Nonlocal Symmetries and Interaction Solutions for Potential Kadomtsev Petviashvili Equation

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1 Commn. Theor. Phs. 65 (16) Vol. 65, No. 3, March 1, 16 Nonlocal Smmetries and Interaction Soltions for Potential Kadomtsev Petviashvili Eqation Bo Ren ( ), Jn Y ( ), and Xi-Zhong Li ( ) Institte of Nonlinear Science, Shaoing Universit, Shaoing 31, China (Received November, 15; revised manscript received December 16, 15) Abstract The nonlocal smmetr for the potential Kadomtsev Petviashvili (pkp) eqation is derived b the trncated Painlevé analsis. The nonlocal smmetr is localized to the Lie point smmetr b introdcing the ailiar dependent variable. Thanks to localization process, the finite smmetr transformations related with the nonlocal smmetr are obtained b solving the prolonged sstems. The inelastic interactions among the mltiple-front waves of the pkp eqation are generated from the finite smmetr transformations. Based on the consistent tanh epansion method, a nonato-bäcklnd transformation (BT) theorem of the pkp eqation is constrcted. We can get man new tpes of interaction soltions becase of the eistence of an arbitrar fnction in the nonato-bt theorem. Some special interaction soltions are investigated both in analtical and graphical was. PACS nmbers: 5.5.Yv,.3.Jr,.3.Ik Ke words: potential Kadomtsev Petviashvili eqation, nonlocal smmetr, front wave, consistent tanh epansion method 1 Introdction A large nmber of sefl methods have been proposed to find soliton soltions for nonlinear partial differential eqations. Some of the most important methods are the inverse scattering transformation, [1] the Hirota s bilinear method, [] smmetr redctions, [3] the Darbo transformation, [] the Painlevé analsis method, [5] the Bäklnd transformation (BT), [6] the separated variable method, [7] etc. [8] Among these methods, it is still qite difficlt to obtain the interaction soltions among different nonlinear ecitations. Recentl, these interaction soltions were directl obtained b sing the localization procedre related with the nonlocal smmetr and a consistent tanh epansion (CTE) method. [9 1] In this paper, we shall appl the localization procedre and CTE method to std the potential Kadomtsev Petviashvili (pkp) eqation. Some interesting reslts are discssed which might be applicable to eplain the relevant phsical processes. The paper is organized as follows. In Sec., the nonlocal smmetr for the pkp eqation is obtained with the trncated Painlevé method. To solve the initial vale problem of the nonlocal smmetr, the nonlocal smmetr is localized b prolongation the pkp eqation. The finite smmetr transformations are presented b solving the initial vale problem of the Lie s first principle. The mlti-front waves and the inelastic interactions of two front waves are analzed b sing the finite smmetr transformations. In Sec. 3, a CTE method is developed to the pkp eqation. It is proved that the pkp eqation is CTE solvable sstem. The CTE method for pkp eqation leads to a nonato-bt theorem. In Sec., some special interaction soltions are given de to the entrance of an arbitrar fnction in the nonato-bt theorem. The last section is a simple smmar and discssion. Nonlocal Smmetr and Mlti-front Waves of the pkp Eqation The (+1)-dimensional pkp eqation reads t =, (1) which describes the dnamics in two-dimensional. (1) is derived in varios phsical contets assming that the wave is moving along and all changes in are slower than in the direction of motion. [13] Varios eact soltions, inclde traveling wave soltions, linear solitar wave soltions, soliton-like soltions and some nmerical soltions have been given. [1 17] Recentl, the periodic soliton soltion, dobl periodic soltion and smmetr invariant soltions are investigated. [18 ] In this section, we shall consider the nonlocal smmetr and mlti-front wave soltion of the pkp eqation from the Painlevé analsis. The soltion of (1) can be trncated abot the singlarit manifold φ(,, t) as [5] = φ + 1, () where and 1 are fnctions with respect to the spacetime variables. B sbstitting the epansion () into (1) and balancing the coefficients of powers of φ 5 and φ independentl, we get = φ, (3) Spported b the National Natral Science Fondation of China nder Grant Nos , and 11511, the Natral Science Fondation of Zhejiang Province of China nder Grant No. LQ13A51 renbos @163.com c 16 Chinese Phsical Societ and IOP Pblishing Ltd

