Solitary Wave Solutions of KP equation, Cylindrical KP Equation and Spherical KP Equation

Transcription

1 Commun. Theor. Phs. 67 (017) Vol. 67 No. Februar Solitar Wave Solutions of KP equation Clindrical KP Equation and Spherical KP Equation Xiang-Zheng Li ( 李向正 ) 1 Jin-Liang Zhang ( 张金良 ) 1 and Ming-Liang Wang ( 王明亮 ) 1 1 School of Mathematics and Statistcs Henan Universit of Science and Technolog Luoang China School of Mathematics and Statistcs Lanzhou Universit Lanzhou China (Received September 016; revised manuscript received December ) Abstract Three (1)-dimensional equations KP equation clindrical KP equation and spherical KP equation have been reduced to the same KdV equation b different transformation of variables respectivel. Since the single solitar wave solution and -solitar wave solution of the KdV equation have been nown alread substituting the solutions of the KdV equation into the corresponding transformation of variables respectivel the single and -solitar wave solutions of the three (1)-dimensional equations can be obtained successfull. PACS numbers: Yv 04.0.Jb DOI: / /67//07 Ke words: -Solitar wave solution KP equation clindrical KP equation spherical KP equation transformation of variables 1 Introduction Man famous nonlinear evolution equations such as Korteweg-de Vries (KdV) Modified KdV (mkdv) Kadomstev Perviashvili (KP) Coupled KP and Zaharov Kuznetsov (ZK) have been obtained in nonlinear propagation of dust-acoustic wave especiall the dustacoustic solitar wave (DASW) in space and laborator plasma. 1] It is nown that the transverse perturbations alwas exist in the higher-dimensional sstem. Anisotrop is introduced into the sstem and the wave structure and stabilit are modified b the transverse perturbation. Recent theoretical studies for ion-acoustic/dustacoustic waves show that the properties of solitar waves in bounded nonplanar clindrical/spherical geometr differ from that in unbounded planar geometr. A dissipative clindrical/spherical KdV is obtained b using the standard reductive perturbation method in Ref. ]. The clindrical KP equation (CKP) has been introduced b Johnson 3 4] to describe surface wave in a shallow incompressible fluid. A spherical KP (SKP) equation is obtained b using the standard reductive perturbation method. 5] We consider the classical KP equation in the form (u t 6uu x u xxx ) x su 0 (1) where s is a constant. The KP equation is also derived using reductive perturbation method in superthermal dust plasma and the stead state solution has been given. 6] The clindrical KP equation (CKP) (u t 6uu x u xxx 1 t u )x 3α t u 0 () where α is a constant has also been investigated to obtain deca mode solutions b means of Hirota method and simplified homogeneous balance method respectivel. 7 8] We also consider the classical spherical KP equation (SKP) in the form (u t 6uu x u xxx 1 ) t u 1 x v 0 t (u 1 u ) 0 (3) where v 0 is a constant. An exact solitar wave solution of SKP that demonstrates the amplitude and wave velocit of solitar wave are uniquel determined b the parameters of the sstem and onl depending on the initial conditions has been obtained in Ref. 5]. The KdV equation has been researched b man authors. The multi-soliton solutions of the KdV equation with general variable coefficients have been completel obtained b homogeneous balance principle. 9 10] Some solutions which possess movable singular points while their energies are onl redistributed without dissipation of KdV equation have been obtained through the modified bilinear Bäclund transformation. 11] An n-soliton solution with the bell shape has been obtained in Ref. 1] whose stationar height is an arbitrar constant c. The rational solutions solitar wave solutions triangular periodic solutions Jacobi periodic wave solutions and implicit function solutions of KdV equation have been constructed b means of an extended algebraic method. 13] If a transformation of variables between the KdV equation and other equation can be constructed the results of KdV equation above can be used directl. Supported b the National Natural Science Foundation of China under Grant No and the Doctoral Foundation of Henan Universit of Science and Technolog under Grant No and the Science and Technolog Innovation Platform of Henan Universit of Science and Technolog under Grant No. 015XPT001 Corresponding author @qq.com c 017 Chinese Phsical Societ and IOP Publishing Ltd

