On integrability of the vector short pulse equation

On in tegrabilit y of the v ector short pulse equation Ser g ei Sak ov ich Institute of Ph ysics, National Academ y of Sciences, 2200 72 Mins k , Belarus E-mail: saks@tut.b y Abstract Using the P ainlev ´ e analysis preceded b y appropriate transforma- tions of n onlinear systems under in vestig ation, we disco v er t wo new cases in whic h the Pietrzyk–Kanatt ˇ sik o v–Bandelo w vec tor short pulse equation must b e int egrable due to the results of the P ainlev´ e test. Those cases are tec h nologically i mp ortant b ecause they corresp ond to the p ropagation of p olarized ultra-short light pu lses in u sual isotropic silica optical fib ers. The short puls e equation (SPE), whic h has the form u xt = u + 1 6  u 3  xx (1) up to a scale transformation of its v ariables, was in tro duced r ecen tly by Sc h¨ afer and W a yne [1] as a mo del equation des cribing the propagation of ultra-short lig h t pulses in silica optical fib ers ( no te, how ev er, that f or the first time this equation app eared in differen tia l geometry , a s one of Rab elo’s equa- tions describing pseudospherical surfaces [2]). Unlike the celebrated no nlin- ear Sc hr¨ odinger equation whic h mo dels the ev olution o f slo wly v arying w av e trains, the SPE is w ell applicable when the pulse sp ectrum is not nar ro wly lo calized around the carrier frequency , that is when the pulse is as short as a few cycles of its cen tral frequency . Suc h ultra-short pulses are ve ry imp or- tan t fo r future tec hnolo gies of ultr a -fast optical transmission o f information. 1 The SPE (1) is an integrable e quatio n p ossessing a Lax pair [3, 4] of the W adati–Konno–Ic hik aw a t yp e [5]. The tra nsfor ma t io n b etw een the SPE and the sine-Gordon equation w as disco v ered in [4 ], and later it w a s used in [6] for obtaining exact lo op and pulse solutions of the SPE from t he w ell- known kink and breather solutions of the sine-Gordon equation. The deriv ation of that transformation w as considerably simplified in [7], where analogous transformations to w ell-studied equations were a lso found for the other three equations of Ra b elo. The recursion op erato r [4], Hamiltonian structures a nd conserv ed quan tities [8, 9], Hiro t a’s bilinear represen tat ion [10], m ultisoliton solutions [11] and p erio dic solutions [1 2, 13] of the SPE (1) were fo und and studied a s we ll. V ery recen tly , Pietrzy k, Kanatt ˇ sik o v and Bandelow [14] introduced the v ector short pulse equation ( VSPE), a tw o-comp onen t nonlinear w av e equa- tion that generalizes the s cala r SPE (1) and desc r ib es the propagation of p olarized ultra-short ligh t puls es in cubically nonlinear anis otropic optical fib ers, which can b e written a s U n,xt = c ni U i + c nij k ( U i U j U k ) xx , (2) where n, i, j , k = 1 , 2, and the summation o v er the repeated indices is as- sumed. Since the constan t co efficien ts c ni and c nij k are determined b y optical prop erties of the fib er’s material, there is a wide v ariet y of mathematically differen t cases of the VSPE (2), and it is in t eresting to find out whic h of them are integrable systems of coupled no nlinear w av e equations. Three integrable cases of the VSPE (2) were obtained in [14] b y direct construction of their Lax pa ir s, namely , u xt = u + 1 6  u 3 + 3 uv 2  xx , v xt = v + 1 6  3 u 2 v + v 3  xx , (3) u xt = u + 1 6  u 3 − 3 uv 2  xx , v xt = v + 1 6  3 u 2 v − v 3  xx , (4) and u xt = u + 1 6  u 3  xx , v xt = v + 1 2  u 2 v  xx , (5) where u and v denote the p olarizatio n comp onen ts U 1 and U 2 . The following v a luable remark w as made in [14] on the nature of the system (5): this case of the VSPE describes the propagation of a small p erturbation v on the bac kground of a s o lution u of the scalar SPE (1) . Con trary to what w as prop osed in [14], how ev er, we cannot consider the VSPE (3) as a short pulse coun terpart of t he Manak o v syste m [15] of coupled nonlinear Schr¨ odinger 2 equations. Indeed, in the new v ariables p = u + v and q = u − v t he equations of the system (3) b ecome uncoupled and turn into t w o scalar SPEs (1) for p and q separately , whereas the p ola rization mo des in the Manak ov system do interact nonlinearly [16]. In the v ariables p = u + iv and q = u − iv the equations of the system (4) b ecome uncoupled as w ell. It is easy to pro ve that the 4 × 4 zero-curv ature represen tations, f o und in [14] for (3) and (4), can b e broug ht by gauge transformations into the blo ck -diagonal form with the 2 × 2 diagonal blo ck s corresp onding to the zero-curv ature represen tation of the scalar SPE (1) for p o r q . In the presen t pa p er, w e show that there are at least t w o more cases of the VSPE (2) whic h can b e strongly exp ected to b e integrable due to the analytic pro p erties o f their general solutions, na mely , u xt = u + 1 6  u 3 + uv 2  xx , v xt = v + 1 6  u 2 v + v 3  xx , (6) and u xt = u + 1 6  u 3  xx , v xt = v + 1 6  u 2 v  xx . (7) The VSPE (6) represen ts the tec hnologically imp o rtan t case, where the fib er is made of a cubically nonlinear isotropic o ptical material, suc h as the widely used silica glass, but the ultra-short light pulse is not linearly p olarized; if in tegrable, this system can b e interes ting a s a short pulse coun terpart o f the Manak ov system. The VSPE (7) can b e considered as the limiting case of the system (6) for small v alues of v , that is the case of almost linearly p o la rized pulses. W e disco ver these systems (6) and ( 7) by applying the P ainlev´ e test for integrabilit y of partial differen tial equations [17, 18, 19] to t he following t wo one-parameter classes of VSPEs: u xt = u + 1 6  u 3 + cuv 2  xx , v xt = v + 1 6  cu 2 v + v 3  xx , (8) and u xt = u + 1 6  u 3  xx , v xt = v + c 6  u 2 v  xx , (9) where c is the parameter. Of course, this is far not the complete test for in tegrability of the whole v ariet y of systems (2) but ra ther the first a t tempt to searc h for new inte grable VSPEs systematically . In what follows , w e mak e the computations using the Mathematica computer algebra system [2 0] and omit their inessen tial bulky details. Let us consider the class o f systems (8) first. It is easy to see that the P ainlev ´ e test cannot b e applied to the VSPE (8) directly , for the reason of 3 an inappropriate do minan t b ehav ior o f solutions near a mov able singularity manifold, a nd w e mus t appropriat ely transform the nonlinear system under in v estigation in o r der to improv e this b ehavior and star t the test. This is a p oint of crucial imp ortance in our study . W e follow the w ay of tra nsformation similar to the wa y used in [7] for the scalar SPE ( 1) and other Rab elo’s equations. W e mak e the c hange of the indep enden t v ariable x , u ( x, t ) = f ( y , t ) , v ( x, t ) = g ( y , t ) , y = y ( x, t ) , (10 ) and determine the function y ( x, t ) by the relation y t = 1 2  u 2 + v 2  y x . (11) In this relatio n (11), the p olynomial u 2 + v 2 is tak en for the reason of sym- metry b et wee n u and v ; one could use there a general quadratic p olynomial in u and v instead, but t his w ould ha v e no effect on the dominant b eha vior of solutions, p ositions of resonances a nd compatibility o f recursion relations, found during the P ainlev ´ e analysis. Then, inv erting y = y ( x, t ) as x = x ( y , t ), w e obta in fr o m (8) a nd (11) the follo wing system of three coupled equations for f ( y , t ), g ( y , t ) and x ( y , t ): 2 x t + f 2 + g 2 = 0 , 6 x 2 y f y t + (3 − c ) g 2 x y f y y − 2 cf g x y g y y +  ( c − 3) g 2 f y + 2 cf g g y  x y y +(6 − 4 c ) g f y g y x y − 2 cf g 2 y x y − 6 f x 3 y = 0 , 6 x 2 y g y t + (3 − c ) f 2 x y g y y − 2 cg f x y f y y +  ( c − 3) f 2 g y + 2 cg f f y  x y y +(6 − 4 c ) f g y f y x y − 2 cg f 2 y x y − 6 g x 3 y = 0 . (12) Note that the fact of correspo ndence b et w een the fifth- o rder system (12) and the fourt h-order system (8) (w e mean here the tota l order o f a system, or the n umber of a r bit r a ry functions in its gene ral solution) c a n b e explained b y the inv ariance o f (12) with resp ect to an arbitrary transformation y 7→ Y ( y ), whic h just means that the solutions of (12) represen t the s o lut io ns of (8) parametrically , with y b eing the parameter. Substitution of the expansions x = x 0 ( y , t ) φ ( y , t ) α + · · · + x r ( y , t ) φ ( y , t ) r + α + · · · , f = f 0 ( y , t ) φ ( y , t ) β + · · · + f r ( y , t ) φ ( y , t ) r + β + · · · , g = g 0 ( y , t ) φ ( y , t ) γ + · · · + g r ( y , t ) φ ( y , t ) r + γ + · · · (13) 4 to the system (12) determines the dominan t b eha vior o f solutions in the neigh b o r ho o d of a manifo ld φ ( y , t ) = 0, i.e. the admis sible v alues of α , β , γ , x 0 , f 0 and g 0 , and t he corresp o nding p ositions of resonances r , where some arbitrary functions can en ter the expansions . If w e assume that at least one of the exp onen ts α , β , γ is negativ e, we immediately get α = β = γ = − 1 fo r all v alues o f c except c = − 1 (w e do not kno w at presen t ho w to transform t he syste ms ( 8 ) and (9) with c = − 1 in order to start the P ainlev´ e test for them, and w e do not consider t he case of c = − 1 in this paper). W e hav e to ex clude the manifolds φ = 0 with φ y φ t = 0, for whic h no w ell- p osed recursion r elations app ear for the co efficien ts x n , f n , g n of the expansions (13); the reason to exclude suc h characteristic manifolds consists in that arbitra rily nasty singularities of solutions can o ccur along c haracteristics, eve n for in tegrable equations [21, 22]. F or non- c haracteristic manifolds, without loss of generality , we choose φ y ( y , t ) = 1 with φ t 6 = 0 and set all the co efficien ts x n , f n , g n in (13) to b e functions of t only . Then we find fo r c 6 = − 1 , 1 , 3 that x 0 = − (1 + c ) φ t , f 0 = ± i √ 1 + c φ t , g 0 = ± i √ 1 + c φ t , r = − 1 , 1 , 4 , 1 2  5 − q 27+23 c 3 − c  , 1 2  5 + q 27+23 c 3 − c  , (14) where the ± signs in the expressions for f 0 and g 0 are indep enden t, and i 2 = − 1. The case o f c = 3, i.e. the VSPE (3), is not of in t erest b ecause the equations can b e easily uncoupled. And f o r c = 1 we find that x 0 = − 2 φ t , f 2 0 + g 2 0 = − 4 φ 2 t , r = − 1 , 0 , 1 , 4 , 5 , (15) where g 0 ( t ) (or f 0 ( t )) is ar bitrary due to the resonance r = 0. In the case of relations (14), there a re only three p ossibilities to hav e four resonances in integer non-negativ e p ositions, namely , c = − 1 , 0 , 1. How ev er, the v alues c = − 1 , 1 hav e b een excluded, whereas t he v alue c = 0 corr esp o nds to the unin teresting case of uncouple d equations in the VSPE (8). On the other hand, in t he case of relations (15 ) corresp onding to c = 1, w e hav e a lr eady got the a dmissible p ositions of re sonances. Con tin uing to study this case, w e deriv e from the equations (12) with c = 1 the r ecursion relations fo r the co efficien ts of expansions (13) with α = β = γ = − 1, che ck t he compatibility of those relatio ns at the resonances, and find t hat all the compatibilit y con- ditions are satisfied identically . Th us, in this case, the singular expansions of solutions turn out to be some generalized Lauren t series con taining no 5 non-in teger p ow ers and no lo garithmic terms, the arbitrary functions in the series b eing g 0 ( t ), x 1 ( t ), g 4 ( t ), g 5 ( t ), a nd ψ ( t ) in φ = y + ψ ( t ). T o complete the P ainlev ´ e te st for the s ystem (12) w it h c = 1, we also ha ve to study the expansions (13 ) with α = β = γ = 0. Suc h expansions, whic h star t like T ay lo r series, exist for ev ery partial differen t ia l equation. Usually , they are the T a ylor expansions of regular solutions gov erned by the Cauc h y–Kov alevsk a y a theorem [23]. In some cases, how ev er, suc h T aylor- lik e expans io ns can con tain non-dominan t singularities; this ma y happ en, for example, when the Kov alevsk ay a form of a studied equation is singular for some of Cauc h y dat a [24, 25]. Substituting the expansions (13) with α = β = γ = 0 t o the system (12) with c = 1 , w e find that no we ll-p osed recursion r elations app ear for the co efficien ts x n , f n , g n if t he ma nif o ld φ = 0 is determined b y the conditions φ y φ t = 0 or ( f 2 0 + g 2 0 + 6 x 0 ,t ) φ y = 6 x 0 ,y φ t . W e exclude suc h c hara cteristic manifolds, c ho osing φ y = 1 with φ t 6 = 0 and setting all t he co efficien ts x n , f n , g n in ( 1 3) to b e functions of t only , with f 2 0 + g 2 0 + 6 x 0 ,t 6 = 0. Then w e find that the p ositions o f resonances dep end on whether the Cauch y data satisfy the conditio n f 2 0 + g 2 0 + 2 x 0 ,t = 0. When f 2 0 + g 2 0 6 = − 2 x 0 ,t , the arbitrary functions in the expansions are x 0 ( t ), f 0 ( t ), g 0 ( t ), f 1 ( t ), g 1 ( t ), as w ell as ψ ( t ) in φ = y + ψ ( t ) (the appeara nce of one