Approximate symmetry reduction approach: infinite series reductions to the KdV-Burgers equation

Appro ximate symmetry reduction approac h: infinite series reductions to the KdV-Burgers equation Xiao yu Jiao 1 , R uo xia Y ao 1 , 2 , 3 , Sh unli Z ha ng 4 and S. Y. Lou 1 , 2 1 Dep artment of Physics, Shanghai Jiao T ong University, Shanghai, 200240, Ch ina 2 Dep artment of Ph ysics, N ingb o University, Ningb o, 315211, Ch ina 3 Scho ol of Computer Scienc e, Shaanxi Normal University, Xi’an , 710062, Chi na 4 Dep artment o f Mathematics , Northwest University, Xi’an, 710069, China Abstract F or w eak disp ersion a nd wea k d issipation cases, the (1+1)-dimensional KdV-Burgers equ ation is in v estigated in terms of appro ximate symmetry reduction app roac h. The formal coherence of similarit y reduction solutions a nd similarit y reduction e quations of differen t orders enables series reduction solutions. F or weak dissipation c ase, zero-order similarit y solutions satisfy the P ainlev ´ e I I, P ainlev ´ e I and Jacobi elliptic fu nction equations. F or w ea k d isp ersion case, zero-order simila rit y solutions are in the form of Kummer, Airy and hyp erb olic tangen t functions. Higher order similarit y solutions can b e obtained b y solving linear ord inary differential equations. P ACS num be rs: 02.30 .Jr Keywords: KdV-Burge r s equation, approximate symmetry reduction, series r eduction solutions 1 I. INTR ODUCTIO N Nonlinear problem s arise in man y fields of scienc e and enginee ring. Lie g r oup t heory [1, 2, 3] greatly s implifies many nonlinear partial differen tial e quations. Exact analytical solutions are nonetheless difficult to study in general. P erturbation theory [4, 5, 6] w as th us dev eloped and it pla ys an ess en tial role in nonline ar scienc e, es p ecially in finding approximate analytical solutions to p erturb ed partia l differential equations. The inte gration of Lie group theory and p erturbation theory yields t w o distinct approxi- mate symmetry reduction methods. The first metho d due to Baik o v, et al [7, 8] generalizes symmetry group generators to perturbation f orms. F o r the second metho d prop osed b y F ushc hic h, et al [9], dependen t v aria bles a r e expanded in p erturbation series and approxi- mate symmetry of the original equation is decomp o sed into exact symmetry of the system resulted f rom p erturbat io n. The second metho d is sup erior to the first one from t he com- parison in Refs. [10, 11]. The w ell kno wn K ortew eg-de V ries-Burgers (KdV- Burgers) equation u t + 6 uu x + µu xxx + ν u xx = 0 , (1) with µ and ν cons tant co efficien ts, is widely used in the many ph ysical fields esp ecially in fluid dynamic s. The effec ts of nonlinearit y (6 uu x ), dispersion ( µ u xxx ) and dis sipation ( ν u xx ) are incorp orated in this equation whic h sim ulates the propagation of w a v es on an elastic tub e filled