Continuous Families of Embedded Solitons in the Third-Order Nonlinear Schrödinger Equation

Transcription

1 Coninuous Families of Embedded Solions in he Third-Order Nonlinear Schrödinger Equaion By J. Yang and T. R. Akylas The nonlinear Schrödinger equaion wih a hird-order dispersive erm is considered. Infinie families of embedded solions, parameerized by he propagaion velociy, are found hrough a gauge ransformaion. By applying his ransformaion, an embedded solion can acquire any velociy above a cerain hreshold value. I is also shown ha hese families of embedded solions are linearly sable, bu nonlinearly semi-sable. 1. Inroducion The nonlinear Schrödinger (NLS) equaion wih a hird-order dispersive erm iφ + φ xx + 2 φ = iβφ xxx (1) arises in a wide variey of physical sysems. For insance, propagaion of pico-second opical pulses near he zero second-order dispersion poin in an opical fiber is governed by his equaion [1 3]. Femo-second pulses in a fiber laser caviy are modeled by his equaion as well [1, 2]. Equaion (1) also arises in waer waves near a causic [4]. Wih he rescaling of variables φ = βφ, x = x/β, = /β 2, and dropping he primes, Equaion (1) is normalized as iφ + φ xx + 2 φ = iφ xxx. (2) Address for correspondence: T. R. Akylas, Deparmen of Mechanical Engineering, MIT, Cambridge, MA 2139; [email protected] STUDIES IN APPLIED MATHEMATICS 111: C 23 by he Massachuses Insiue of Technology Published by Blackwell Publishing, 35 Main Sree, Malden, MA 2148, USA, and 96 Garsingon Road, Oxford, OX4 2DQ, UK.

2 36 J. Yang and T. R. Akylas This is he hird-order nonlinear Schrödinger (TNLS) equaion we will sudy in his paper. Noe ha his equaion is Hamilonian. Soliary waves of he TNLS equaion and heir sabiliy properies, raise imporan issues from boh physical and mahemaical poins of view. Physically, if sable soliary waves exis in his equaion, hey could be used as informaion bis in communicaion sysems near he zero dispersion wavelengh, where he fiber s second-order dispersion is relaively small. Mahemaically, due o he hird-order dispersive erm, any soliary wave of he TNLS equaion, if i exiss, resides inside he coninuous specrum of ha equaion, and is hus an embedded solion [5]. Classificaion of embedded solions in he TNLS equaion and characerizaion of heir sabiliy properies pose nonrivial mahemaical challenges. Some progress has been made on hese problems. For insance, Jang and Benney [6], Wai e al. [3], and Haus e al. [7] have shown ha in he presence of hird-order dispersion, he familiar NLS solion wih he sech profile canno remain saionary and will always shed coninuous-wave radiaion and lose energy. Akylas and Kung [4], Klauder e al. [8], and Calvo and Akylas [9, 1] have discovered an infinie number of isolaed embedded solions wih muli-hump profiles. Wih regard o sabiliy, he numerical works by Klauder e al. [8] and Calvo and Akylas [1] have suggesed ha embedded solions are linearly unsable, bu nonlineariy has a sabilizing effec. However, as discussed below, hose numerical resuls invie a differen inerpreaion from wha hese auhors proposed. Despie he above progress, several imporan quesions sill remain open: are embedded solions in he TNLS equaion isolaed, or do hey exis as coninuous families? Are embedded solions indeed linearly unsable as previously claimed [8, 1]? Are hese solions nonlinearly sable? From a broader perspecive, here are many recen resuls in he lieraure ha bear upon hese quesions. For insance, embedded solions have been discovered in various physical sysems, such as he fifh-order Koreweg de Vries (KdV) equaions [9, 11 14], he exended NLS equaions [15, 16], he coupled KdV equaions [17], he second-harmonic-generaion (SHG) sysem [5, 18], he massive Thirring model [19, 2], he hree-wave sysem [21], and ohers [22, 23]. A common feaure of all hose embedded solions is ha hey exis a isolaed parameer poins. For such isolaed embedded solions, heurisic argumens [5], as well as rigorous solion-perurbaion [13] and inernal-perurbaion [14, 18] calculaions have shown ha hey are always nonlinearly semi-sable if hey are linearly sable. These analyical calculaions are fully suppored by direc numerical simulaions [5, 14, 18, 22, 23]. An ousanding open quesion, however, is wheher embedded solions can exis as coninuous families. If hey do, ha would have imporan implicaions for heir nonlinear sabiliy properies: firs, he previous argumens for semi-sabiliy of isolaed embedded solions no longer apply. Second, when coninuous embedded solions are perurbed, hey may, in principle, shed some energy via

