Exact accelerating solitons in nonholonomic deformation of the KdV equation with two-fold integrable hierarchy

Exact accelerating s olitons in nonholono mic deformatio n of the KdV equation wit h t w o-fold in tegr able hierarc h y Anjan Kundu Theory Group & CAM CS, Sa ha Institute of Nuclear Ph ysic s Calcutta, INDIA anjan.kundu@saha.ac.in Septem b er 27, 2021 Short Title: Solitons in deforme d KdV and its two-fold inte gr able hier ar chy P A CS: 02.30.lk, 02.30.jr, 05.45.Yv, 11.10.Lm, Key wor ds : Nonholonomic deformation of KdV, accelerating soliton, in ve r se scattering m etho d, AK NS Lax pair, in tegrable hierarchies. Abstract Recently prop osed no nholonomic defor mation of the KdV equatio n is solved through inverse scat- tering metho d b y co nstructing AKNS t yp e La x pair. Exac t N-soliton so lutions are found for the basic field a nd the defo r ming function showing un usual ac(de)celerated motio n. Two-fold integrable hierar - ch y is revealed, one with usual hig he r orde r disp e rsion and the other with no vel higher nonholonomic deformations. 1 In tro du ction Nonholonomic constrain ts on field mo dels are receiving incr easing atten tion in recent yea r s nh oln ¸ . As a notable ac hievemen t suc h n onholonomic deformation (NHD) has b een ap p lied to an in tegrable sys tem, namely to the KdV equation preserving its in tegrabilit y 6kdv ¸ . F or this system, equiv alen t to a 6th order KdV equation, certain particular tra ve ling wa v e solutions are f ound, a linear problem is formulate d and sev eral conjectures on imp ortant issues are put forw ard in 6kdv ¸ . The main exp ectatio n s for this KdV equation with nonholonomic constraint are: i) existence of an in finite set of higher conserv ed quantit ies, ii) form ulation of the Lax pair iii) app lication of the inv erse scattering metho d (ISM), iv) N-soliton solutions for the basic and the deforming field, v) elastic n ature of soliton scattering etc. Among th ese conjectures only the fir st one sho wing the existence of an int egrable hierarch y w ith us ual higher d isp ersion is pr o v ed r ecen tly kup 08 ¸ . Our aim here is to establish the r est w ith explicit resu lt. In particular w e construct AKNS t yp e matrix Lax pair for this NHD of the KdV equation, revea ling an imp ortant connection b et wee n the time ev olution of the Jost fun ction and the NHD of the nonlinear equation. Applying su bsequent ly the ISM w e find the exact N-soliton solutions f or b oth the field and the deforming function of the deformed KdV, which exh ibit the usu al prop erty of solito n scattering, but with an unusual accelerat in g (or decelerating) soliton motion. Finally we un ra v el a n o v el tw o-fold in tegrable 1 hierarc hy for this system, one with th e u sual higher disp ersion kup 08 ¸ and th e other yielding a new t yp e of deformed Kd V with increasingly higher order nonholonomic constrain ts. 2 ISM for the deformed KdV Recen tly pr op osed 6kdv ¸ NHD of the KdV equation u t − u xxx − 6 uu x = w x , (1) w xxx + 4 uw x + 2 u x w = 0 , (2) can b e written by eliminating deforming function w , also as a 6th order KdV equation for v x = u : ( ∂ 3 xxx + 4 v x ∂ x + 2 v xx )( v t − v xxx − 3 v 2 x ) = 0 . (3) 2.1 AKNS type La x pair W e in tend to fi nd the exact N-soliton solutions to the deformed KdV equation (1-2) for the field u and the deformin