Classification of Single Traveling Wave Solutions to the Generalized Strong Nonlinear Boussinesq Equation without Dissipation Terms in P = 1

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1 Journal of Applied Matheatics and Physics 5-59 Pulished Online Feruary ( Classification of Single Traveling Wave Solutions to the Generalized Strong Nonlinear Boussinesq Equation without Dissipation Ters in P Xinghua Du Departent of Matheatics Northeast Petroleu University Daqing China Eail: xinghuadu@6co Received January 5 ; revised Feruary 5 ; accepted Feruary Copyright Xinghua Du This is an open access article distriuted under the Creative Coons Attriution License which perits unrestricted use distriution and reproduction in any ediu provided the original wor is properly cited In accordance of the Creative Coons Attriution License all Copyrights are reserved for SCIRP and the owner of the intellectual property Xinghua Du All Copyright are guarded y law and y SCIRP as a guardian ABSTRACT By the coplete discriination syste for polynoial ethod we otained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq equation without dissipation ters in p KEYWORDS Coplete Discriination Syste for Polynoial; Traveling Wave Solution; Generalized Strong Nonlinear Boussinesq Equation without Dissipation Ters Introduction There are any ethods of otaining the exact solutions for nonlinear evolution equations such as the hoogeneous alance ethod [] the inverse scattering ethod [] Hirotas ilinear transforation [] the extended tanh-function ethod [] the sech-function ethod [5] and so on Liu introduced coplete discriination syste for the polynoial ethod to otain the classification of traveling wave solutions to soe nonlinear evolution equations [6-8] In [9] the generalized strong nonlinear Boussinesq equation without dissipation ters was given y tt xxxx p+ p+ ( u + u u+ u + u ( xx > p > p are constants When p Equation ( ecoes tt xxxx ( u + u u+ u ( xx Equation ( is an iportant odel equation in physics It descries the wave propagation in the wealy nonlinear and dispersive edia When > or < the Equation ( ecoes good Boussinesq equation [] or ad Boussinesq equation [] The good Boussinesq equation and ad Boussinesq equation have een studied y any authors [-7] But the classification of single traveling wave solutions to these equations hasn't een studied In the present paper we consider the following generalized strong nonlinear Boussinesq equation without dissipation ters in p : tt xxxx ( u + u u+ u + u ( xx > are constants By using Liu s ethod the classification of single traveling wave solutions to Equation ( is otained

2 X H DU ET AL 5 The Traveling Wave Solutions to the Equation ( Tae wave transforation ( ( u x t u ξ ξ x ωt ( Sustituting Equation ( into Equation ( yields the following nonlinear ordinary difference equation: ω (5 ( ( u u u u u Integrating Equation (5 once with respect to ξ and setting the integration constant to zero yields: ω ( (6 u u u u u Integrating Equation (6 twice with respect to ξ yields: ω u u + u + u + cu + c (7 ( c and c are aritrary constants In order to find the traveling wave solutions to the Equation ( let us solve Equation (7 In this article there are two cases to discuss the exact solutions of Equation (7 according to the aritrary constant c Case c then Equation (7 ecoes ω u u u + u + u+ c ( Integrating Equation (8 once yields If ( du ( ε ( ξ ξ ± (9 εuf u ( ω c F u u + u + u+ ( > we tae ε ; if < we tae ε The coplete discriination syste for the third F u is given as follows: order polynoial ( ( c 68 ( + ω ( 8 8ω D 66ω In order to otain the solutions to the Equation (9 according to the coplete discriination syste for the F u there are four cases to e discussed third order polynoial ( D < F( u ( u α ( u β Case αβ are real constants α β β > If ε when α > β and u > β fro Equation (9 we give the solution of Equation (7 as follows: when α < and u < β we have ± α α β ξ ξ ( u u( α β α β ( ( ln u β ± α α β ξ ξ ( u u( α β β α ( ( ln u β (8 ( ( (