3 Coninuous Families of Embedded Solions 361 radiaion and approach nearby embedded solions. Thus, here is a possibiliy ha coninuous embedded solions may be nonlinearly sable a resul, which would be very significan physically. So far, quesions of wheher coninuous families of embedded solions are possible in physical sysems and heir sabiliy properies, have no received much aenion in he lieraure. Here, hese issues will be discussed in he conex of he TNLS Equaion (2). In he presen aricle, we firs show ha infinie families of embedded solions can be found in he TNLS equaion hrough a gauge ransformaion. By applying his ransformaion, an embedded solion can acquire any velociy above a cerain hreshold value. Moreover, all hese families of embedded solions are linearly sable, conrary o previous claims abou heir linear insabiliy. Lasly, we numerically esablish ha hese embedded solions are sill nonlinearly semi-sable. In oher words, even hough hese solions exis as coninuous families, hey sill suffer nonlinear insabiliy for cerain ypes of perurbaions. 2. Infinie families of embedded solions In his secion, we sudy soliary waves of he TNLS equaion (2). I is easy o see ha due o he hird-order dispersive erm, any soliary wave of he TNLS equaion is embedded in he coninuous specrum of he linear par of ha equaion, hus is an embedded solion [5]. We look for embedded solions of he form φ(x, ) = ψ(x v)e ikx+iλ, where ψ is a complex funcion, while v, k, and λ are velociy, wavenumber, and frequency consans. This furnishes moving embedded solions of a raher general form, as more complicaed phase and ampliude funcions generally lead o nonsaionary propagaion of he wave ampliude profile. For insance, inclusion of a chirp (ix 2 erm) in he phase funcion induces ampliude breahing [24]. Wih a redefiniion of he funcion ψ and frequency λ, he above solion form can be rewrien as where φ(x, ) = ψ()e iλ, (3) = x v. (4) In oher words, he wavenumber k in he original solion form can be normalized o zero. Then ψ() saisfies he equaion ψ λψ + ψ 2 ψ = i(ψ + vψ ). (5) I is noed ha if ψ() is a soluion, so are ψ()e iδ, ψ( ), and ψ ( ). Here δ and are arbirary consans, and ψ is he complex conjugae of ψ.

