Coupled system description of perturbed KdV equation

Coupled system desc ription of pert u rbed KdV equ ation Yair Z armi Jacob Blaus tein Institutes for Desert Research & Physics Department Ben-Gurion University of the Negev Midreshet Ben-Gurion, 84990 Israel Abstract In the multiple-soliton cas e, the freedom in the expansion of t he solution of the pertur bed KdV equation is exploited so as to transform the equation into a system of two equations: T he ( inte- grable) Normal Form for KdV-type solitons, which obey the usual infi nity of KdV- conservation laws, and an auxiliar y equation that de scribes the contribution of obstacles to a symptotic inte- grability, which arise from the second order onwards. The ana lysis ha s been carried through the third order in the expansion. Within that order, the solution of the auxiliary equation is a con- served quantity. Key words: Pert urbed KdV equation; Pertur bation expans ion; T wo-component description PACS 02. 30.IK, 02.30.Mv, 05.45.-a MSC 35Q58, 41A58, 35Q51 1. Int roduction The generic form of the KdV equation, pertu rbed through second order, is [1-6]: w t = 6 w w 1 + w 3 + ! 30 " 1 w 2 w 1 + 10 " 2 w w 3 + 20 " 3 w 1 w 2 + " 4 w 5 ( ) + ! 2 140 # 1 w 3 w 1 + 70 # 2 w 2 w 3 + 280 # 3 w w 1 w 2 + 14 # 4 w w 5 + 70 # 5 w x 3 + 42 # 6 w 1 w 4 + 70 # 7 w 2 w 3 + # 8 w 7 $ % & ' ( ) + O ! 2 ( ) ! « 1 , w p * + x p w ( ) . (1) One expands w in powers of ε , w t , x ( ) = u t , x ( ) + ! u 1 ( ) t , x ( ) + ! 2 u 2 ( ) t , x ( ) + O ! 3 ( ) , (2) Eq. ( 1) is int egrable through O ( ε ) [1-6]. Namely, if one terminates the analysis at O ( ε ), then the zero-order app roximation, u , is determi ned by a Normal Form that is integrable, u t = 6 u u 1 + u 3 + ! " 4 S 3 u [ ] + O ! 2 ( ) S 3 u [ ] = 30 u 2 u 1 + 10 u u 3 + 20 u 1 u 2 + u 5 ( ) , (3) and the f irs t-order corr ection, u (1) , has a closed-form expression as a diff erential polynomial in u : u 1 ( ) = a 1 u 2 + a 2 u 1 q + a 3 u 2 q t , x ( ) = ! x " 1 u t , x ( ) ( ) , (4) where a 1 = ! 5 2 " 1 + 5 3 " 3 + 5 6 " 4 , a 2 = ! 10 3 " 2 + 10 3 " 4 , a 3 = ! 5 " 1 + 5 3 " 2 + 10 3 " 4 . (5) In E q. (3) , S 3 [ u ] is a symmetry of the KdV equation [1-9]. Eq. (3) has the same single- and mul tiple-soliton s olutions as the unp erturbed KdV equation. De- noting t he wave number of a soliton by k i , the only ef fect of E q. (3) is to update the velocity of each soliton according to v i = 4 k i 2 + ! " 4 4 k i 2 ( ) 2 + O ! 2 ( ) . (6) However, this scheme cannot be extended to O ( ε 2 ), unless [2-6] µ = 5 3 3 ! 1 ! 2 + 4 ! 2 2 " 18 ! 1 ! 3 + 60 ! 2 ! 3 " 24 ! 3 2 + 18 ! 1 ! 4 " 67 ! 2 ! 4 + 24 ! 4 2 ( ) + 7 3 # 1 " 4 # 2 " 18 # 3 + 17 # 4 + 12 # 5 " 18 # 6 + 12 # 7 " 4 # 8 ( ) = 0 . (6) If Eq. (6) is satisfied, then u (2) , the second-order corr ection in Eq. (2), can be also solved for in closed f orm as a dif ferential polynomial in the zero-or der approximation, u , and the Nor mal Form Eq. (3) is updated through O ( ε 2 ) int o: u t = 6 u u 1 + u 3 + ! " 4 S 3 u [ ] + ! 2 # 8 S 4 u [ ] + O ! 2 ( ) . (7) Eq. (7) has the same soliton solutions as the unperturbed KdV equation, with the velocity o f each soliton now updated according to v i = 4 k i 2 + ! " 4 4 k i 2 ( ) 2 + ! 2 # 8 4 k i 2 ( ) 3 + O ! 3 ( ) . (8) However, if µ ≠ 0, then the requirement that u (2) be a differential polynomial in u , spoils the inte- grability of the Norma Form [2-5]. I nstead of Eq . (7), u obeys the fol lowing equation: u t = 6 u u 1 + u 3 + ! " 4 S 3 u [ ] + ! 2 # 8 S 4 u [ ] + µ R 2 ( ) u [ ] ( ) + O ! 