g fu nction w by th e ISM, for wh ic h we construct fir st the AKNS typ e Lax pair U ( λ ) , V ( λ ) to form u late the linear prob lem Φ x = U Φ , Φ t = V Φ. W e observ e remark ably , that such a Lax pair for the deformed K dV can b e build u p from the kno w n pair U k dv ( λ ) , V k dv ( λ ) for the standard K d V equation solit1 ¸ , by deforming only its time-Lax op erator: U ( λ ) = U k dv ( λ ) , V ( λ ) = V k dv ( λ ) + V def ( λ ) , (4) Where the deform in g op erator is giv en as V def ( λ ) = 1 2 ( λ − 1 G (1) + λ − 2 G (2) ) (5) with G (1) = iwσ 3 − w x σ + G (2) = w x 2 σ 3 + iw σ − + eσ + , e x = iuw x . (6) Here u ( x, t ) is the KdV field and w ( x, t ) is the deforming function with its asymptotic lim | x |→∞ w = c ( t ), b eing an arbitrary fu nction in time. W e can c hec k that th e flatness condition U t − V x + [ U, V ] = 0 of the Lax pair (4) yields th e deformed Kd V equation (1 -2 ), where the un deformed part is g iven by the standard p air s olit1 ¸ U k dv ( λ ) = i ( λσ 3 + U (0) ) , U (0) = u ( x, t ) σ + + σ − (7) V k dv ( λ ) = iU (0) xx − 4 iλ 3 σ 3 + 2 σ 3 λ ( − U (0) x + i ( U (0) ) 2 ) − 4 iU (0) λ 2 + 2 i ( U (0) ) 3 − [ U (0) , U (0) x ] , (8) while the deformed part with th e nonholonomic constrain t is generated by the n ew add ition (5 ). Ther efore w e may draw an intriguing conclusion that the NHD of the K dV (1-2) can b e lin k ed to the d eformation in the time ev olution of the Jost fun ction, which is related to V def ( λ ) as giv en b y (5-6). W e see in the sequel that this fact leads to an u n us u al solitonic p rop erty in the deform ed KdV, namely the p ossibilit y of ac(de)celerated soliton motion. 2 2.2 Exact soliton solutions F or the deformed Kd V equation (1-2) n o exact solution, except a few particular solutions, could b e found 6kdv ¸ . W e d eriv e f or the s ame equation exact N-soliton solutions, whic h is a clear signature of complete in tegrabilit y of a n onlinear system. It is im p ortan t to note that, the ev olution of the basic KdV field ∂ t u is sustained here from t wo differen t sources: ∂ t 0 u = ∂ t u | c ( t )=0 and ∂ t d u = c ( t ) ∂ ˜ c ( t ) u . The first one is generated b y the standard d isp ersive and nonlinear terms in equation (1), wh ile the seco n d one is sourced b y the deformin g term w x . Therefore the deforming fun ction w satisfying the non h olonomic constr aint (2), can b e determined self-consisten tly through the KdV field as w ( x, t ) = c ( t ) Z u ˜ c ( t ) dx + c ( t ) , (9) where the arb itrary fun ction c ( t ) acting as a f orcing term sitting at the space b ound aries lim x →±∞ w ( x, t ) = c ( t ) arises as an inte gration constan t. Therefore we can find by applying the I SM the exact soliton solutions for the basic as w ell as for the p erturb ing field, in terd ep endent on eac h other. Recall that in using the ISM through the associated linear problem, the sp ace-Lax op erator U ( λ ) describin g the scattering of the Jost functions, plays th e k ey role. Only at the fi nal stage we need to fix the time ev olution of the solitons through the time-dep end en ce of the sp ectral data, determined in turn by the asym p totic v alue of the time-Lax op erator V ( λ ). Note that since in the case of th e deformed KdV equ ation the space-Lax op erator (4) is giv en b y the same oper ator as in the standard K d V (7), the steps for its I S M follo ws the same initial path as for the Kd V equation solit1 ¸ . Th erefore referring the readers to the original literatures f or details we pr o duce the explicit form of the N-soliton for the KdV fi eld u ( x ) as an exact solution to the deformed s y s tem (1-2), or equiv a lently to the 6th order KdV (3) as u N ( x ) = 2 d 2 dx 2 ln detA ( x ) (10) where the matrix fun ction A ( x ) is expressed th rough its elements as A nm = δ nm + β n κ n + κ m e − ( κ n + κ m ) x (11) Here parameters κ n , n = 1 , 2 , . . . , N , denote the time-indep endent zeros of the s cattering matrix elemen t a ( λ = λ n ) = 0, along the imaginary axis: λ n = iκ n and β n ( t ) = b ( λ = λ n ) are th e time-depen den t sp ectral data to b e determin ed from V ( λ ) = V k dv ( λ ) + V def ( λ ), at x → ±∞ . W e notice that due to u → 0 , w → c ( t ) at x → ±∞ , th e asymptotic v alue of (8): V k dv ( λ ) → − 4 iλ 3 σ 3 is th e usual one, while that of V def ( λ ) → i 2 λ − 1 c ( t ) σ 3 determines the cru cial effect of the deformation. As a result w e obtain β n ( t ) = β n (0) e − (8 κ 3 n t − ˜ c ( t ) κ n ) , ˜ c t = c (12) determining fi nally the evo lution of the soliton through (11). The exact N-soliton solution for the deforming fu nction w ( x, t ), indu ced throu gh (9) by the solution of the basic field (10), therefore can b e give n by w N ( x, t ) = 2 c ( t ) ∂ 2 ∂ x∂ ˜ c ( t ) (ln detA ( x, t )) + c ( t ) , (13) where A ( x, t ) is the same matrix fu nction (11) with its time-dep endence (12). 3 T o examine the deform in g effect on solitons in more detail we analyze p articular cases of solution (10) and (13) for N = 1 , 2. 1- soliton solution of NHD of the KD V equation as reduced from (10-13) can b e expressed as u 1 ( x, t ) = v 0 2 sec h 2 ξ , ξ = κ ( x + v t ) + φ, (14) w 1 ( x, t ) = c ( t )(1 − sec h 2 ξ ) , (15) with v = v 0 + v d , where v 0 = 4 κ 2 is the usual constant KdV soliton v elocit y , while v d = − 2˜ c ( t ) v 0 t is its unusual time dep endent part induced b y the deformation. W e stress again th at the deformin g fun ction is determined by the dynamics of the field u , whic h in turn is forced self-consisten tly b y the deforming field. Inserting the explicit soliton solutions (14, 15 ) for b oth u and w in the nonholonomic deformation of the KdV (1-2) one can d irectly c hec k the v alidit y of these exact solutions. Notice that the time-dep endent asymptotic v alue of the d eformation: c ( t ) acts h ere lik e a forcing term sitting at the space b oundaries, whic h for c ( t ) = c 0 t with c 0 > 0 forces the soliton to accelerate, while with c 0 < 0 mak es the soliton to decelerate and finally rev ert its direction (see Fig. 1). It can also b e noted that less the original soliton v elo cit y v 0 , more is the d eforming v elo cit y v d , whic h is p h ysically co n sisten t since the forcing te r m in general must h a v e more pr ominen t effect on slow mo vin g solitons. The exact 2- soliton in the deformed KdV can b e deriv ed similarly from (10-13) with N = 2 in the explicit form: u 2 ( x, t ) = 2 D 2 ( D xx D − D 2 x ) , D = 1 + e − ξ 1 + e − ξ 2 + p 12 e − ( ξ 1 + ξ 2 ) (16) w 2 ( x, t ) = c ( t )  1 + 2 D 2 ( D ˜ D x − ˜ DD x )  , ˜ D = 1 κ 1 e − ξ 1 + 1 κ 2 e − ξ 2 + p 12 ( 1 κ 1 + 1 κ 2 ) e − ( ξ 1 + ξ 2 ) (17) where the s cattering amplitude p 12 =  κ 1 − κ 2 