3 5 X H DU ET AL when β > α > we have ± α β α ξ ξ ( u + u( ( u α β α β ( ( arcsin β α If ε when α > β and u > β fro Equation (9 we give the solutions of Equation (7 when α < and u < β we have when β > α > we have ± α β α ξ ξ ( u+ u( α β β α ( ( ln u β ± α β α ξ ξ ( u u( α + β α β ( ( ln u β ± ( u+ + u( ( u α β β α α ( α β( ξ ξ arcsin β α ( (5 (6 (7 Case D F u u α α is real constant If ε when u > α we have ( ( If ε when u < α we have α u + α α ξ ξ ( α u + α α ξ ξ ( Case > D < αβγ are different real constants If ε when u < γ we have β γαsn ( α β ( ξ ξ u ( β γsn ( α γ ( ξ ξ ( α γ F( u ( u α( u β( u γ ( β ( γ γ α β α β γ ( α β sn ( α β ( ξ ξ β( α γ u β γsn ( α γ ( ξ ξ ( α γ If ε when γ < u < β we have (8 (9 ( β βαsn ( α β ( ξ ξ + αβ u β αsn ( α γ ( ξ ξ + β u γβ β β γ sn α γ ξ ξ β ( ( ( ( (

4 X H DU ET AL 5 α β β α ( γ ( γ Case < we have F u uα u l + s α ls are all real constants and α > ls > ( ( ( s a α( c d α( d c c α l sα acn ( ξ ξ + u sα ccn ( ξ ξ + d Case c In order to solve Equation (7 when ( α s l l sα ( d l s E E ± E + + u ξ w + ξ Coining the expression (7 with Equation (5 yields ( ω p ( w F w w pw qw r > we tae the transforation as follows (5 ξ ( ω + 7 q 7 And + 6 6ω 8c r + c 5 When < we tae the following transforation: u ξ w + ξ Coining the expression (7 with Equation (7 yields p ( ω and ( ( w F w w pw qw r (7 ξ ( ω + 7 q ω + 8c r c 5 The coplete discriination syste for the fourth order polynoial F ( w w + pw + qw + r as follows:

5 5 X H DU ET AL D D p D 8rp p 9 q 7 (9 D p r p q + 6prq r p q + 6 r E 9p pr In order to otain the solutions to Equation (6 and Equation (8 according to the coplete discriination syste for the fourth order polynoial F( w there are nine cases to e discussed Case D < D and D then F( w (( w l + s s > For > the solution of Equation (7 is Case D D and ( ( ξ ξ u stan s + l D then F( w w For ( ξ ξ u > the solution of Equation (7 is ( ( Case D > D D and > when w > α or w < β the solution of Equation (7 is α > β For E F ( w ( w α ( w β ( α β( ξ ξ β α β u coth + when β < w < α the solution of Equation (7 is ( α β( ξ ξ β α β u tanh + > ( ( Case D > D D and w> α w> β or w< α w< β the solution of Equation (7 is E then F( w ( w α ( w β when u + ( β α ( ξ ξ ( α β α > ( when < w> α w< β or w< α w> β the solution of Equation (7 is u + ( β α ( ξ ξ ( α β α (5

6 X H DU ET AL 55 Case 5 D > D > and and w > β or when α < γ and w < γ we have D then F( w ( w α ( w β( w γ If > exp ± ( α β( α γ( ξ ξ u β ( α γ u γ + + ( α β u α + when α > β (6 when α > β and w < γ or when α < γ and w < β we have exp ± ( α β( α γ( ξ ξ u β ( γ α u γ + + ( β α u α + when β > α > γ we have ± sin ( β α( α γ( ξ ξ u+ β ( α γ + u+ γ ( α β u + α ( β α (7 (8 If < when α > β and w > β or when α < γ and w < γ we have ± exp ± ( α β( α γ( ξ ξ u+ β ( γ α u+ γ ( β α u α u + + α (9

7 56 X H DU ET AL when α > β and w < γ or when α < γ and w < β we have ± exp ± ( α β( α γ( ξ ξ u+ β ( α γ u+ γ ( α β u α u + + α ( when β > α > γ we have ± sin ( β α( α γ( ξ ξ γ α l ( α l + s Case 6 > > u+ + β α γ + + γ α β u + α ( β α ( u ( α( α l and ( α l + s ( α l + s D F( w ( wα ( wα ( wα ( w α D D and > α > α > α > α If > when w > α or w < α the solution of Equation (7 is ( α α( α α α( αα sn ( ξ ξ α ( α α u ( α α( α α ( α α sn ( ξ ξ ( α α when α < w < α the solution of Equation (7 is ( ( If ( α α( α α α( α α sn ( ξ ξ α ( α α u ( α α( α α ( α α sn ( ξ ξ ( α α < when α > w > α the solution of Equation (7 is (