with a viscous fluid [1 2], and the flow of liquids containing gas bubbles [13] and turbulence [14, 1 5], etc. Johnson [12] inspected the tra v elling w a v e solutions to the w e ak dissipation ( ν ≪ 1) KdV- Burgers equation (1) in the phase plane by a p erturbation metho d and dev eloped formal asymptotic expansion for the solution. T anh function metho d w as applied to Eq. (1) in the limit of weak disp ersion ( µ ≪ 1) in a p erturbative w a y [16]. In R efs. [17, 1 8], p erturbatio n analysis w as also applied to the p erturb ed KdV equations η t + 6 η η x + η xxx + α c 1 η xx + α c 2 η xxxx + α c 3 ( η η x ) x = 0 , α ≪ 1 , (2) and u t + uu x + u xxx = ǫ αu + ǫβ u xx , ǫ ≪ 1 , (3) resp ectiv ely . 2 Eq. (1) can also b e manipulated b y means of appro ximate sy mmetry reduction appro ac h. Section I I and se ction I I I are dev oted to a pplying approximate sym metry reduction approach to Eq. (1) under the case of w eak dissipation ( ν ≪ 1) and w eak dispersion ( µ ≪ 1), resp ectiv ely . Section IV is conclusion and discussion of the results. I I. APPR O XIMA TE SYMMETR Y REDUC T ION APPR O A CH T O WEAK DI SSI- P A TION KDV-BUR GERS EQUA TION According to the p erturbation theory , solutions to p erturb ed partia l diffe ren tial equations can b e expressed as a series containing a small para meter. Sp ecifically , we supp ose that the w eak dissipation ( ν ≪ 1) KdV-Burgers equation (1) has the solution u = ∞ X k =0 ν k u k , (4) where u k are functions of x a nd t , and solv e t he follo wing system u k ,t + 6 k X i =0 u k − i u i,x + µu k ,xxx + u k − 1 ,xx = 0 , ( k = 0 , 1 , · · · ) (5) with u − 1 = 0, whic h is obtained b y inserting Eq. (4) into Eq . (1) and v anishing the co efficien ts of different p ow ers of ν . The next crucial step is to study symm etry reduction of the abov e system via the Lie symmetry approach [19]. T o that end, w e construct the Lie p oint symmetries σ k = X u k x + T u k t − U k , ( k = 0 , 1 , · · · ) , (6) where X , T a nd U k are functions with res p ect to x , t and u i , ( i = 0 , 1 , · · · ). The linearized equations for Eq. (5) are: σ k ,t + 6 k X i =0 ( σ k − i u i,x + u k − i σ i,x ) + µσ k ,xxx + σ k − 1 ,xx = 0 , ( k = 0 , 1 , · · · ) (7) with σ − 1 = 0. Eq. (7) means that Eq. (5) is inv ariant under the t r ansformations u k → u k + εσ k ( k = 0 , 1 , · · · ) with a n infinitesimal para meter ε . There ar e infinite num b er o f equations in Eqs. (5) and (7) and infinite n um b er of argu- men ts in X , T and U k ( k = 0 , 1 , · · · ). T o simplify the problem, w e begin the discussion from finite num b er of equations. 