2 3 Commnications in Theoretical Phsics Vol. 65 1, = φ t + φ + φ φ 3φ φ. () Sbstitting the epressions (), (3) and () into (1), the field φ satisfies the following Schwarzian pkp form ( φt + φ ) ( {φ; } + 3 φ + φ ) 3 ( φ ) φ =, (5) where {φ; } = (φ /φ ) (3/)(φ /φ ) is the Schwarzian derivative. The Schwarzian form (5) is invariant nder the Möbios transformation φ aφ + b, ac bd. (6) cφ + d For the special case a = d = 1, b =, c = ǫ, the smmetr of (5) reads as σ φ = φ. (7) B sbstitting the Möbios transformation smmetr σ φ into the linearized eqation of (3), the nonlocal smmetr of the pkp eqation (1) is σ = φ. (8) The nonlocal smmetr (8) is the residal of the singlarit manifold φ. This nonlocal smmetr is ths called as the residal smmetr (RS). [1] The RS (8) can also be read ot b the trncated Painlevé epansion (). For the nonlocal smmetr (8), the corresponding initial vale problem is dū dǫ = φ, ū ǫ= =. (9) It is difficlt to solve the initial vale problem of the Lie s first principle (9) de to the intrsion of the fnction φ and its differentiations. [1] To solve the initial vale problem (9), we prolong the pkp sstem (1) sch that RS becomes the local Lie point smmetr for the prolonged sstem. B localization the nonlocal smmetr (8), the potential field of φ is introdced as φ = g. (1) The local Lie point smmetr for the prolonged sstems (1), () and (1) reads as σ = g, σ φ = φ, σ g = φg. (11) Correspondingl, the initial vale problem becomes dū dǫ = g, ū ǫ= =, d φ dǫ = φ, φ ǫ= = φ, dḡ dǫ = φg, ḡ ǫ= = g. (1) The soltion of the initial vale problem (1) for the enlarged pkp sstem (1), () and (1) is given as ū = + ǫg ǫφ + 1, φ φ = ǫφ + 1, ḡ = g (ǫφ + 1). (13) Using the finite smmetr transformations (13), one can obtain another soltion from an initial soltion. We take the trivial soltion = for (1). The mlti-front waves soltion for () and (5) is spposed as N φ = 1 + ep(k n + l n + ω n t), (1) n=1 where k n, l n and ω n are arbitrar constants. The mltifront waves soltion (1) is the soltion of () and (5) onl with the relations k 1 + 3l1, n = 1, k 1 ω n = k n (6k1 3k n + 6k 1 k n l 1 3k1 k 1 k n 6k 1 l 1 3l1 ) k1, n, l n = k n(kn 1 k n 1k n + l n 1 ), n. (15) k n 1 The mltiple-front waves of Eq. (1) present in the following form sing (1), (13) and (1) N n=1 = ǫk n ep(k n + l n + ω n t) 1 + ǫ + N n=1 ǫ ep(k n + l n + ω n t). (16) For the interaction behaviors of the mltiple-front waves (16), we discss the details of interactions between two front waves soltion as the specific and tpical. The interactions are classified into two cases, i.e., k 1 k < and k 1 k >. [1 ] It represents the head-on coalescence and overtaking coalescence respectivel. We show the evoltion of two fronts with the parameters n =, k 1 = 1, l 1 = 1, k = 1, ǫ = 1/ in Fig. 1. It illstrates two fronts along the opposite propagation direction in -ais coalescing into one large front in their interaction region of the (, )-plane. As comparison, Fig. shows coalescence of two fronts with the same propagation direction in -ais k 1 k >, in which the large-amplitde front with faster velocit overtakes the small-amplitde one. The parameters are n =, k 1 = 1, l 1 = 1, k = 3, ǫ = 1/. It is obviosl that the two front waves soltion ma combine into a single front wave after collision (soliton fsion). Neither elastic scattering nor fission does eist in both interaction modes. Remark The mltiple-front waves are established with variet of powerfl methods, sch as the Cole-Hopf transformation and pertrbation epansion method, the mltiple ep-fnction method and the Hirota s bilinear method. [1 5] In this paper, we obtain the mltiple-front waves with trncated Painlevé epansion related nonlocal smmetr.

3 No. 3 Commnications in Theoretical Phsics 33 (a) (b) c Fig. 1 Plot the propagation of two front waves epressed b (16) with the parameters n =, k 1 = 1, l 1 = 1, k = 1, ǫ = 1/. (a) t = 6; (b) t = ; (c) t = 6. It is obviosl that two fronts propagate with the opposite direction in -ais and coalesce into one large front in their interaction region. (a) (b) (c) Fig. Plot the propagation of two front waves epressed b (16) with the parameters n =, k 1 = 1, l 1 = 1, k = 3, ǫ = 1/. (a) t = 1; (b) t = ; (c) t = 1. It shows that two fronts propagate with the same direction in -ais and the large-amplitde front with faster velocit overtakes the small-amplitde one. 3 CTE Solvabilit for pkp Sstem The consistent tanh epansion writes as the following form b sing the leading order analsis [1] = + 1 tanh(f), (17) where, 1 and f are arbitrar fnctions of (t,, ). B sbstitting (17) into the pkp sstem (1) and vanishing the coefficients of powers of tanh 5 (f), tanh (f) and tanh 3 (f), we obtain 1 = f,, = f t + f 3f + f f f. (18) Collecting the coefficient of tanh (f) and sing (18), the consistent condition reads f t f f f f t 3 f f 1 f f 3 f 3 + f f + 3 f f + f f f =. (19) The coefficients of tanh 1 (f) and tanh (f) are identicall zero b sing (19). The consistent condition (19) can also be written as ( S + C + 3 R) + 3R f f =, () where C = f t /f, R = φ /φ, S = {φ; } = φ /φ (3/)(φ /φ ). The pkp sstem for, 1 and f is consistent, or, not over-determined, the epansion (17) is called a CTE and the pkp sstem (1) is CTE solvable sstem. [9] The nonato-bt theorem for the pkp eqation (1) is given b sing the CTE approach. Nonato-BT Theorem If the soltion f satisfies the consistent condition (19), then for eqation (1) is also a soltion of the pkp sstem (1) = f tanh(f) + ( φ t + φ 3φ + φ φ φ ) d + G(, t), (1) where G(, t) is arbitrar fnctions of and t. B means of the nonato-bt theorem, we can obtain some special eact soltions of the pkp eqation, in particlarl the interaction soltions among solitons and other kinds of complicated waves. In the net section, some concrete interesting eamples are given via above nonato-bt theorem. Interaction Soltions for pkp Sstem A qite trivial straight line soltion of (19) has the form f = k + l + ωt, () where k, l and ω are the free constants. Sbstitting the trivial soltion () into (1), the eact soltion of pkp sstem ields = k tanh(k + l + ωt) + G(, t) (3l + ωk + k ) 6k. (3) The nontrivial soltion of the pkp eqation is given from some qite trivial soltion of (). Actall, the soltion of the pkp eqation ma have qite rich strctres

4 3 Commnications in Theoretical Phsics Vol. 65 de to an arbitrariness fnction of (3). The parameters are k = 1, l =, ω = both in Figs. 3(a) and 3(b). One kink soliton copled to the periodic wave backgrond shows in Fig. 3(a) with the arbitrar fnction G(, t) = sin( + t). Figre 3(b) plots the interaction between one kink and one soliton soltion with the arbitrar fnction G(, t) = 1/[1 + ( + t) ]. It is obvios that the interaction behavior is different with selecting different parameters. To find the other tpes of interaction soltions, we can look for the soltions with one straight line () pls an ndetermined waves for the f field. The interaction soltions between solitons and mltiple resonant soliton soltions of the pkp eqation assme f = k+l +ωt+ 1 n ) (1+ ln ep(k i +l i +ω i t), () i=1 where k i are arbitrar constants while l i and ω i are determined b the relations l i = ± k i k (k ik + k ± l), ω i = k i k (k i k + 6k i k + k 3 ± 3k i l ± 6kl ω). (5) The soltion of pkp eqation will be obtained b sbstitting () into (1). Becase the epression of is qite complicated, we neglect write it down here and onl plot their figres nder the special vales of the parameters. Figre displas the special interaction soltion with the smbol ± in (5) as + and the parameters selected as n =, k =, l = 1, ω =, k 1 = 1, k =. The arbitrar fnction G(, t) is chosen as 1/( + t) and sin( + t) for Fig. (a) and Fig. (b) respectivel. For the interaction soltion between solitons and cnoidal periodic waves, the interaction soltion reads as f = k + l + ωt + F(X), X = k + l + ω t, (6) where k, l, ω, k, l and ω are all the free constants. Sbstitting the epression (6) into (19), we obtain an eqation abot F 1 (X) as where F 1,X F 1 + a 1F a F 1 + a 3F 1 + a =, F X = F 1, (7) a 1 = C 1 k 8k k, a 3 = 6C 1 k C k ω k 3, a = k (C 1 k C ) k + kω 6ll k + 6kl k 5 a = 6C 1 kk C k k kω + l k k C 1 and C are arbitrar constant. The soltion F 1 in (7) can be written as [11 1], + k(kω ll ) k 5 + 5k l k 6 r (r 3 r 1 ) F 1 = r 3 +. (8) S(r 1 r ) r 1 + r 3 The interaction soltion between solitons and cnoidal periodic waves of (6) reads ( ( r f = k + l + ωt + 1 r 3 (r 3 r )E π S, r 1 r ) ( r 1 r ) ) 3, m r 3 E F S, m, (9) (r1 r 3 )(r r ) r 1 r r 1 r 3 r 1 r where S is the Jacobi elliptic fnction S = sn( (r 1 r 3 )(r r )X, m), m = [(r 1 r )(r r 3 )]/[(r 1 r 3 )(r r )], E F and E π are the first and third incomplete elliptic integrals and r 1, r, r 3, r are related with a 1, a, a 3, a in the following relations a 1 = (r 1 + r + r 3 + r ), a = (r 1 r + r 1 r 3 + r 1 r + r r 3 + r r + r 3 r ), a 3 = (r 1 r r 3 + r 1 r r + r 1 r 3 r + r r 3 r ), a = r 1 r r 3 r. (3), Fig. 3 (a) Plot of one kink soliton in the periodic wave backgrond epressed b (3) with the arbitrar fnction G(, t) = sin( + t); (b) Plot of the interaction between one kink and one soliton soltion b (3) with the arbitrar fnction G(,t) = 1/[1 + ( + t) ]. The parameters are k = 1, l =, ω =.

