Approximate perturbed direct homotopy reduction method: infinite series reductions to two perturbed mKdV equations

Appro ximate p erturb ed direct homotop y reduction metho d: infinite series r edu cti ons to t w o p erturb ed mKdV equations Xiao yu Jiao a , Ruo xia Y ao a,b,c and S. Y. Lo u a,b,d, 1 a Department of Physics, Shangha i Jiao T ong Universit y , Shanghai, 200 030, China b Department of Physics, Ningb o Universit y , Ningb o , 31521 1, China c School o f Co mputer Science, Shaanxi Normal University , Xi’an, 7100 6 2, China d School o f Mathema t ics, F udan Universit y , Shanghai, 2 00433, China Abstract: An approxi mate p erturb ed direct homotop y red uct ion metho d is prop osed and applied to tw o p erturb ed mo dified Korteweg -de V ries (mKdV) equations with f o urth ord e r disp ersion and second order dissipation. The similarity redu c tion equations are deriv ed to arbitrary orders. The metho d is v alid not only for sin g le soliton solution but also for the Pa inlev ´ e I I wa v es and p erio dic w a ves exp ressed by Jacobi elliptic fun c tions for b oth fourth order disp ersion and second order dissipation. The metho d is v alid also for strong p erturbations. P A CS num b ers: 02.30. Jr Key W ords: p erturb ed mKdV equations, approxima te direct homotop y r e duction metho d, series reduction solutions It is v ery difficult to study n o nlinear phenomena lies in the fact that there are v arious non- linear systems which are usually noninteg rable. F or some types of idea cases so-called integrable mo dels o ne ma y use some t yp es of p o werful metho ds (such as the symmetry reduction metho d [1], the Darb oux transformation [2], the nonlinearization [3] or sym m e try constrain t metho d [4] etc) to fin d some kinds of exact solutions thanks to there u sually exist infi n it ely many symmetries. Ho wev er, for real nonin tegrable ph ysical systems, there are only a little of s ymmetrie s or ev en there is no sym m e try at all. In man y cases, the nonin tegrable sector of a ph ysical sys tem ma y compan y with s ome small parameters. In these cases, one may use the p ertur b a tion theory to treat the p roblems via differen t appr o ac hes. Among these app roa c hes, the approximate sym - metry redu c tion metho d m ay b e one of th e b est w a ys [5, 6]. T o fi nd symmetry reductions, one ma y use the classical, nonclassical approac hes [ ? , 1] and/or the Clarkson-Kru sk al’s (CK’s) direct 1 Corresponding Author: S. Y. Lou , sylou@sjtu.edu .cn 1 metho d [7, 8]. The CK’s direct metho d is simplest one and can b e u sed to fi nd man y group in v arian t solutions without using group theory . F u r thermore, in m ore general cases, the p ertur- bations may not b e w eak at all. F or s trong p erturb ations, some other t yp es of approac hes, such as the h omotomy analysis metho d (HAM) [9] and th e Linear [10] and n onlinear [11] n onsensitiv e homotop y approac hes etc., ha v e to b e used. In this letter, we try to com bined the C K ’s d irect symmetry redu ction metho d and HAM to an app ro ximate homotop y direct reduction approac h (AHDRA). The celebrated mo dified Kortewe g-de V ries (mKdV) equation app ears in many branc hes of nonlinear s cience. As one f orm of ap p ro ximation, the singularly p erturb ed form, u t + 6 au 2 u x + u x 3 = ǫ ( u x 2 + u x 4 ) , (1) where a = 1 or a = − 1, th e subs cr ip ts x n mean the differentiati ons with r esp ect to x in n times, has arisen in a n umber of ph ysical fields, such as mo dels of sh allo w w ater