3 Confining the range of k to { k | k = 0 , 1 , 2 } in Eqs. (5), (6) and (7), w e see that X , T , U 0 , U 1 and U 2 are functions with res p ect to x , t , u 0 , u 1 and u 2 . In this case, the determining equations can be obtained b y substituting Eq. (6) in to Eq. (7), eliminating u 0 ,t , u 1 ,t and u 2 ,t in terms of Eq. (5) and v anishing all co efficien ts of different partial deriv ativ es of u 0 , u 1 and u 2 . So me of the determining equations read T x = T u 0 = T u 1 = T u 2 = 0 , from whic h w e hav e T = T ( t ). Considering this condition, we c ho ose the simplest equations for X X u 0 = X u 1 = X u 2 = 0 , from whic h w e ha v e X = X ( x, t ). Considering this condition, we c ho ose the simplest equations for U 0 , U 1 and U 2 U 0 ,xu 2 = U 0 ,u 0 u 0 = U 0 ,u 0 u 1 = U 0 ,u 0 u 2 = U 0 ,u 1 u 1 = U 0 ,u 1 u 2 = U 0 ,u 2 u 2 = 0 , U 1 ,u 0 u 0 = U 1 ,u 0 u 1 = U 1 ,u 0 u 2 = U 1 ,u 1 u 1 = U 1 ,u 1 u 2 = U 1 ,u 2 u 2 = 0 , U 2 ,u 0 u 0 = U 2 ,u 0 u 1 = U 2 ,u 0 u 2 = U 2 ,u 1 u 1 = U 2 ,u 1 u 2 = U 2 ,u 2 u 2 = 0 , of whic h the solutions are U 0 = F 1 ( x, t ) u 0 + F 2 ( x, t ) u 1 + F 3 ( t ) u 2 + F 4 ( x, t ) , U 1 = F 5 ( x, t ) u 0 + F 6 ( x, t ) u 1 + F 7 ( x, t ) u 2 + F 8 ( x, t ) , U 2 = F 9 ( x, t ) u 0 + F 10 ( x, t ) u 1 + F 11 ( x, t ) u 2 + F 12 ( x, t ) . Under these relations, the determining equations a re simplified to X xx = F 2 = F 3 = F 5 = F 7 = F 8 = F 9 = F 10 = F 12 = F 1 ,x = F 4 ,xx = F 4 ,t = F 6 ,x = F 11 ,x = 0 , T t = 3 X x , X t = 6 F 4 , F 1 ,t = − 6 F 4 ,x , F 6 ,t = − 6 F 4 ,x , F 11 ,t = − 6 F 4 ,x , T t = X x − F 1 , T t = 2 X x − F 1 + F 6 , T t = X x + F 11 − 2 F 6 , T t = 2 X x − F 6 + F 11 , whic h pro vide us with X = 6 at + cx + x 0 , T = 3 ct + t 0 , U 0 = − 2 cu 0 + a, U 1 = − cu 1 , U 2 = 0 , where a , c , x 0 and t 0 are arbitrary constan ts. 4 Similarly , limiting the range of k to { k | k = 0 , 1 , 2 , 3 } in Eqs. (5), (6) and (7) where X , T , U 0 , U 1 , U 2 and U 3 are functions with respect to x , t , u 0 , u 1 , u 2 and u 3 , w e rep eat the calculation as b efore and obtain X = 6 at + cx + x 0 , T = 3 ct + t 0 , U 0 = − 2 cu 0 + a, U 1 = − cu 1 , U 2 = 0 , U 3 = cu 3 , where a , c , x 0 and t 0 are arbitrary constan ts. With more similar calculation considered, w e see that X , T and U k ( k = 0 , 1 , · · · ) are formally coheren t, i.e., X = 6 at + cx + x 0 , T = 3 ct + t 0 , U k = ( k − 2) cu k + aδ k , 0 , ( k = 0 , 1 , · · · ) (8) where a , c , x 0 and t 0 are arbitrary constants. T he notation δ k , 0 satisfying δ 0 , 0 = 1 and δ k , 0 = 0 ( k 6 = 0) is a do pt ed in the follo wing text. Subsequen tly , s olving the c haracteristic equations d x X = d t T , d u 0 U 0 = d t T , d u 1 U 1 = d t T , · · · , d u k U k = d t T , · · · (9) leads to the similarit y solutions to Eq. (5) whic h can b e distinguished in the f o llo wing tw o sub cases. A. Symmetry reduction of the P ainlev ´ e I I solutions When c 6 = 0, for brevit y of the results, w e rewrite the constan ts a , x 0 and t 0 as ca , cx 0 and ct 0 , resp ectiv ely . Solving d x X = d t T in Eq. (9) leads to the inv ariant I ( x, t ) = ξ = ( x − 3 at + x 0 − 3 at 0 )(3 t + t 0 ) − 1 3 . (10) In the same w ay , we get other in v