2 08 Communications in Theoretical Phsics Vol. 67 In the present paper we aim to find solitar wave solutions of KP Eq. (1) CKP Eq. () and SKP Eq. (3). The paper is organized as : In Sec. the KP Eq. (1) CKP Eq. () and SKP Eq. (3) are reduced to the same KdV equation b different transformation of variables respectivel. In Sec. 3 the single solitar wave solutions and -solitar wave solutions of KP Eq. (1) CKP Eq. () and SKP Eq. (3) can be obtained in terms of the corresponding transformation of variables respectivel since the solutions of the KdV equation have been nown alread. In Sec. 4 some conclusions are made. Reduction of KP CKP and SKP.1 Reduction of KP In Eq. (1) we assume that u(x t) w(ξ t) ξ x q 1 ( t) (4) where q 1 q 1 ( t) is a function to be determined later. Substituting Eq. (4) into Eq. (1) ields an equation as ξ (w t 6ww ξ w ξξξ ) sq 1 w ξ (sq 1 q 1t )w ξξ 0. (5) Setting the coefficients of w ξ and w ξξ to zero ields sq 1 0 sq 1 q 1t 0. (6) The sstem (6) admits the following solution: q 1 ( t) λ sλ t (7) where λ is a nonzero arbitrar constant. Using Eq. (7) the expression (4) becomes u w(ξ t) ξ x λ sλ t. (8) And after integrating (5) with respect to ξ once and taing the constant of integration to zero Eq. (5) becomes the classical KdV equation for w w(ξ t) w t 6ww ξ w ξξξ 0. (9) From the discussion above we come to the conclusion that the KP Eq. (1) for u u(x t) is reduced to the KdV Eq. (9) for w w(ξ t) b using the transformation of variables (8) if w(ξ t) is a solution of KdV Eq. (9) substituting it into Eq. (8) then we have the exact solution of the KP Eq. (1).. Reduction of CKP In Eq. () we assume that u(x t) w(ξ t) ξ x q ( t) (10) where q q ( t) is a function to be determined later. Substituting Eq. (10) into Eq. () ields an equation as ξ (w t 6ww ξ w ξξξ ) t 6α q t w ξ ( 3α ) t q q t w ξξ 0. (11) Setting the coefficients of w ξ and w ξξ to zero ields t 6α 3α q 0 t q q t 0. (1) The sstem (1) admits a solution: q ( t) 1 1α t t 3α t. (13) Using Eq. (13) the expression (10) becomes u w(ξ t) ξ x 1 1α t t 3α t. (14) And after integrating (11) with respect to ξ once and taing the constant of integration to zero Eq. (11) becomes the classical KdV Eq. (9) for w(ξ t). From the discussion above we come to the conclusion that the CKP Eq. () for u u(x t) is reduced to the KdV Eq. (9) for w w(ξ t) b using the transformation of variables (14) if w(ξ t) is a solution of KdV Eq. (9) substituting it into (14) then we have the exact solution of CKP Eq. ()..3 Reduction of SKP In Eq. (3) we assume that u(x t) w(ξ t) ξ x q 3 ( t) (15) where q 3 q 3 ( t) is a function to be determined later. Substituting Eq. (15) into Eq. (3) ields an equation as ξ (w t 6ww ξ w ξξξ ) q 3 (v 0 t q 3 ) v 0 t w ξ ( 1 ) v 0 t q 3 q 3t w ξξ 0. (16) Setting the coefficients of w ξ and w ξξ to zero ields 1 q 3 (v 0 t q 3 ) 0 v 0 t q 3 q 3t 0. (17) The sstem (17) admits a solution: q 3 ( t) v 0 t µ (18) where µ is a nonzero arbitrar constant. Using Eq. (18) the expression (15) becomes u w(ξ t) ξ x v 0 t µ (19) and after integrating Eq. (16) once taing the constant of integration to zero Eq. (16) becomes the classical KdV Eq. (9). From the discussion above we come to the conclusion that the SKP Eq. (3) for u u(x t) is reduced to the KdV Eq. (9) for w w(ξ t) b using the transformation of variables (19) if w(ξ t) is a solution of KdV Eq. (9) and substituting it into Eq. (19) we have the exact solution of the SKP Eq. (3). 3 Solitar Wave Solutions of KP CKP and SKP In previous section the KP Eq. (1) CKP Eq. () and SKP Eq. (3) have been reduced into the same classical KdV Eq. (9) b the transformation (8) (14) and (19) respectivel. The KdV Eq. (9) is of phsicall importance and its solutions have been nown for man researchers

