The short pulse equation and associated constraints

LETTER TO THE EDITOR The short puls e equation and asso ciated cons train ts Theo doros P . Horikis Department of Co mputer Scie nce a nd T ec hnology , Univ ersity of Peloponnese, T rip olis 22100 , Greece E-mail: horiki s@uop. gr Abstract. The short pulse e quation (SPE) is consider ed as an initial-b oundar y v a lue problem. It is found that the solutions of the SP E must satisfy an integral r e lation otherwise the tempo ral der iv ative exhibits discontin uities. This integral rela tion is not necess ary for a so lution to exist. An infinite num b er of such constra int s ca n b e dynamically gener a ted by the evolution equa tion. P ACS num b ers: 02.30 .Ik , 0 2 .30.Jr Submitted to : J. Phys. A: Math. Gen. L etter to the Editor 2 The standard mo del for describing propagation of a pulse-shap ed complex field en v elop e in no nlinear disp ersiv e media is the nonlinear Sc hr¨ odinger (NLS) equation. In the con text of nonlinear o pt ics, the main assumption made when deriving the NLS equation fro m Maxwe ll’s equations is that the pulse-width is large as compared to the p erio d of the carrier frequency . When this a ssumption is no longer v alid, i.e., f o r pulse duration of the order of a few cycles of the carrier, the ev olution of suc h “short pulses” is b etter described by the so-called short- pulse equation (SPE) [1]. The SPE can b e expressed in the follo wing dimensionless form, u xt = u + 1 6 ( u 3 ) xx (1) where subscripts denote partial deriv ativ es. The SPE forms an initial-b oundary problem when accompanied b y the initial dat a u (0 , x ) = u 0 , and sufficien tly fa st deca ying b oundary conditions u ( t, ±∞ ) = 0. Muc h like the NLS equation, the SPE is inte gra ble [2 ] and exhibits soliton solutions in the form of lo op- solitons [3]. Ho w ev er, when it is formed as an ev olution equation certain conditions m ust apply otherwise, as sho wn below, the temporal deriv ativ e exhibits discon tin uities. Despite t he fact that the equation is in tegrable via the inv ers e scattering transform [4], there are certain subtleties that need to b e clarified. In tegration of Eq. (1) in tro duces the op eratio n ∂ − 1 x u ( t, x ) = Z x −∞ u ( t, x ′ ) d x ′ Clearly as x approac hes −∞ , ∂ − 1 x u = 0, consisten t with rapidly decayin g data. How ev e r, as x appro a c hes + ∞ , for u a nd its time and space deriv atives to deca y , a constraint seems to b e necessary (see the discussion b elo w), namely Z + ∞ −∞ u ( t, x ) d x = 0 (2) Indeed, writing the SPE in ev olution t yp e form w e hav e u t = ∂ − 1 x u + 1 6 ( u 3 ) x = Z x −∞ u d x ′ + 1 6 ( u 3 ) x and imp osing the b o undar y condition as x → + ∞ , one results to Eq. (2). In f a ct, this constrain t induces f urther constraints obta ined b y success iv ely taking the time deriv ativ e of the integral and using Eq. (1). F or example, the next constrain t is giv en b y Z + ∞ −∞ ∂ − 1 x u d x = 0 (3) Ho w ev er, Eqs . (2), (3), along with the rest of the family of infinite constrain ts generated as ab ov e, are not generically true. One migh t surmise that constrain ts are required at all t imes for a solution to exist. Ho w ev er, as discussed b elow, this is not the case. Extra constraints on the initial data ar e not necessary , but the solution suffers from L etter to the Editor 3 a temp oral discon tinuit y . F or smo oth initial data not satisfying Eq. (2), u t ( t, x ) has at t = 0 differen t left a nd right limits and the rest of the family o f constrain ts cannot b e generated dynamically at that p o in t. The same issues arise in the con text of the Kadom tsev-P etviash vili ( K P) equations and w ere studied in R efs. [5, 6]. Our analysis starts by taking the F ourier transfor m (FT) of Eq. (1), ik ˆ u t = ˆ u − k 2 6 b u 3 (4) where the F T pair is defined as ˆ u ( t, k ) = F { u ( t, x ) } = Z + ∞ −∞ u ( t, x ) e ik x d x u ( t, x ) = F − 1 { ˆ u ( t, k ) } = 1 2 π Z + ∞ −∞ ˆ u ( t, k ) e − ik x d k Define ˆ U = b u 3 and write Eq. (4) in the form of a first o r der differential equation in t , ˆ u t − 1 ik ˆ u = ik 6 ˆ U (5) whic h can b e readily in tegrated with the use of integrating factors to giv e ˆ u ( t, k ) = ˆ u 0 e t/ik + ik 6 e t/ik Z t 0 ˆ U e − τ /ik d τ (6) where ˆ u 0 = ˆ u 0 ( k ) = F { u 0 ( x ) } , is the FT of the initial data. Using Eq. (6) w e calculate the temp oral deriv ativ e to b e ˆ u t ( t, k ) = 1 ik ˆ u 0 e t/ik + ik 6 ˆ U + 1 6 Z t 0 ˆ U e ( t − τ ) /ik d τ (7) Clearly , from Eq. (5), as k tends to zero, w e should demand that so will ˆ u . This translates to [7] ˆ u ( t, 0) = 0 ⇔ Z + ∞ −∞ u ( t, x ) d x = 0 Ho w ev er, as t → ± 0, we hav e that ˆ u (0 , x ) = ˆ u 0 from Eq. (6), and ˆ u t = 1 ik − sign( t )0 ˆ u 0 + ik 6 ˆ U This is b ecause the function exp( t/ik ) defines a distribution, dep ending con tinuously on t , in t he Sc h w artz space of the v ariable k [8] with ∂ ∂ t e t/ik = 1 ik − sign( t )0 e it/k , t = 0 and ∂ ∂ t e t/ik = 1 ik e t/ik , t 6 = 0 This suggests that although there is no discon tin uit y in the solution, there is one in the deriv a tiv e. Indeed, taking the in v erse FT o f Eq. (7), at t → ± 0, we hav e u t ( t → ± 0 , x ) = 1 2 π lim t →± 0 Z + ∞ −∞ 1 ik ˆ u 0 e t/ik e − ik x d k + 1 6 ( u 3 ) x ( t → ± 0 , x ) L etter to the Editor 4 The nonlinear term is straig h tforw ard to handle so w e fo cus on the linear par t, I ( x ) = 1 2 π lim t →± 0 Z + ∞ −∞ 1 ik ˆ u 0 e t/ik e − ik x d k = 1 2 π lim t →± 0 Z + ∞ −∞ 1 ik  ˆ u 0 e − ik x + ˆ u 0 (0) − ˆ u 0 (0)  e t/ik d k = 1 2 π lim t →± 0 Z + ∞ −∞ 1 ik  ˆ u 0 e − ik x − ˆ u 0 (0)  e t/ik d k + 1 2 π lim t →± 0 Z + ∞ −∞ 1 ik ˆ u 0 (0) e t/ik d k (8) Using the prop ert y Z + ∞ −∞ 1 ik e t/ik d k = − π sign( t ) the second integral of Eq. (8) is reduced to − ˆ u 0 (0) π sign( t ) / 2. F urthermore, w e write ˆ u 0 (0) = Z + ∞ −∞ δ ( k ) ˆ u 0 ( k ) e − ik x d k so that finally I ( x ) = 1 2 π Z + ∞ −∞  P  1 ik  − π sign( t ) δ ( k )  ˆ u 0 ( k ) e − ik x d k = 1 2 π Z + ∞ −∞ ˆ u 0 ( k ) ik − 0 sign( t ) e − ik x d k = Z x sign( t ) ∞ u 0 ( x ′ ) d x ′ where P denotes principal v alue. Th us, at t = 0 , Eq. (7) translates in to ph ysical space as u t = Z x sign( t ) ∞ u 0 ( x ′ ) d x ′ + 1 6 ( u 0 ) x As also mentioned in Refs. [7 , 6], the op erator ∂ − 1 x = Z x sign( t ) ∞ d x ′ and its relativ e a v erage ∂ − 1 x = 1 2  Z x −∞ d x ′ + Z ∞ x d x ′  are equiv alent, meaning tha t one can ch o ose either one of t hem. If t 6 = 0 w e ha v e that R + ∞ −∞ u d x = 0, hence b oth choice s a r e v alid. A t t = 0 there is a discon tin uit y in the temp oral deriv ativ e. F or the ev olution of the SPE, Eq. (2) is not preserv ed in time and as suc h leads to the infinite n umber of further constrain ts. Indeed, if Eq. (2) holds then an infinite n um b er of constrain ts, dynamically generated using t he SPE, hold during the ev olution. Within the ph ysical framew ork of the SPE these constrain ts are neither “natural” nor L etter to the Editor 5 necessary . Solutions of the SPE can exist without satisfying this condition, the most prominen t example b eing t he lo op-soliton [3]. This solution, how ev er, in addition to t he p ossible temp oral discon tin uities, suffers from discon tinuities in its spatial deriv ative s, u x ( t, x ), and extra care may b e needed when t he ab ov e fo rmalism is applied. W e conclude with a note on the so-called regularized SPE ( R SPE) mo del, recently deriv ed in Ref. [9]. The latter has b een deriv ed b y including a regularization term, based on the next term in the expansion o f the dielectric’s susceptibilit y . In tha t case, the pulses (of the real comp onen t of the electric field) ar e describ ed by: u xt = u + 1 6 ( u 3 ) xx + β u xxxx where β is a small parameter. Without the regularizatio n term, β u xxxx , i.e., in the case of the SPE –cf. Eq. (1)–, tra v eling pulses in t he class of piece wise smo oth functions with one discon tinuit y do not exist. Ho w ev er, when the regularization term is added, and for a particular par a meter regime, the RSPE supp or t s smo ot h tr av eling w a v es whic h ha v e structure similar to solitary w a v es o f the mo dified KdV equation [9]. The regularization term do es not alter the analysis for the SPE. Indeed, in the F ourier domain the term is written as F { β u xxxx } = β k 4 ˆ u and th us when dividing with ik from the left-hand- side the resulting p ow er of k is contin uous at k = 0. The linear part of the RSPE, m uch lik e the linear part of the K P- I equation [5, 8], deserv es mor e study and the analysis will b e presen ted in a future comm unication. Ac kno wledgmen ts I wish to thank Mark J. Ablo witz for bringing the KP analysis to m y atten tion and Barbara Prinari, Dimitr i J. F rantze sk akis and P anay otis G. Kevrekidis for man y useful discussions . References [1] T. Sch¨ afer and C.E. W ayne. Pro pagation o f ultr a -short o ptica l pulses in cubic nonlinear media . 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