ariants I 0 ( x, t, u 0 ) = P 0 = 1 2 (3 t + t 0 ) 2 3 (2 u 0 − a ) (11) and I k ( x, t, u k ) = P k = u k (3 t + t 0 ) − 1 3 ( k − 2) , ( k = 1 , 2 , · · · ) (12) from d u 0 U 0 = d t T and d u k U k = d t T ( k = 1 , 2 , · · · ) respectiv ely . Viewing P k ( k = 0 , 1 , · · · ) as functions of ξ , w e get the similarit y solutions to Eq. (5) u k = (3 t + t 0 ) 1 3 ( k − 2) P k ( ξ ) + 1 2 aδ k , 0 , ( k = 0 , 1 , · · · ) (1 3) 5 with the similarit y v ariable ξ = ( x − 3 at + x 0 − 3 at 0 )(3 t + t 0 ) − 1 3 . (14) Accordingly , the series reduction solution to Eq. (1) is deriv ed u = a 2 + ∞ X k =0 ν k (3 t + t 0 ) 1 3 ( k − 2) P k ( ξ ) , (15) and the related similarity reduction equations a r e µP k ,ξ ξ ξ + 6 k X i =0 P k − i P i,ξ − ξ P k ,ξ + ( k − 2) P k + P k − 1 ,ξ ξ = 0 , ( k = 0 , 1 , · · · ) (16) with P − 1 = 0. When k = 0, Eq . (16) is equiv alent to the P ainlev ´ e II t yp e equation. The n th ( n > 0) similarity reduction equation is actually a third or der line ar ordinary differen tial equation of P n when the previous P 0 , P 1 , · · · , P n − 1 are kno wn, since Eq. (16) is just µP k ,ξ ξ ξ + 6( P 0 P k ,ξ + P k P 0 ,ξ ) − ξ P k ,ξ + ( k − 2) P k = f k ( ξ ) , ( k = 1 , 2 , · · · ) (16 ′ ) where f k is a only f unction of { P 0 , P 1 , · · · , P k − 1 } f k ( ξ ) = − 6 k − 1 X i =1 P k − i P i,ξ − P k − 1 ,ξ ξ . B. Symmetry reduction of the P ainlev´ e I solutions When c = 0 and t 0 6 = 0, we rewrite a and x 0 as at 0 and x 0 t 0 , respective ly . The similarit y solutions are u k = ( at + x 0 6 ) δ k , 0 + P k ( ξ ) , ( k = 0 , 1 , · · · ) (17) with the similarit y v ariable ξ = − x + 3 at 2 + x 0 t . Hence, the series reduction solution to Eq. (1) is u = at + x 0 6 + ∞ X k =0 ν k P k ( ξ ) , (18) in whic h P k ( ξ ) satisfys µP k ,ξ ξ + 3 k X i =0 P k − i P i − P k − 1 ,ξ − aδ k , 0 ξ + A k = 0 , ( k = 0 , 1 , · · · ) (19) where P − 1 = 0 and A k are arbitrary integral constan ts. 6 When k = 0, Eq. (19) is equiv alent to the Painlev ´ e I ty p e equation pro vided that a 6 = 0. When k = 0, a = 0 a nd A 0 = 4 3 µ 2 p 4 1 ( m 2 − 1 − m 4 ), Eq. (19) can b e solv ed by the Jacobi elliptic function, P 0 = 2 3 µp 2 1 (1 − 2 m 2 ) + 2 µm 2 p 2 1 cn 2 ( p 1 ξ + p 2 , m ) , (20) where p 1 , p 2 and m are arbitra ry constan ts. When k > 0, an equiv alent form of Eq. (19) is µP k ,ξ ξ + 6 P k P 0 = g k ( ξ ) , ( k = 1 , 2 , · · · ) (19 ′ ) where g k ( ξ ) is a function of { P 0 , P 1 , · · · , P k − 1 } as follows g k ( ξ ) = − 3 k − 1 X i =1 P k − i P i + P k − 1 ,ξ − A k . F rom Eq. (1 9 ′ ), w e see that Eq. (19 ) is a second order linear ordinar y differen tial equation of P k and can b e in tegra t ed out step by step when a = 0. The results read P k = P 0 ,ξ [ C k + µ − 1 Z P − 2 0 ,ξ ( B k + Z P 0 ,ξ g k d ξ )d ξ ] , (21) with arbitrary integral constan ts B k and C k . I I I . APP R OXIMA TE SYMMETR Y REDUCTION APPR OA CH TO WE AK DIS- PERSION KD