on tilted planes [12]. Soliton p erturb ation pr op ert y of the mKdV equation w as analyzed in [13–15]. In this letter, we consider t wo sp ecial form s of the ab ov e equation u t + 6 au 2 u x + u xxx = ǫu x 3 ± 1 , (2) with fourth order disp ersion (the u p sign case) and second order diss ipation (the lo w er s ign case) in terms of APDRA w hic h is a com bination of p er tu rbation theory , direct metho d and HAM [7–9]. It sh ould b e emphasize that in real physical case, the p erturb ation terms, s a y , the parameter ǫ in (2), may n ot b e small. Wh en th e p er tu rbations are n ot w eak, the HAM may b e s uccessfully applied by in tro du cing a h omotop y H ( u, q ) = 0 of the original mo del A ( u ) = 0. When the homotop y p arameter, q = 0, th e h omotop y m o del H 0 ( u ) = H ( u, 0 ) = 0 should b e solv ed via kno wn approac h. Usu ally , H 0 ( u ) is selected as a linear system. In this pap er, w e select H 0 ( u ) as an integ rable nonlinear system. Concretely , for th e p erturb ed m KdV system (2), we in tro du ce the follo win g linear homotop y mo d el (linear for the homotop y parameter), (1 − q )( u t + 6 au 2 u x + u xxx ) − q ( u t + 6 au 2 u x + u xxx − ǫu x 3 ± 1 ) = 0 . (3) It is clear that when q = 0, (3) is the well known in tegrable mKdV equation whic h can b e solv ed via man y m etho ds. When q = 1, it is just the original mo del (2). No w we can s olv e (3) via p ertur bation approac hes by taking q as a p erturbation parameter no m atter ǫ is small or not. 2 F or Eq. (3), according to p ertur b ation theory , the solution can b e expressed as a s er ies of q u = ∞ X j =0 q j u j , (4) with u j b eing fun ctions of x and t . Substituting Eq. (4) into Eq. (2) and v anishing the co efficien ts of all differen t p ow ers of q , we get O ( ǫ 0 ) : u 0 t + 6 au 2 0 u 0 x + u 0 x 3 = 0 , (5a) O ( ǫ 1 ) : u 1 t + 6 a ( u 2 0 u 1 x + 2 u 0 u 1 u 0 x ) + u 1 x 3 − ǫu 0 x 3 ± 1 = 0 , (5b) O ( ǫ 2 ) : u 2 t + 6 a ( u 2 0 u 2 x + u 2 1 u 0 x + 2 u 0 u 2 u 0 x + 2 u 0 u 1 u 2 x ) + u 2 x 3 − ǫu 1 x 3 ± 1 = 0 , (5c) · · · · · · · · · O ( ǫ j ) : u j t + 6 a j X k =0 k X l =0 u l u k − l u j − k ,x + u j x 3 − ǫu j − 1 ,x 3 ± 1 = 0 , (5d) · · · · · · · · · . The similarit y solutions for th e ab ov e equ ation are of the form u j = U j ( x, t, P j ( z ( x, t ))) , ( j = 0 , 1 , · · · ) , (6) where U j , P j and z are fu nctions with resp ect to the ind icated v ariables and P j ( z ) satisfy a system of ordinary differentia l equations, wh ic h can b e obtained b y substituting Eq. (6) into Eq. (5 ). After th e substitution, it is easily seen that the co efficien ts for P j,z z z and P j,z z P j,z are U j,P j z 3 x and 3 aU j,P j P j z 3 x resp ectiv ely . W e reserv e u pp ercase Greek lette rs for undetermined functions of z from now on. Because the fun ctions P j ( z ) d ep endent only on the v ariable z , th en the ratios of the co efficien ts are only fun ctions of z , namely , 3 aU j,P j P j z 3 x = U j,P j z 3 x Γ j ( z ) , ( j = 0 , 1 , · · · ) , (7) with the solution U j = F j ( x, t ) + G j ( x, t )e 1 3 a Γ j ( z ) P j − → F j ( x, t ) + G j ( x, t ) P ′ j ( z ). Hence, it is sufficien t to seek similarit y reductions of Eq. (5 ) in the sp ecial form u j = α j ( x, t ) + β j ( x, t ) P j ( z ( x, t )) , ( j = 0 , 1 , · · · ) , (8) instead of the general form Eq. (6). R emark : T hree freedoms in the d etermin ation of α j ( x, t ), β j ( x, t ) and z ( x, t ) can b e notified: (i) If α j ( x, t ) has the f orm α j ( x, t ) = α ′ j ( x, t ) + β j Ω( z ), then one can tak e Ω( z ) = 0; 3 (ii) If β j ( x, t ) has the f orm β j ( x, t ) = β ′ j ( x, t )Ω( z ), then one can tak e Ω( z ) = constan t; (iii) If z ( x, t ) is determined by Ω( z ) = z 0 ( x, t ), where Ω ( z ) is an y in v ertible function, then one can tak e Ω( z ) = z . Substituting Eq. (8 ) into Eq. (5a), w e can s ee that the co efficien ts for P 0 z z z , P 0 z P 2 0 , P 0 z P 0 and P 0 z z are β 0 z 3 x , 6 aβ 3 0 z x , 12 aα 0 β 2 0 z x and 3 β 0 x z 2 x + 3 β 0 z x z xx , resp ectiv ely . W e require that 6 aβ 3 0 z x = β 0 z 3 x Ψ 0 ( z ) , 12 aα 0 β 2 0 z x = β 0 z 3 x Φ 0 ( z ) , 3 β 0 x z 2 x + 3 β 0 z x z xx = β 0 z 3 x Ω 0 ( z ) , (9) and th us, applying R emark (i), (ii) and (iii), w e ha v e α 0 ( x, t ) = 0 , β 0 ( x, t ) = z x , z ( x, t ) = θ ( t ) x + σ ( t ) . (10) Eq. (5a) is th en s im p lified to θ 4 P 0 z z z + 6 θ 4 P 2 0 P 0 z + ( θ t z − θ t σ + θ σ t ) P 0 z + θ t P 0 = 0 . (11) F rom the co efficien ts of P 0 z z z , z P 0 z and P 0 z of the ab o v e equation, it is easily seen that θ t = Aθ 4 , − θ t σ + θ σ t = B θ 4 , (12) with A an d B b eing arbitrary constan ts. When A 6 = 0, Eq. (12 ) has the solution θ = − (3 A ( t − t 0 )) − 1 3 , σ = − B A + s 0 ( t − t 0 ) − 1 3 , ( 13) where s 0 and t 0 are arbitrary constants. On substitution of Eq. (8) into the general form Eq. (5d), the co efficien ts of P j,z 3 , P j − 1 ,z 3 ± 1 and P 0 z P 0 are β j z 3 x , ∓ ǫβ j − 1 z 4 x and 12 aα j β 2 0 z x , resp ectiv ely , wh ic h lead to β j − 1 z 4 x = ∓ ǫβ j z 3 x Ψ j ( z ) , 12 aα j β 2 0 z x = β j z 3 x Φ j ( z ) , ( j = 1 , 2 , · · · ) . (14) By β 0 = z x , R emark (i) an d (ii), we obtain α j = 0 , β j = z 1 ± j x , ( j = 0 , 1 , · · · ) . (15) Eq. (4), (8), (10), (13) and (15) determine the p ertu rbation series solution to Eq. (2) u = ∞ X j =0 ( − 1) j +1 (3 A ( t − t 0 )) − 1 3 (1 ± j ) ǫ j P j ( z ) , (16) 4 with the similarit y v ariable z = − x (3 A ( t − t 0 )) − 1 3 + s 0 ( t − t 0 ) − 1 3 − B A and the similarit y reduction equations are P j,z 3 = − ( j + 1) AP j − ( Az + B ) P j,z − 6 a j X k =0 k X l =0 P l P k − l P j − k ,z + ǫP j − 1 ,z 3 ± 1 , ( j = 0 , 1 , · · · ) , (17) with P − 1 = 0. When j = 0, Eq. (17) degenerates to Painlev ´ e I I t yp e equation. When A = 0, Eq. (12 ) has the solution θ = t 0 , σ = B t 3 0 t + s 0 , (18) where s 0 and t 0 are arb itrary constants. F rom Eq . (10), Eq. (18) implies an equiv alen t tra v elling w a v e f orm z = x + ct , so that w e obtain the p erturb ation series tra v elling w av e solution to Eq. (2) u = ∞ X j =0 ǫ j P j ( z ) , z = x + ct, (19) where all P j ( z ) satisfy cP j + 2 a j X k =0 k X l =0 P j − k P k − l P l + P j,z 2 − ǫP j − 1 ,z 2 ± 1 + a j = 0 , ( j = 0 , 1 , · · · ) , (20) with P − 1 = 0. T aking j = 0, it is ob vious that Eq. (20) b ecomes the zeroth order equation, the w ell k n o wn mKdV equation whic h has the general solution Z P 0 d p p c 0 − ap 4 − cp 2 − 2 a 0 p = ± ( z − z 0 ) (21) with arbitrary constants c, a 0 , c 0 and z 0 . It is also in teresting that for th e series tra v elling w a v e solution, it is n ot difficu lt to fi nd th e solution of P j can b e exp r essed b y P 0 , result r eads P j = q c 0 − 2 a 0 P 0 − cP 2 0 − aP 4 0 " c 1 j + c 2 j Z P 0 d p p ( c 0 − ap 4 − cp 2 − 2 a 0 p ) 3 − Z P 0 Z p ′ F j ( p ) p ( c 0 − ap ′ 4 − cp ′ 2 − 2 a 0 p ′ ) 3 d p d p ′ # , (22) where c 1 j and c 2 j are arbitrary constants wh ile F j ( P 0 ) ≡ 2 a j − 1 X k =1 k X l =0 P j − k P k − l P l + 2 aP 0 j − 1 X l =1 P j − l P l − ǫP j − 1 ,z 3 ± 1 + a j . (23) 