5 No. 3 Commnications in Theoretical Phsics 35 In the ocean, there are some tpical nonlinear waves sch as interaction soltions between solitons and cnoidal periodic waves. [11] We introdce the interaction soltions ma be sefl for stding the ocean waves. Remark The CTE method has been sccessfll applied to lots of nonlinear integrable [6 8] and even nonintegrable sstems. [9 3] The interaction soltions between a soliton and the cnoidal waves, Painlevé waves, Air a waves, Bessel waves are generated with the CTE method. For the pkp sstem, an arbitrar fnction is inclded in a nonato-bt theorem. There eist man abndant interaction soltions of pkp eqation b selecting the different arbitrar fnction. The interaction behaviors for pkp eqation are ths different from other sstems via the CTE method. (b) t t Fig. (a) Plot of special interaction between one soliton and two resonant soliton soltions with the arbitrar fnction G(, t) = 1/[1 + ( + t) ]; (b) Plot of the interaction between one soliton and two resonant soliton soltions in the periodic wave backgrond with the arbitrar fnction G(, t) = sin( + t). The parameters are n =, k =, l = 1, ω =, k 1 = 1, k =. 5 Conclsion In smmar, the nonlocal smmetr of the pkp eqation is obtained with the trncated Painlevé method. To solve the initial vale problem related b the nonlocal smmetr, we prolong the pkp eqation sch that nonlocal smmetr becomes the local Lie point smmetr for the prolonged sstem. The finite smmetr transformations of the prolonged pkp sstem are derived b sing the Lie s first principle. The mlti-front wave soltion and their interaction behaviors for pkp eqation are stdied with the finite smmetr transformations. The inelastic interactions among the two-front wave are stdied which were not reported for the pkp eqation. Then, the CTE method is applied to the pkp eqation. The CTE method for the pkp sstem leads to a nonato-bt theorem. Abndant interaction soltions between solitons and other tpes of solitar waves for the pkp sstem are obtained with a nonato-bt theorem. These tpes of interaction soltions can also be given throgh smmetries redction of the prolonged sstems. In this paper, we discss the localization procedre for one particlar nonlocal smmetr, i.e., the residal smmetries. There are man methods to obtain the nonlocal smmetr, sch as the Darbo transformation, [11 1,31 3] the Bäcklnd transformation, [33] and the nonlinearizations. [3 35] How to appl these varios nonlocal smmetries to obtain new interaction soltions is an important topic. References [1] C.S. Gardner, J.M. Greene, M.D. Krskal, and R.M. Mira, Phs. Rev. Lett. 19 (1967) 195. [] R. Hirota, The Direct Method in Soliton Theor, Cambridge Universit Press, Cambridge (). [3] P.J. Olver, Application of Lie Grop to Differential Eqation, Springer-Verlag, Berlin (1986); G.W. Blman and S.C. Anco, Smmetr and Integration Methods for Differential Eqations, Springer-Verlag, New York (). [] V.B. Matveev and M.A. Salle, Darbo Transformations and Solitons, Springer, Berlin (1991). [5] J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phs. (1983) 5. [6] C. Rogers and W.K. Schief, Bäcklnd and Darbo Transformations Geometr and Modern Applications in Soliton Theor, Cambridge Tets in Applied Mathematics, Cambridge Universit Press, Cambridge (). [7] X.Y. Tang, S.Y. Lo, and Y. Zhang, Phs. Rev. E 66 () 661. [8] X. Lü, Commn. Nonlinear Sci. Nmer. Simlat. 19 (1) 3969; W.M. Moslem, S. Ali, P.K. Shkla, and X.Y. Tang, Phs. Plasmas 1 (7) 838. [9] S.Y. Lo, Std. Appl. Math. 13 (15) 37. [1] X.N. Gao, S.Y. Lo, and X.Y. Tang, J. High Energ Phs. 5 (13) 9. [11] X.R. H, S.Y. Lo, and Y. Chen, Phs. Rev. E 85 (1) 5667.

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