V-BUR GERS EQUA TION W e searc h for series reduc tion solutions to w eak disp ersion ( µ ≪ 1) KdV-Burgers equation (1). The pro cess is similar to t he section I I. A system of partial differen tial equations u k ,t + 6 k X i =0 u k − i u i,x + ν u k ,xx + u k − 1 ,xxx = 0 , ( k = 0 , 1 , · · · ) (22) with u − 1 = 0, is obtained b y plugging the p erturbation series solution u = ∞ X k =0 µ k u k , (23) in to Eq. (1) and v anishing the co efficien ts of differen t p o w ers of µ . It is easily seen that the linearized equations relat ed to Eq. (22) are σ k ,t + 6 k X i =0 ( σ k − i u i,x + u k − i σ i,x ) + ν σ k ,xx + σ k − 1 ,xxx = 0 , ( j = 0 , 1 , · · · ) (24) 7 with σ − 1 = 0. The Lie p oint s ymmetries (6) satisfy the linearized equations (24) under the appro ximate equations ( 2 2). Restricting the range of k t o { k | k = 0 , 1 , 2 } in Eqs . (6), (22) and (24), w e see that X , T , U 0 , U 1 and U 2 are functions with resp ect to x , t , u 0 , u 1 and u 2 . The determining equations are derive d b y subs tituting Eq. (6) into Eq. (24), eliminating u 0 ,t , u 1 ,t and u 2 ,t in terms of Eq. (22) and v anishing co efficien ts of different partial deriv ativ es o f u 0 , u 1 and u 2 . So me of the determining equations are T x = T u 0 = T u 1 = T u 2 = 0 , from whic h w e get T = T ( t ). Conside ring this condition, the simplest equations for X in the determining equations are X u 0 = X u 1 = X u 2 = 0 , from wh ic h w e get X = X ( x, t ). Considering this condition, w e selec t the sim plest equations for U 0 and U 1 U 0 ,u 0 u 0 = U 0 ,a 1 = U 0 ,a 2 = 0 , U 1 ,xu 0 = U 1 ,u 0 u 0 = U 1 ,u 0 u 1 = U 1 ,u 1 u 1 = U 1 ,u 2 = 0 , with the solution U 0 = F 1 ( x, t ) u 0 + F 2 ( x, t ) and U 1 = F 3 ( t ) u 0 + F 4 ( x, t ) u 1 + F 5 ( x, t ). F rom the reduced determining equations, the simplest equations for U 2 read U 2 ,u 0 u 0 = U 2 ,u 0 u 1 = U 2 ,u 0 u 2 = U 2 ,u 1 u 1 = U 2 ,u 1 u 2 = U 2 ,u 2 u 2 = 0 , leading to U 2 = F 6 ( x, t ) u 0 + F 7 ( x, t ) u 1 + F 8 ( x, t ) u 2 + F 9 ( x, t ). Com bined with these conditions, the determining equations are simplified to X xx = F 3 = F 5 = F 6 = F 7 = F 9 = F 1 ,x = F 2 ,xxx = F 4 ,x = F 8 ,x = 0 , T t = 2 X x , X t = 6 F 2 , F 1 ,t = − 6 F 2 ,x , F 2 ,t = − ν F 2 ,xx , F 4 ,t = − 6 F 2 ,x , F 8 ,t = − 6 F 2 ,x , T t = X x − F 1 , T t = X x − 2 F 4 + F 8 , T t = 3 X x − F 1 + F 4 , T t = 3 X x − F 4 + F 8 , It is easily seen that X = 6 at + cx + x 0 , T = 2 ct + t 0 , U 0 = − cu 0 + a, U 1 = − 2 cu 1 , U 2 = − 3 cu 2 , where a , c , x 0 and t 0 are arbitrary constan ts. 8 In lik e manner, we o btain X = 6 at + cx + x 0 , T = 2 ct + t 0 , U 0 = − cu 0 + a, U 1 = − 2 cu 1 , U 2 = − 3 cu 2 , U 3 = − 4 cu 3 , where a , c , x 0 and t 0 are arbitrary constan ts. Rep eating similar calculation sev eral times, w e summarize the solutions to the determin- ing equations X = 6 at + cx + x 0 , T = 2 ct + t 0 , U k = − ( k + 