2 ( ) S 4 u [ ] = 140 u 3 u 1 + 70 u 2 u 3 + 280 u u 1 u 2 + 14 u u 5 + 70 u 1 3 + 42 u 1 u 4 + 70 u 2 u 3 + u 7 ( ) . (9) In Eq. ( 9 ), S 4 [ u ] is the next symmetry of t he KdV equation [ 1-8] and R (2) [ u ] is t he second-order obstacle to asymptotic integrability [2-5 ]. Whereas the value of µ is unique, the structur e of R (2) [ u ] is not, owing to the freedom inherent in the expans ion scheme. T he obstacle, R (2) [ u ], is not a symmet ry of t he KdV equation. Therefore, i t spoils the integrability of E q. ( 9). As a r esult, soliton parameters i n the zero-order ter m, u , develop higher -order time de- pendence, non-KdV solitons ar e generated in u , a nd the elastic sc attering picture of soliton colli- sions is lost in u [3-5]. The difficulti es reviewed above may b e interpr eted dif ferently: That whereas u (1) , the first-or der term in Eq. (2), can be constructed as a dif ferential polynomial in the zero-or der term, u (2 ) ( t , x ), the second-order corr ection - may not [10, 11] . Whe n µ ≠ 0, one mu st allow f or a non-pol ynomial term in u (2) ( t , x ), and write it as u 2 ( ) t , x ( ) = ! u 2 ( ) u [ ] + ! 2 ( ) t , x ( ) . (10) In E q. ( 9), ! u 2 ( ) u [ ] is the dif ferential-polynomial part, and η (2) ( t , x ) is the non-polynom ial part. The effect of the obstacle to i ntegrability, R (2) [ u ], is accounted for by η (2) ( t , x ), and the integrable Nor- mal For m, Eq. (8) is r ecovered . The most general expres sion for ! u 2 ( ) u [ ] that i s localized along soliton t rajectories is ! u 2 ( ) = b 1 u 4 + b 2 u 3 q + b 3 u 2 q 2 + b 4 u 1 q 3 + b 5 u 1 q 3 ( ) + b 6 u q 4 + b 7 u q q 3 ( ) + b 8 u q 4 ( ) + b 9 u u 2 + b 10 u 1 2 + b 11 u u 1 q + b 12 u 2 q 2 + b 13 u 3 q = ! x " 1 u ( ) , q 3 ( ) = ! x " 1 u 2 ( ) , q 4 ( ) = ! x " 1 u 2 q ( ) ( ) . (11) There is ample fr eedom i n t he choice of b k , 1 ≤ k ≤ 13. The Normal Form, Eq. (8), is recovered and the dynamical equation for η (2) ( t , x ) has exceptiona l characteristics with the following choice: b 1 = 5 72 135 ! 1 2 " 24 ! 1 ! 2 " 12 ! 2 2 " 36 ! 1 ! 3 " 200 ! 2 ! 3 + 92 ! 3 2 " 114 ! 1 ! 4 + 216 ! 2 ! 4 + 20 ! 3 ! 4 " 77 ! 4 2 # $ % & ' ( " 7 6 6 ) 1 " 3 ) 2 " 16 ) 3 + 14 ) 4 + 9 ) 5 " 15 ) 6 + 9 ) 7 " 4 ) 8 ( ) , (12) b 2 = 25 9 3 ! 1 ! 2 " 2 ! 2 ! 3 " 3 ! 1 ! 4 + ! 2 ! 4 + 2 ! 3 ! 4 ( ) " 14 3 # 4 " # 8 ( ) , (13) b 3 = 50 9 ! 2 " a 4 ( ) 2 , (14) b 5 = 5 3 9 ! 1 ! 2 + 2 ! 2 2 + 6 ! 1 ! 3 " 20 ! 2 ! 3 + 8 ! 3 2 " 6 ! 1 ! 4 " ! 2 ! 4 + 2 ! 4 2 ( ) " 7 # 1 + 2 # 2 " 6 # 3 + # 4 + 4 # 5 " 6 # 6 + 4 # 7 ( ) , (15) b 9 = 5 9 135 ! 1 2 " 39 ! 1 ! 2 " 12 ! 2 2 " 6 ! 1 ! 3 " 190 ! 2 ! 3 + 72 ! 3 2 " 129 ! 1 ! 4 + 221 ! 2 ! 4 + 40 ! 3 ! 4 " 92 ! 4 2 # $ % & ' ( " 14 3 12 ) 1 " 6 ) 2 " 32 ) 3 + 27 ) 4 + 18 ) 5 " 27 ) 6 + 18 ) 7 " 10 ) 8 ( ) , (16) b 10 = 5 9 90 ! 1 2 " 3 ! 1 ! 2 " 9 ! 2 2 " 12 ! 1 ! 3 " 160 ! 2 ! 3 + 64 ! 3 2 " 93 ! 1 ! 4 + 162 ! 2 ! 4 + 30 ! 3 ! 4 " 69 ! 4 2 # $ % & ' ( " 7 3 18 ) 1 " 9 ) 2 " 48 ) 3 + 43 ) 4 + 27 ) 5 " 48 ) 6 + 32 ) 7 " 15 ) 8 ( ) , (17) b 11 = 100 3 ! 1 ! 2 " ! 4 ( ) " 28 # 4 " # 8 ( ) , (18) b 13 = 25 9 18 ! 1 2 " 9 ! 1 ! 2 " 2 ! 2 2 + 6 ! 1 ! 3 " 20 ! 2 ! 3 + 8 ! 3 2 " 18 ! 1 ! 4 + 25 ! 2 ! 4 " 8 ! 4 2 # $ % & ' ( " 7 3 15 ) 1 " 10 ) 2 " 30 ) 3 + 27 ) 4 + 20 ) 5 " 30 ) 6 + 20 ) 7 " 12 ) 8 ( ) , (19) b 4 = b 6 = b 7 = b 8 = b 12 = 0 . (20) With this choice f o r b k , t he equation f o r η (2) ( t , x ) becomes in t his order ! t " 2 ( ) = ! x 6 u " 2 ( ) ( ) + ! x 2 " 2 ( ) + µ u u 3 + u u 2 # u 1 2 ( ) { } . (21) If u is a multiple-soliton solution of Eq. (8), then a solution of Eq. (19), which is bounded for fixed t , obeys the conservation law d dt ! 