4 362 J. Yang and T. R. Akylas Thus, wihou loss of generaliy, we require he soluion ψ o possess he following symmery: ψ( ) = ψ (), (6) i.e., Re(ψ) is symmeric, and Im(ψ) anisymmeric. Numerically, ψ can be deermined by a shooing procedure wih he following boundary condiions imposed: ψ(),, (7) Re(ψ ) = Im(ψ) = Im(ψ ) =, =. (8) Embedded solions in Equaion (5) depend on wo real parameers: he velociy v, and he frequency λ. We will show ha embedded solions exis on an infinie number of coninuous curves in he (v, λ) parameer plane. In oher words, infinie coninuous families of moving embedded solions exis. These resuls grealy generalize he discree se of embedded solions repored in [8, 9]. Firs, we employ a ransformaion of variables, so ha one of he wo free parameers (v and λ) in Equaion (5) is removed. I may be emping o jus ry rescaling (ψ, ) variables, for insance, defining ψ/ v and v (or ψ/ λ and λ) as new variables. This does no work however: while v (or λ) is normalized o 1, a new parameer appears in fron of he ψ erm, hus no parameer reducion is achieved. A successful variable ransformaion is he following: ψ() = ( v + 1 ) 3 4 e 1 3 i ( ), (9) 3 = ( v ) 1 2, (1) = ( ) v ( 2 λ v + 27) 2. (11) Under his gauge ransformaion, Equaion (5) becomes + 2 = i( + ). (12) We see ha only one parameer,, remains now. In addiion, he erm has disappeared. Noe ha if ψ() possesses he symmery (6), so does ( ). Equaion (12) allows an infinie number of isolaed, double-hump embedded solions a a discree se of parameer values = n,(n = 1, 2,...). These n values can be inferred from previous work on he relaed equaion u u + u 2 u = iɛ(u u ), (13) where embedded solions have been found a discree ɛ values ɛ n (n = 1, 2,...) [8, 9]. Firs, we inroduce new variables û = ɛ n u and ˆ = /ɛ n. Then

5 Coninuous Families of Embedded Solions 363 Equaion (13) becomes he same as Equaion (5) wih λ = ɛ 2 n and v = ɛ 2 n. Subsiuing hese λ and v values ino expression (11), we find ha he Equaion (12) for admis embedded solions a n = ( 1 3 ) 3 ( ɛ2 2 2 n 3 ɛ2 n + 27) 2. (14) As n, ɛ n. In his limi, n The firs few ɛ n values were no explicily given in [8, 9]. Thus, in order o ge he corresponding n values, we have used a shooing echnique on Equaion (12) wih boundary condiions similar o (7) and (8). The firs four n are found o be 1 =.8619, 2 =.6959, 3 =.6316, and 4 = Embedded solions a hese four n values are displayed in Figure 1. We see ha a larger n, he solion becomes lower, and he wo humps end o separae. Embedded solions in Equaion (5) for ψ can now be obained from hose in Equaion (12) hrough he gauge ransformaion (9) (11). We find ha 1.5 (a) 1.5 (b) 1 n=1 1 n=2 Ψ.5 Ψ Θ Θ 1.5 (c) 1.5 (d) 1 n=3 1 n=4 Ψ.5 Ψ Θ Θ Figure 1. The firs four embedded solions ( ) in Equaion (12). The corresponding n values (1 n 4) are.8619,.6959,.6316, and.5939, respecively. Solid lines: ; dashed lines: Re( ); dash-doed lines: Im( ).

6 364 J. Yang and T. R. Akylas.6 (a) (b).4 n= 2 v.2.2 n=4 n=1 c d E n=4 n= λ λ.6 (c).6 (d).4 n=1.4 n=2 ψ.2 ψ Figure 2. (a, b) Families of curves in he parameer planes (λ, v) and (λ, E), where embedded solions ψ exis in he TNLS Equaion (5). (c, d) Profiles of embedded solions a velociy and frequency values marked as c and d in (a), respecively. Solid lines: ψ ; dashed lines: Re(ψ); dash-doed lines: Im(ψ). embedded solions ψ exis on he following infinie number of coninuous curves in he (v, λ) plane: ( ) 3 λ = n v v (15) The firs four such curves (1 n 4) as well as he limi curve (n ) are shown in Figure 2(a). On each of hese curves, a coninuous family of embedded solions ψ exiss. Noe ha he velociies of hese families of embedded solions have a common lower bound, v 1, bu here are no upper bounds. 3 Thus, for any velociy v> 1, a discree infinie se of embedded solions can 3 be found. If v< 1, hese solions disappear. Frequency λ also has a lower 3 bound, whose value depends on he individual soluion family. Bu λ holds for all families. On each soluion curve, when λ< 1 bu above he lower λ 27 bound of ha family, wo embedded solions wih differen velociies exis.

