndb 2 ppering in Boussinesq eqution (5)or(6) re the min fctors which determine the chnge of physicl structures of solutions. More precisely exponent n nd number ( 1 μ η 2 )/(b 1 μ 2 +b 2 η 2 ) positive or negtive directly result in the qulittive chnges in the physicl structures of solutions. 6. Conclusion In this pper we solve the generlized Boussinesq wter equtions with positive or negtive exponents. The nlyticl expressions of the trvelling wve solutions for the nonliner equtions presented by (3) (6)reestblished. For the (1+1)-dimensionl Boussinesq equtions two outcomes re chieved. One is tht the wve number μ in the nlyticlly expressions of the trvelling wves solutions keeps s n rbitrry nonzero constnt when the corresponding phse equls zero. In other words the trvelling wve

6 6 Mthemticl Problems in Engineering solutions re independent of the wve number μ if the phse of the solutions is zero. The other is tht the exponent n together with the rtio /b positive or negtive directly determines the qulittive chnges in the physicl structures of solutions. For the (2+1)-dimensionl generlized Boussinesq equtions with positive or negtive exponents we obtin nlyticl expressions for the compctons solitons solitry ptterns nd periodic solutions. It is shown tht the exponent n the wve numbers μ nd η ndthecoefficients 1 2 b 1 ndb 2 ppering in Boussinesq eqution (5) nd(6) re the min fctors which determine the chnges of physicl structures of solutions. More precisely the exponent n nd the fctor ( 1 μ η 2 )/(b 1 μ 2 +b 2 η 2 ) positive or negtive directly led to the qulittive chnges in the physicl structures of the solutions. References 1 A. Elgryhi nd A. Elhnbly New exct trveling wve solutions for the two-dimensionl KdV-Burgers nd Boussinesq equtions Physics Letters A vol. 343 no. 1 3 pp O. Nwogu Alterntive form of Boussinesq equtions for nershorewvepropgtion JournlofWterwyPortCostl &OcenEngineeringvol.119no.6pp D. Ky The exct nd numericl solitry-wve solutions for generlized modified Boussinesq eqution Physics Letters A vol. 348 no. 3 6 pp S.Y.LindY.H.Wu ThesymptoticsolutionoftheCuchy problem for generlized Boussinesq eqution Discrete nd Continuous Dynmicl Systems B vol.3no.3pp S.Y.LiY.H.WundX.Yng Thegloblsolutionofninitil boundry vlue problem for the dmped Boussinesq eqution Communictions on Pure nd Applied Anlysisvol.3no.2pp P. Rosenu Nonliner dispersion nd compct structures Physicl Review Lettersvol.70no.5pp P.J.OlverndP.Rosenu Tri-Hmiltonindulitybetween solitons nd solitry-wve solutions hving compct support Physicl Review Evol.53no.2pp P. Rosenu On clss of nonliner dispersive-dissiptive interctions Physic D vol. 123 no. 1 4 pp A.-M. Wzwz Existence nd construction of compcton solutions Chos Solitons nd Frctls vol. 19 no. 3 pp M. S. Ismil nd T. R. Th A numericl study of compctons Mthemtics nd Computers in Simultion vol.47no.6pp Y. Zheng nd S. Li Pekons solitry ptterns nd periodic solutions for generlized Cmss-Holm equtions Physics Letters Avol.372no.23pp A.-M. Wzwz Nonliner vrints of the improved Boussinesq eqution with compct nd noncompct structures Computers nd Mthemtics with Applictions vol.49no.4pp

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