8 X H DU ET AL 57 ( α α( α α α( αα sn ( ξ ξ α ( αα u ( α α( α α ( α α sn ( ξ ξ ( αα when α < w < α the solution of Equation (7 is ( α α( α α α( αα sn ( ξ ξ α ( α α u ( α α( α α ( α α sn ( ξ ξ ( α α ( αα( α α ( α α ( α ξ Case 7 DD and < The solution of Equation (7 (when ( D ( ( α ( β ( F w w w w l + s α > β and s > > we tae the positive sign; when < s ( acn α β ± ( ξ ξ + s ( α β ccn ( ξ ξ ± + d u ± a ( α + β c ( α β d ( α β ( α β E (( α β s ( α β s + l l E± E + Case 8 DD and > solution of Equation (7 is E + d c + D then ( ( we tae the negative is s c α l d α l s ( (( ( (5 (6 F w w l + s w l + s s s > The ( η( ξ ξ + ( η( ξ ξ ( ( + ( ( asn cn c dcn u ± sn η ξ ξ η ξ ξ ( + ( + s c d c d η c + d ( l l + s + s ss E+ E Case 9 DD < and a lc + sd ld s sc c s d l l ( D then ( ( α ( F w w w l + s α l and s are real (7 nuers If > we have

9 58 X H DU ET AL ± ( ( l l + s ξξ e γ + ( α l + s ( γ ± ( l l + s ( ξξ u e γ (8 γ α l ( α l + s Conclusion By the coplete discriination syste for polynoial ethod we have otained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq without dissipation ters in p These solutions include trigonoetric periodic solutions rational function solution hyperolic funtion solutions Jacoi elliptic function solutions and so on This ethod is siple and efficient Acnowledgeents The project is supported y Scientific Research Fund of Education Departent of Heilongjiang Province of China under Grant No 59 REFERENCES [] Fan EG (998 A Note on the Hoogenous Balance Method Physics Letters A [] Alowitz MJ and Clarson PA (99 Solitons Non-Linear Evolution Equations and Inverse Scattering Transfor Caridge University Press Caridge [] Hirota R (97 Exact Envelope-Soliton of a Nonlinear Wave Equation Journal of Matheatical Physics [] Ma WX and Fuchssteiner B (996 Explicit and Exact Solutions to a Kologorov-Petrovsii-Pisunov Equation International Journal of Non-Linear Mechanics [5] Ma WX (99 Travelling Wave Solutions to a Seventh Order Generalized KdV Equation Physics Letters A 8 - [6] Liu CS ( Applications of Coplete Discriination Syste for Polynoial for Classifications of Traveling Wave Solutions to Nonlinear Differential Equations Coputer Physics Counications [7] Liu CS (7 Classification of All Single Traveling Wave Solutions to Calogero-Focas Equation Counications in Theoretical Physics (Beijing [8] Liu CS (6 Direct integral ethod coplete discriination syste for polynoial and applications to classifications of all single travelling wave solutions to nonlinear differential equations: a survey arxiv: nlin/6958v [9] Zhang WG and Tao T (8 Analysis of Solitary-Wave Shape and Solutions of the Generalized Strong Nonlinear Boussinesq Equation Acta Matheatica Sientia 8A [] Whitha GB (97 Linear and Nonlinear Wave Springer New Yor [] Zhaarov VE (97 On Stochastization of One-Diensional Chains of Nonlinear Oscillation Soviet Physics-JETP 8 8- [] McKean HP (98 Boussinesq s Equation on the Circle Pure and Applied Matheatics [] Manoranjan VS et al (985 Nuerical Solution of the Good Boussinesq Equation SIAM: SIAM Journal on Scientific Coputing [] Weiss J (985 The Painlevé Property and Baclund Transforation for the Sequence of Boussinesq Equations Journal of Matheatical Physics [5] Hu XG Wu YH and Li L ( New Traveling Wave Solutions of the Boussinesq Equation Using a New Generalized

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