7 Coninuous Families of Embedded Solions 365 The energy of an embedded solion is an imporan quaniy. Here we define he energy as E(λ, v) = ψ() 2 d. (16) Uilizing he gauge ransformaion (9), we easily find ha ( E(λ, v) = β n v + 1 3), (17) where β n = ( ) 2 d. The firs four β n values (1 n 4) are , , , and , respecively. Formula (17) indicaes ha he energy of an embedded solion increases linearly wih velociy. Eliminaing v from Equaions (15) and (17), we find ha he energy E is relaed o he frequency λ as ( ) 3 E 2 E λ = n + 1 β n 3β n 27. (18) The firs four (λ, E) curves are displayed in Figure 2(b). Noe ha on each energy curve, when λ< 1, wo embedded solions wih differen energies exis. 27 The profiles of embedded solions ψ() in he above soluion families can be readily deduced from he embedded solions ( ) hrough he gauge ransformaion (9) (11). To illusrae ypical embedded solions ψ of Equaion (5), we selec wo poins in Figure 2(a) marked as c and d, which belong o he firs and second soluion families respecively. The coordinaes of hese wo poins are approximaely (λ, v) = (.8,.3) and (.8,.674). Embedded solions a hese wo poins are shown in Figure 2(c) and (d). We see ha heir ampliude profiles ( ψ ) are similar o hose in Figure 1(a) and (b) save for horizonal and verical rescalings, while heir phase disribuions are differen. The reason is apparenly due o he gauge ransformaion (9) (11). I is ineresing o noe ha all soluion families in Figure 2(a) emanae from he single poin (λ c,v c ) = ( 1 27, 1 ), which, according o (9), corresponds 3 o he linear limi ( ψ ) of hose solion soluions. The significance of his criical poin may also be seen by considering infiniesimal normal-mode disurbances e iκ and examining he linear specrum of Equaion (5): F(κ; λ, v) κ 3 + κ 2 vκ + λ =. (19) Clearly, (19) has eiher hree real or one real and a pair of complex-conjugae roos. Based on prior experience [25 27], one hen would expec small-ampliude soliary waves, in he form of wave packes, o bifurcae from infiniesimal sinusoidal disurbances having (real) wavenumber κ c ha corresponds o a riple roo of he linear specrum: F(κ c ; λ c,v c ) =, F (κ c ; λ c,v c ) =, F (κ c ; λ c,v c ) =. (2) The criical poin (λ c, v c ) in Figure 2(a) is consisen wih hese condiions, and he value of he criical wavenumber is found o be κ c = 1. The second 3

8 366 J. Yang and T. R. Akylas v n=4 b n=1 (a) ψ (b) n= λ Figure 3. Embedded solions near he bifurcaion poin (λ c, v c ) = (1/27, 1/3). (a) Enlargemen of (λ, v) curves for embedded solions near his poin; (b) an embedded solion of he fourh family a he poin marked b in (a). Solid line: Re(ψ); dashed line: Im(ψ). of condiions (2), in paricular, implies ha he phase speed v of linear sinusoidal waves is saionary a criical condiions, and hence is equal o he group velociy here. (Noe: in his inerpreaion of phase speed and group speed, he frequency shif λ inroduced in Equaion (3) is reaed as a free parameer.) Accordingly, small-ampliude soliary wavepackes, close o he bifurcaion poin, may also be inerpreed as envelope solions wih saionary cress [25 27]. Figure 3 displays one such embedded-solion wavepacke, which belongs o he fourh family. One difference beween wave packes here and hose in [25 27], however, is ha he presen wave packes are embedded solions, while hose in [25 27] are no. We remark here ha he infinie families of embedded solions displayed in Figure 2 are no he only possible embedded solions ψ of Equaion (5). Equivalenly, he infinie discree se of embedded solions, he firs four of which are displayed in Figure 1, are no he only possible embedded solions of Equaion (12). The solions we have sudied above have wo major humps in heir ampliude profiles. Calvo and Akylas [9] have shown ha embedded solions wih hree or more major humps exis as well. In his paper, we shall no discuss hose (more complicaed) solions. 3. Linear and nonlinear sabiliy of embedded solions We now urn o he sabiliy properies of he families of embedded solions found in he las secion. This problem has been considered in [8, 1] for cerain isolaed embedded solions. I was suggesed ha hey are linearly weakly unsable, bu nonlineariy has a sabilizing effec, permiing hose solions o propagae for a long ime wihou breakup [1]. On he oher hand,