5 The general solution (21) can b e rewr itten as s ome t yp es of Jacobi elliptic fu nctions [16]. F or some sp ecial selections of the constants, it can b e written as some t yp es of soliton solutions or p erio d ic wa v e solutions, for instance, if w e select a = − 1 , c = 2 k 2 , a 0 = 0 , c 0 = k 4 for the up sign (the diss ipativ e case), then w e ha v e the h yp erb olic tangent shap e kin k soliton solution P 0 = k tanh ( k x + 2 k 3 t ) ≡ k T . (24) R emark: Th e con v ergence of infinite ser ies solution Eq. (16 ) is sup erior to the fourth order disp ers ion case (the up sign case), b ecause the general terms of Eq. (16) b ecome infinitesimal for su fficien tly large time t , | 3 A ( t − t 0 ) | ≫ 1 . F or the infinite series solution (16) with the lo w er sign (the dissipativ e case), the series will b e con v ergen t for not ve ry large time, i.e. for | 3 A ( t − t 0 ) | ≪ 1 . More sp ecifically , for th e d ark solitary wa ve solution (26), we can easily fin d the closed forms for the h igher order homotop y p erturbation solutions. Here is a exp licit f orm of the eigh th order appro ximate solution (the en d p oint condition q = 1 has b een used) u = k T + ǫ 6 + S 2 T − 2 T 48 k ǫ 2 − 2 T + B S 2 ( B T + 1) 2034 k 3 ǫ 4 − 12 T + 3 B 3 S 4 − 2 B ( B 2 − 3) S 2 33177 6 k 5 ǫ 6 + 3 B 3 S 4 ( B T − 1) − B S 2 ( T B 3 + 15 − 3 B T − 2 B 2 ) − 30 T 15925 248 k 7 ǫ 8 + · · · , (25) where S ≡ p 1 − T 2 , B ≡ ln 1 − T 1 + T . It should b e emphasized that homotop y app ro ximation con v ergence quite w ell not only for weak p ertur bation (small ǫ ) but also for strong p ertu rbations. Fig. 1 shows the sc hematic plots of the fi rst fiv e appro ximan ts with resp ect to the orders 1, 2, 4, 6 an d 8 resp ectiv ely from upp er to lo wer of th e righ t side of th e figur e wh ile the parameters are fixed as ǫ = 2 . 9 , k = 1 . (26) F rom the figure, we find that the lines of the sixth order and the eigh th order are almost stuck together though th e “p erturb ed parameter” ǫ = 2 . 9 which is n ot a s mall one! 6 –0.5 0 0.5 1 1.5 u –8 –6 –4 –2 2 4 6 8 X Figure 1: The plots of th e p ertur b ed kink solitary wa ve solution for the orders 1, 2, 4, 6 and 8 resp ectiv ely fr om u pp er to lo wer of the right side of the figu r e w h ile the p arameters are fixed as ǫ = 2 . 9 , k = 1. Similar to the HAM, the APDRA is app licable to other p erturb ed nonlinear partial dif- feren tial equations w ith and with ou t small parameters and it is though t-pro v oking to explore a general principle f or the p erturb ed n onlinear partial differential equations holdin g similar resu lts. Differen t from the HAM, we tak e the zeroth order as an nonlinear in tegrable system instead of a linear one, w hic h largely mo dified the con v ergence rate. Here we tak e the dir ect m etho d as a to ol to find the app ro ximate symmetries and symm etry r eductions. T he similar r esu lts can also b e obtained via app r o ximate classical and nonclassical sym metry reduction approac hes w hic h ma y b e used to the Kd V-Burgers equation [6], the p erturb ed n onlinear Schr¨ odinger systems [ ? ] and the p erturb ed Boussinesq system [18]. Ac knowledgemen t The work was supp orted by the National Natural S cience F ound ations of China (Nos. 10735 030, 10475 055, 10675 065 and 90503006), National Basic Researc h Pr ogram of China (973 Program 2007CB8 14800) and PCSIR T (IR T 0734), and the Researc h F un d of P ostdo ctoral of China (No.200 704107 27). 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