3 Communications in Theoretical Phsics No. for instance according to Ref. 9] the KdV Eq. (9) has single solitar wave η ξ 3 t x0 (0) w(ξ t) (1 ) w(ξ t) 09 where and x0 are arbitrar parameters. Eq. (9) also has -soliton solution 1 e η1 e η (1 ) e η1 η a1 ( e η1 η 1 e η1 η ) (1 e η1 e η a1 e η1 η ) And KdV (1) where i and xi are arbitrar parameters ηi i ξ i3 t xi (i 1 ) a1 (1 ) /(1 ). 3.1 Solitar Wave Solutions of KP Substituting Eq. (0) into Eq. (8) we have the single solitar wave solution for KP Eq. (1) which is expressed b u(x t) η (x λ) (sλ 3 )t x0 () (1 e η ) where and x0 are arbitrar parameters. Substituting Eq. (1) into Eq. (8) we have the -solitar solution for KP Eq. (1) which is expressed b e η1 e η (1 ) e η1 η a1 ( e η1 η 1 e η1 η ) u(x t) 1 (3) (1 e η1 e η a1 e η1 η ) where i and xi are arbitrar parameters ηi i (x λ) (i sλ i3 )t xi (i 1 ) a1 (1 ) /(1 ). 3. Solitar Wave Solutions of CKP Substituting Eq. (0) into Eq. (14) we have the single solitar wave solution for CKP Eq. () which is expressed b u(x t) (1 ) ( 1 η x 3α t x0 1α where and x0 are arbitrar parameters. The solution (4) is shown in Figs. 1 4 with 0. x0 3 α 1. Fig. 1 Plots of solution (4) with t 0. Fig. 3 Plots of solution (4) with t 0.5. Fig. Plots of solution (4) with t 0.1. Fig. 4 Plots of solution (4) with t 1. (4)

4 Communications in Theoretical Phsics 10 Substituting Eq. (1) into Eq. (14) we have the -solitar solution for CKP Eq. () which is expressed b e η1 e η (1 ) e η1 η a1 ( e η1 η 1 e η1 η ) u(x t) 1 (1 e η1 e η a1 e η1 η ) where i and xi are arbitrar parameters (( 1 ) (1 ) ηi i x 3α. t x (i 1 ) a i 1 i 1α (1 ) The solution (5) is shown in Figs. 5 8 with α 1 x1 x Fig. 5 Plots of solution (5) with t 0. Fig. 6 Plots of solution (5) with t 0.5. Fig. 7 Plots of solution (5) with t 1. Fig. 8 Vol. 67 (5) Plots of solution (5) with t. 3.3 Solitar Wave Solutions of SKP Substituting Eq. (0) into Eq. (19) we have the single solitar wave solution for SKP Eq. (3) which is expressed b ) ] (v 0 t µ x0 (6) u(x t) η x (1 e η ) where and x0 are arbitrar parameters. Substituting Eq. (1) into Eq. (19) we have the -solitar solution for SKP Eq. (3) which is expressed b e η1 e η (1 ) e η1 η a1 ( e η1 η 1 e η1 η ) u(x t) 1 (1 e η1 e η a1 e η1 η ) where i xi are arbitrar parameters (( v ) (1 ) 0 ηi i x i t xi (i 1 ) a1. (1 ) (7) 4 Conclusion balance method9 10] to obtain single solitar wave solu- In this paper b maing corresponding transformation of variables the KP equation clindrical KP equation and spherical KP equation are all reduced to the same classical KdV equation which can be solved b using homogeneous tion and -soliton solution. Substitutiong the solitar solutions of the KdV equation into the corresponding transformation of variables we have the solitar wave solutions of the KP equation clindrical KP equation and spheri-

5 No. Communications in Theoretical Phsics 11 cal KP equation respectivel. It is interesting to research CKP equation and SKP equation but avoid the singularit point analsis when t 0. The analsis in the present paper ma be extended to other wors to mae further progress. Acnowledgments The authors are ver grateful to the referees for their invaluable comments. References 1] H.Y. Wang and K.B. Zhang J. Sichuan Normal Univ. 36 (013) 911 (in Chinese). ] J.K. Xue Phs. Lett. A 3 (004) 5. 3] R.S. Johnson J. Fluid Mech. 97 (1980) ] R.S. Johnson A Modern Introduction to the Mathematical Theor of Water Waves Cambridge Universit Press Cambridge (1997). 5] J.K. Xue Phs. Lett. A 314 (003) ] N.S. Saini Nimardeep Kaur and T.S. Gill Advan. Space Res. 55 (015) ] S.F. Deng Appl. Math. Comput. 18 (01) ] M.L. Wang J.L. Zhang and X.Z. Li Appl. Math. Lett. 6 (016) 9. 9] M.L. Wang and Y.M. Wang Phs. Lett. A 87 (001) ] W.P. Zhong R.H. Xie M. Belié N. Petrovié G. Chen and L. Yi Phs. Rev. A 78 (008) ] Z.Y. Chen J.B. Bi and D.Y. Chen Commun. Theor. Phs. 41 (004) ] F.K. Guo and Y.F. Zhang Commun. Theor. Phs. 46 (006) ] X.L. Yang and J.S. Tang Commun. Theor. Phs. 48 (007) 1.