κ 1 + κ 2  2 and ξ n = 2 κ n ( x + v n t ) + φ n , n = 1 , 2 with v n = v 0 n + v dn . The usual constant soliton ve lo cities v 0 n = 4 κ 2 n , n = 1 , 2 of the KdV equation is b o osted here b y time- dep endent vel o cities v dn = − 2˜ c ( t ) v 0 n t , n = 1 , 2, caused by the nonholonomic deformation. The scattering of solitons for the fi eld u as describ ed by the solution (16) with ˜ c ( t ) = c 0 2 t 2 is depicted in Fig. 2, whic h sho ws the usual elastic collision of solitons as conjectured in 6kdv ¸ , b ut with an unusual dyn amics due to the accelerating m otion of the solitons. 3 Tw o-fold in tegrable hierarc h y for the deformed KdV equation The we ll kno wn in tegrable hierarc hy of the standard Kd V equation is giv en by solit1 ¸ u t = B 1 ( δH kdv n +1 δu ) = B 2 ( δH kdv n δu ), with B 1 = ∂ ∂ x and B 2 = ∂ 3 + 2( u∂ + ∂ u ) , where H k dv n , n = 1 , 2 , . . . are the higher Hamiltonians of th e KdV hierarc hy , e.g. H k dv 1 = u, H k dv 2 = u 2 2 , H k dv 3 = 1 3 u 3 − 1 2 u 2 x etc., w ith n = 2 yielding the KdV equation. As has b een sh o wn in kup08 ¸ an in tegrable hierarch y w ith the same Hamiltonians exists also for the d eformed KdV equation with NHD of the equations as u t = B 1 ( δ H k dv n +1 δ u − w ) , B 2 ( w ) = 0 , (18) F or n = 2 one ob viously reco v ers the kno wn deformed KdV equation (1-2). This usual t yp e of h ierarc hy with higher disp ersions can b e generated from the AKNS Lax p air, where the space- Lax op erator remains same as the original one (7), b ut the time-Lax op er ator is c hanged to V ( λ ) = V ( n ) k dv ( λ ) + V def ( λ ). Here 4 the deforming part V def ( λ ) is the same as (5-6), while V ( n ) k dv ( λ ) is th e h igher generalizati on of (8 ), where p olynomial in sp ectral parameter λ up to n-th p o we r app ears. Suc h higher order time-Lax op erator can b e constr u cted by expand in g this matrix in the p o wers of λ and d etermining the matrix co efficient s recursiv ely from the flatness condition, as done in the standard AKNS treatmen t solit1 ¸ . Another simpler solution for this problem based on the dimensional analysis and iden tification of the b uilding blo c ks of the Lax op erators h as b een prop osed r ecen tly kun 08 ¸ . W e disco ver, apart from the us u al in tegrable hierarc hy giv en ab ov e, an unusual hierarc hy for the nonholonomic deformation of the KdV equation, wh ic h can b e represent ed by the same d eformed Kd V (1) bu t w ith higher order n onholonomic constrain ts on the deforming f u nction w . This no ve l integ r ab le hierarc hy can b e generat ed as the flatness condition of a Lax pair, w here th e space-Lax op erator remains as (7), while in the time-Lax operator only the deform in g part c hanges as V ( λ ) = V k dv ( λ ) + V ( n ) def ( λ ). Unlik e the ab ov e KdV hierarc h y , V ( n ) def ( λ ) = 1 2 P n j λ − j G ( j ) , j = 1 , 2 , . . . n con tains only − v e p ow ers of λ up to n , with n -num b er of deforming matrix co efficien ts G ( j ) . The consistency condition generates this new in tegrable hierarc h y of n on h olonomic d eformations giv en by the same deformed evolutio n equation (1), but where constraint (2) is generalized n o w to n -th order through a set of coup led differen tial equations: u t − u xxx − 6 uu x = G (1) 12 , G (1) x = i [ U (0) , G (1) ] + i [ σ 3 , G (2) ] , . . . . . . , G ( n − 1) x = i [ U (0) , G ( n − 1) ] + i [ σ 3 , G ( n ) ] , G ( n ) x = i [ U (0) , G ( n ) ] . (19) Clearly this hierarc h y reduces to NHD of the KdV (1-2) for n = 2, with deforming op erator V (2) def ( λ ) reducing to (5-6 ). 