extra ar bit r a ry function in T a ylor- lik e expansions of solutions w as discussed in [24]). When f 2 0 + g 2 0 = − 2 x 0 ,t , the arbitrary functions in the expansions are x 0 ( t ), g 0 ( t ), g 1 ( t ), g 2 ( t ), as w ell as ψ ( t ) in φ = y + ψ ( t ). In b ot h cases, the p ositions of resonances are inte g er, and the recursion relations ar e com- patible a t the resonances. Therefore the expansions are some T ay lor series con taining no non- in teger p o w ers and no logarithmic terms. Consequen tly , the system (12) with c = 1 has passed the P ainlev ´ e test, and the VSPE (6) can b e strongly exp ected to b e integrable. Let us pro ceed now to the class of systems (9 ). Making the same trans- formation ( 1 0), setting y t = 1 2 u 2 y x , (16) and inv erting y = y ( x, t ) as x = x ( y , t ), w e obtain from (9) and (16) the follo wing system of three coupled equations for f ( y , t ), g ( y , t ) and x ( y , t ): 2 x t + f 2 = 0 , f y t − f x y = 0 , 6 x 2 y g y t + (3 − c ) f 2 x y g y y − 2 cg f x y f y y +  ( c − 3) f 2 g y + 2 cg f f y  x y y +(6 − 4 c ) f g y f y x y − 2 cg f 2 y x y − 6 g x 3 y = 0 . (17) 6 Then, using the exp ansions (13) , c ho osing φ y = 1 w ith φ t 6 = 0 to exclude c haracteristic manifolds φ = 0, setting a ll the co efficien ts x n , f n , g n to b e functions of t only , and assuming that at least one of the exp onents α , β , γ is negative, w e find from the system (17) that α = β = − 1 , γ 2 − 3 γ + 2 = 6 /c, x 0 = − 2 φ t , f 0 = ± 2i φ t , r = − 1 , 0 , 1 , 4 , 3 − 2 γ , (18) where the resonance r = 0 correspo nds to the arbitrariness of g 0 ( t ). The resonance r = 3 − 2 γ m ust corresp ond to the arbitrary co efficien t g 3 − 2 γ ( t ), due to the structure of recursion relatio ns whic h follow from (17) and (13). Denoting the resonance p osition 3 − 2 γ as m and using (1 8), w e hav e γ = (3 − m ) / 2 , c = 24 / ( m 2 − 1) . (19) The admissible v alues of m are m = 2 , 3 , 4 , 5 , . . . , since m = 0 is excluded b ecause it leads to the double r esonance r = 0 , 0 whic h indicates that the ex- pansion for g m ust con ta in a logarithmic t erm, and m = 1 is excluded b ecause it implies c = ∞ . Though the ev en v alues of m corresp o nd t o non-in t eger v a lues of γ , these cases should not b e excluded, b ecause b y intro ducing the new v ariable h = g 2 one can impro ve the dominant b ehav io r of solutions. W e hav e fo und infinitely many cases of the system (17) , whic h are all c har - acterized by some admissible dominan t b ehav ior of solutions and a dmissible p ositions o f resonances. Unfortunately , we cannot c heck the compat ibility of the recursion relat io ns at t he resonances for the whole infinite set of those cases at once. This situation is quite similar to the one observ ed in the P ainlev ´ e analysis of triangular systems of coupled Kor t ew eg–de V ries equa- tions [26]. On a v a ilable computers, we w ere able to complete the P ainlev ´ e test for the cases m = 2 , 3 , . . . , 9 , 10 of the system (1 7) with c given b y (19 ) . The recursion relations turn out to b e compatible only in the cases m = 3 and m = 5, whereas some non tr ivial compatibilit y conditions app ear in all other cases at the resonance r = m as an indication of non- dominan t log - arithmic singularities of solutions. The case m = 3 with γ = 0 and c = 3 corresp onds to the in tegrable VSPE (5) discov ered in [14]. The case m = 5 with γ = − 1 a nd c = 1 corresponds to our new VSPE (7). Con tinuing to study the system (17) with c = 1, w e consider the expansions (1 3) with α = β = γ = 0, and they turn out to b e some T a ylor series con taining no non-in teger p o wers and no logarithmic terms. Conse quently , the system (17) 7 with c = 1 has passed the P a inlev ´ e test, and the VSPE (7) can b e strongly exp ected to b e in tegrable. Let us remind, ho wev er, that the Painlev ´ e pro p ert y do es not prov e the in tegrability of a nonlinear equation but only giv es a strong indication that the equation must b e integrable. C onsequen tly , the new pro bably in tegr a ble nonlinear systems (6) and (7), discov ered in this pap er, deserv e further in ve s- tigation, esp ecially t a king in to accoun t their imp o r tance for phy sics and tech- nology . 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