9 Coninuous Families of Embedded Solions 367 we have seen in he above secion ha when λ< 1, here exis wo branches of 27 embedded solions in he same family having differen energy (see Figure 2(b)). In such a case, i is ypically expeced from saddle-node bifurcaions in dynamical sysems ha one branch of soluions is sable, while he oher branch is unsable [28]. If his holds also for he TNLS equaion, hen one would expec differen sabiliy properies for he wo branches of embedded solions in he same soluion family. Previous work also shows ha while single-hump (fundamenal) solions are ofen linearly sable, muli-hump solions (wheher embedded or no) are ofen linearly unsable [5, 23]. This fac suggess ha embedded solions in he TNLS equaion, or a leas higher families (n 2) of such solions, migh be linearly unsable. If embedded solions are linearly sable, nonlinearly hey could be semi-sable, i.e., wheher hey persis or break up, depends on he ype of iniial perurbaions imposed. This semi-sabiliy propery has been esablished rigorously for isolaed embedded solions in Hamilonian sysems [5, 14, 18, 22, 23]. For he TNLS equaion, embedded solions exis as coninuous families. Thus, when hey are perurbed, hey may be able o shed some energy via radiaion and end o nearby embedded solions in he same soluion family. If his happens, hese embedded solions could be nonlinearly sable. In his secion, we esablish, however, ha he sabiliy properies of embedded solions in he TNLS Equaion (2) do no follow he above common scenarios. In paricular, we show ha for he TNLS equaion, (i) all embedded solions in he same family have idenical linear and nonlinear sabiliy properies; (ii) all families of embedded solions are linearly sable; (iii) embedded solions are nonlinearly semi-sable a leas for he firs and second families. The firs resul can be easily seen from he fac ha all embedded solions in he TNLS equaion are relaed o each oher by he gauge ransformaion (9) (11), hus hey mus be eiher all sable or all unsable. This conrass oher physical sysems, where differen branches in he same soluion family have differen sabiliy properies [28]. The second and hird resuls will be esablished in he following wo subsecions Linear sabiliy of embedded solions To sudy he linear sabiliy of embedded solions (3) in he TNLS Equaion (2), we wrie φ(x, ) = e iλ {ψ() + φ(,)}, (21) where φ is an infiniesimal perurbaion, and is defined in (4). The linearized equaion for φ is i φ λ φ iv φ + φ xx + 2 ψ 2 φ + ψ 2 φ i φ =. (22)