4 Concluding remarks W e pr o v e here a num b er of conjectures on the recen tly prop osed nonholonomic deformation of the KdV equation, unrav eling its sev eral u nexp ected features. In p articular we construct AKNS t yp e matrix Lax pair for this deformed K dV equation, showing an intriguing connection b etw een the deformation of the time ev olution in the asso ciated linear p roblem an d the deformation of the nonlinear equation. Applying the inv erse scattering metho d w e find exact N-soliton solutio n s for the basic as w ell as the deforming field for the deformed Kd V equation, w h ic h giv es also the solution of the 6th ord er Kd V equation. Suc h solitons exhibit in spite of the isosp ectral flo w an un u s ual accelerated motio n , wh ic h is ho wev er consisten t with the particle motion under force. The deforming function w ( x, t ), as seen from (18), en ters in the original hierarc hy of the Kd V equation as a p erturbation toget h er with a nonholonomic differen tial constrain t on it. The driving term sitting at th e space-b oundaries: c ( t ) = w ( ±∞ , t ), whic h can b e an arbitrary fun ction in time, forces the fi eld soliton to accelerate or d ecelerate , with the p erturbing soliton itself created b y the field soliton in a self-consisten t w ay . W e disco v er also an uniqu e tw o-fold integrable hierarc hy for this deformed s ystem, o n e with us ual higher disp ers ion foun d already and the other with new increasingly higher order nonholonomic deforma- tion. Extens ion of n onholonomic deformation to other integ r able mo d els lik e NLS, sin e-Gordon, mKd V etc. is u nder inv estigatio n kun08 ¸ . 5 References [1] S. V acaru, arXiv: 0704.39 86, arXiv: 0707.1519 , arXiv: 0707.16 69 [gr-qc] N. A. F ufaev, J. Appl. Math & Mec h. 70 (2006) 593; O. Kr upko v a, J. Math. Ph ys . 38 (1997) 5098; [2] . Karasu-Kalk anli, A. Karasu, A. Sako vic h, S. S ak o vich, R . T u rhan, arXiv: 0708:3247 [nlin.SI], A new i nte gr able gener alization of the KdV e quation [3] B. A. Kup ershmid t, Phys. Lett., A 372 (2008) 2634 [4] M. Ablo witz, D. J. Kaup, A. C. Newell and H. Segur, S tud. Appl. Math. 53 (1974) 249 M. Ab lo witz and H. S egur, Solitons and Inv erse Scattering T r ansforms (SIAM, Philadelph ia, 1981) S. No vik ov, V. Manak o v, L. Pitaevskii and V. Z akharo v, Theory of Solitons (Cons u ltan ts Bureau, NY, 1984) [5] Anjan Ku ndu, ArXiv: 0711.08 78 [nlin.SI], Nonline arizing line ar e quations to inte gr able systems in- cluding new hier ar chies of nonholo nomic deformation s 6 Figure 1: Exact s oliton solution u 1 ( x, t ) of th e KdV field for the nonholonomically deformed equation (1-2). showing u s ual lo calized form of the solito n , bu t w ith its unusual decelerating motion, as eviden t from th e b en d ing of soliton tr a j ectory with time. Figure 2: Exact 2-soliton solution u 2 ( x, t ) of the Kd V field for the d eformed equation (1-2). or equiv alen tly for (3). Usual elastic soliton scattering with p hase shift is evident, though the dynamics here is dominated b y their unusual accelerati n g motion, reflected in the b end ing of soliton tr a jectories. 7