2 ( ) u [ ] dx "# + # $ = 0 . (22) If this is the cas e, then the perturbed KdV equation has been transformed by t he per turbation scheme described above into a system of two equations: T he Normal Form f or or dinary solitons, which obey the well -know inf inity of conservation laws [1- 9] and Eq. (21) for the ef fect of the ob- stacle to asymptotic integrability. T he latter gener ates a conserved qu antity ( at least in this order of the expansion). That this is the indeed ca se i s se en as follows. T he differ ential polynomi al u u 3 + u u 2 ! u 1 2 ( ) i n Eq. (21) is a local special polynomial . It vanishes identically if the single-soliton solution , u Single t , x ; k ( ) = 2 k 2 cosh k x + v t ( ) ! " # $ , (23) is substituted for u . As a result, u u 3 + u u 2 ! u 1 2 ( ) is localized aroun d the origin, and vanishes ex- ponentially fast in all dir ections in the x – t plane, if computed for a multipl e-soliton solution [10, 11]. Hence, if η (2) ( t , x ) is bounded, then for fixed t , Eq. ( 22 ) i s obeyed. That η (2) ( t , x ) is bounded, and, in fact vanis hes as | x | → ∞ for fixed t , is seen as follows. Define ! 2 ( ) = µ " x # 2 ( ) . (24) If η (2) ( t , x ) is bounded , then the equation for ω (2) is ! x " 2 ( ) = 6 u ! x " 2 ( ) + ! x 3 " 2 ( ) + u u 3 + u u 2 # u 1 2 ( ) . (25) The solution for ω (2) [ u ] is bounded be cause the dr i ving term in Eq. (25) does not r esonate with the homogeneous part of the equ ation [11] and vani sh es exponentially fast in all directions in t he x – t plane. As an example, Eq. ( 25) was solved numerically f or zero-initial data at a large negative value of t and vanishing boundar y values for x , when u is a t wo-soliton solution of the KdV equa- tion, with soliton wave numbers equal to 0.1 and 0.2. F ig. 1 shows the solution for ω (2) [ u ]. It is comprised of a soliton and an anti-soliton, accompanied by a decaying dispersive wave. Within the numer ical accuracy, the soliton and an ti-soliton have t he same par ameters (velocities, wave numbers and phase shifts) a s the zero-order solitons in u . Up to overall amplitudes, deter- mined f rom the numerical solution, the soliton an d t he anti- soliton are indistinguishable f rom the ordinary single-KdV solitons. The dispersive wave had been found in previous numerical works [12-14]. Thi s work identifies the specific term that generates it. With ω (2) [ u ] bounded (in fact, vanishing as | x | → ∞ for f ixed t ), η (2) ( t , x ), clearly, obeys the conser- vation law, Eq. ( 22 ), to lowest o rder. To extend E qs. ( 21) and ( 22) to O ( ε ), requir es that a thir d-order pertur bation is appended to Eq. (1), and that the series for w is computed thr ough t hird order. T his has been perfor med. The free- dom in the expans ion allows one to update E q. (21) so that the result is also conservative (i .e., the right-hand side is a complete differential with resp ect to x ): ! t " 2 ( ) = ! x 6 u " 2 ( ) ( ) + ! x 2 " 2 ( ) + µ u u 3 + u u 2 # u 1 2 ( ) + $ A u , " 2 ( ) % & ' ( + P u [ ] ( ) { } . (26) A [ u , η (2) ] is a differential polynomial in u and η (2) . It is linear in η (2) . (Only the f ou rth-order analy- sis will generate in the extended version of E q. (21) terms that will be quadratic in η (2) .) Moreo- ver, t hanks to the freedom i n the expansion, the driving ter m, P [ u ], can be shaped so that it is also a local special differ ential polynomial in the zero- order term, u . 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