1) c u k + aδ k , 0 , ( k = 0 , 1 , · · · ) (25) where a , c , x 0 and t 0 are arbitrary constan ts. The similarit y solutions to Eq. (22) from solving the c haracteristic equations (9) ar e discus sed in the follo wing t w o sub cases. A. Symmetry reduction of the Kummer function solutions When c 6 = 0, for brevit y of the results, w e rewrite the constan ts a , x 0 and t 0 as ca , cx 0 and ct 0 , resp ectiv ely . Solving d x X = d t T in Eq. (9) results in the inv ariant I ( x, t ) = ξ = ( x − 6 at − 6 at 0 + x 0 )(2 t + t 0 ) − 1 2 . (26) Lik ewise, w e g et other in v ariants I 0 ( x, t, u 0 ) = P 0 = (2 t + t 0 ) 1 2 ( u 0 − a ) (27) and I k ( x, t, u k ) = P k = u k (2 t + t 0 ) 1 2 ( k +1) ( k = 1 , 2 , · · · ) (28) from d u 0 U 0 = d t T and d u k U k = d t T ( k = 1 , 2 , · · · ) resp ectiv ely . Considering P k ( k = 0 , 1 , · · · ) as functions of ξ , w e get the similarit y solutions to Eq. (22) u k = (2 t + t 0 ) − 1 2 ( k +1) P k ( ξ ) + aδ k , 0 , ( k = 0 , 1 , · · · ) (29) with the sim ilarity v ariable ξ = ( x − 6 at − 6 at 0 + x 0 )(2 t + t 0 ) − 1 2 , and the series reduction solution to Eq. (1) is u = a + ∞ X k =0 µ k (2 t + t 0 ) − 1 2 ( k +1) P k ( ξ ) , (30) where P k ( ξ ) are sub ject to ν P k ,ξ ξ + 6 k X i =0 P k − i P i,ξ − ξ P k ,ξ − ( k + 1) P k + P k − 1 ,ξ ξ ξ = 0 , ( k = 0 , 1 , · · · ) ( 3 1) 9 with P − 1 = 0. When k = 0, Eq. (31 ) has the Kummer function solution P 0 = (3 C 1 − 1) ν [3 C 1 C 2 K 1 ( 3 2 (1 − C 1 ) , 3 2 , ξ 2 2 ν ) − 2K 2 ( 3 2 (1 − C 1 ) , 3 2 , ξ 2 2 ν )] 6 ξ [ C 2 K 1 ( 1 2 (1 − 3 C 1 ) , 3 2 , ξ 2 2 ν ) + K 2 ( 1 2 (1 − 3 C 1 ) , 3 2 , ξ 2 2 ν )] + C 1 ν ξ , (32) where C 1 and C 2 are arbitrary constan ts, and the t w o ty p es of Kummer functions K 1 ( p, q , z ) and K 2 ( p, q , z ) solv e the differen tial equation z y ′′ ( z ) + ( q − z ) y ′ ( z ) − py ( z ) = 0 . When ( k > 0), w e rearrange the t erms in Eq. (31) as ν P k ,ξ ξ +6( P 0 P k ,ξ + P k P 0 ,ξ ) − ξ P k ,ξ − ( k +1) P k = − 6 k − 1 X i =1 P k − i P i,ξ − P k − 1 ,ξ ξ ξ , ( k = 1 , 2 , · · · ) (33 ) whic h is a sec ond order linear ordinary differential equation of P k when the previous P 0 , P 1 , · · · , P k − 1 are kno wn. B. Symmetry reduction of Airy function and h yp erb olic tangent function solutions When c = 0 and t 0 6 = 0, w e rewrite the constan ts a and x 0 as at 0 and x 0 t 0 , resp ectiv ely . It is easily seen that the similarit y solutio ns are u k = ( at + x 0 6 ) δ k , 0 + P k ( ξ ) , ( k = 0 , 1 , · · · ) (34) with the similarit y v ariable ξ = − x + 3 at 2 + x 0 t , and the series reduction solution to Eq. (1) is u = at + x 0 6 + ∞ X k =0 µ k P k ( ξ ) , (35) with P k ( ξ ) satisfying ν P k ,ξ − 3 k X i =0 P k − i P i − P k − 1 ,ξ ξ + aδ k , 0 ξ + A k = 0 , ( k = 0 , 1 , · · · ) ( 3 6) where P − 1 = 0 and A k are in tegral constan ts. When k = 0 and a ≡ − b , w e get the Airy function solutio n to Eq. (36) P 0 = (3 bν ) 1 3 [ C 1 Ai(1 , 3 1 3 ( bν ) − 2 3 ( A 0 − bξ )) + Bi( 1 , 3 1 3 ( bν ) − 2 3 ( A 0 − bξ ) ) ] 3[ C 1 Ai(3 1 3 ( bν ) − 2 3 ( A 0 − bξ ) ) + Bi(3 1 3 ( bν ) − 2 3 ( A 0 − bξ ) ) ] , (37) 10 where C 1 is an arbitrary constan t. The Airy w a v e functions Ai( z ) and Bi( z ) are linearly indep enden t solutions for y ( z ) in the equation y ′′ ( z ) − z y ( z ) = 0. Ai( n, z ) a nd Bi( n, z ) are the n th deriv ative s o f Ai( z ) and Bi( z ) ev alua ted at z , resp ectiv ely . The h yp erb olic tangen t function solution o f tra v eling wa v e form P 0 = − √ 3 3 p tanh[ √ 3 ν p ( ξ + d )] , (38) with d an arbit r a ry constan t, can b e obtained from a = 0 and p = √ A 0 . When k > 0, an equiv alent form of Eq. (36) is ν P k ,ξ − 6 P 0 P k = g k ( ξ ) , ( k = 1 , 2 , · · · ) (36 ′ ) where g k ( ξ ) = 3 k − 1 X i =1 P k − i P i + P k − 1 ,ξ ξ − A k . F rom Eq. (36 ′ ), it is easily seen that the k th similarit y reduction equation in Eq. (36 ) is a first order linear ordinary differential equation of P k . F urthermore, all equations in Eq. (36 ′ ) can b e solve d step b y step. The results read P k = e xp( 6 ν Z P 0 d ξ )  1 ν Z g k exp( − 6 ν Z P 0 d ξ )d ξ + B k  , ( k = 1 , 2 , · · · ) (39) where B k are arbitrary integral constan ts. IV. CONCLUSIO N AND DISC USSION In summary , b y applying the approximate symmetry reduction approac h t o (1 +1)- dimensional KdV-Burgers equation under the condition of w eak disp ersion and w eak dis- sipation, w e hav e found that the similarity reduction solutions and similarit y reduction equations of differen t orders are coinciden t in their forms. Therefore, w e summarize the series reduction solutions and general formulae for the similarit y equations. F or we ak dissipation case, zero-or der similarit y solutions are equiv alen t to P ainlev ´ e I I, P a inlev ´ e I type and Jacobi elliptic function solutions. F or w eak disp ersion case, zero-order similarit y solutions ar e in the form of Kummer f unction, Airy function and hy p erb olic tan- gen t function solutions. k -order similarity reduction equations are linear ordinary differential equations with re- sp ect to P k ( ξ ). Esp ecially , for the p erio d solutio ns (expressed by Jacobi elliptic functions) 11 with solitary wa v es as a sp ecial case under w eak dissipation, Airy function and hy p erb olic tangen t f unction solutions under w eak dispersion, all the higher order similarit y solutions can b e solve d simply b y direct in tegra tion. The c onv ergence of infinite series solutions remains as a problem a nd expects further study . The appro ximate s ymmetry reduc tion approac h can b e used to s earc h for similar results of other p erturb ed nonlinear differen tia l equations and it is worth while to summarize a general principle for the p erturb ed nonlinear differen tial equations holding analogous results. 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