10 368 J. Yang and T. R. Akylas ~ ~ peak ampliude Figure 4. Evoluion of infiniesimal perurbaions o he embedded solion of he firs family, shown in Figure 2(c). To deermine he linear sabiliy of hese solions, we numerically simulae he above linearized equaion o see if is soluion φ has exponenially growing modes or no. Because, a whole family of embedded solions has he same sabiliy behavior, we only need o pick one embedded solion from each family and es is linear sabiliy. The numerical scheme we use is he pseudo-specral mehod (FFT) along he x-direcion, and he fourh-order Runge Kua mehod in. For simpliciy, we choose a Gaussian iniial condiion φ(,) = (1 + i)e (23) Oher iniial condiions have been used as well, and he resuls are qualiaively he same. In he firs family of embedded solions (n = 1, see Figure 2), we choose he solion as displayed in Figure 2(c), whose frequency and velociy values are (λ, v) = (.8,.3). For his solion, he evoluion of he disurbance φ is shown in Figure 4. Noe ha Figure 4(a) is qualiaively he same as Figure 3 in [1]. We see ha he disurbance grows. In [1], his was inerpreed as weak exponenial growh. However, Figure 4(b) reveals ha he disurbance grows only linearly. This linear growh jus corresponds o an adjusmen of his solion s frequency and velociy values, and is no a sign of linear insabiliy. Thus, his embedded solion, or equivalenly he firs family of embedded solions, is linearly sable. I is no difficul o deermine he origin of his linear growh in he disurbance φ. In fac, he linearly growing mode φ(,) = α 1 [ iψ + ψ λ ] (λ,v ) [ ψ + α 2 ψ v ], (24) (λ,v ) is a paricular soluion of he linearized Equaion (22). Here α 1 and α 2 are arbirary-real consans, ψ = ψ(, λ, v) is he soluion of Equaion (5) for general (λ, v) values (which is generally nonlocal), and (λ, v ) are

11 Coninuous Families of Embedded Solions ~ ~ peak ampliude Figure 5. Evoluion of infiniesimal perurbaions o he embedded solion of he second family, shown in Figure 2(d). he embedded solion s frequency and velociy parameers. A general iniial perurbaion φ(,) excies his mode, hus he disurbance grows linearly in ime. Bu his mode does no imply exponenial insabiliy. An analogous siuaion is he NLS solion under perurbaions [29, 3]. Are oher families of embedded solions wih n 2 linearly sable? To answer his quesion, we have repeaed he above numerical simulaion for embedded solions in he higher families, and found ha hey are all linearly sable. To illusrae, we consider he second family, and pick he embedded solion as shown in Figure 2(d), where he frequency and velociy parameers are (λ, v) = (.8,.674). Wih he same Gaussian iniial condiion (23), evoluion of he disurbance φ is displayed in Figure 5. We see ha, jus like he firs family, he disurbance grows linearly, which implies ha he second soluion family is linearly sable as well Nonlinear semi-sabiliy of embedded solions As we have shown above, embedded solions in he TNLS Equaion (2) exis as coninuous families, and hey are all linearly sable. Are hey also nonlinearly sable o small perurbaions? If embedded solions are isolaed in a Hamilonian sysem, hese solions are generally semi-sable [5, 14, 18, 22, 23]. The reason can be undersood heurisically as follows [5]. When an isolaed embedded solion is perurbed, i will end o an adjacen sae, which is no locally confined, bu feaures ails of nonzero ampliude. However, i would require infinie energy o creae a nonvanishing ail. Because he energy is conserved in a Hamilonian sysem, and he energy E of he unperurbed embedded solion is finie, here are wo possibiliies: firs, if he iniial perurbed sae has a oal energy E < E, he energy los in an aemp o generae he infinie ail will drive he solion farher away from he iniial sae. As a resul, he perurbed embedded solion can be expeced o evenually decay ino radiaion. On he

12 37 J. Yang and T. R. Akylas oher hand, if he iniial perurbed sae has energy E > E, we may expec he energy los in generaing he ail o drag he pulse back oward he unperurbed embedded solion. Thus, we can anicipae ha he embedded solion is subjec o a nonexponenial one-sided insabiliy, which is he so-called semisabiliy. However, embedded solions in he TNLS Equaion (2) are coninuous raher han isolaed. These solions have a free parameer, which is heir velociy. In addiion, heir energy depends linearly on heir velociy (see Equaion 17), hus heir energy can acquire an arbirary value. Because of his, he above argumen for semi-sabiliy no longer holds for embedded solions in he TNLS equaion. When an embedded solion in he TNLS equaion is perurbed, in principle, i could simply emi some radiaion and adjus is shape o a nearby embedded solion in he same soluion family, as an NLS solion does under perurbaions. This prospec originally led us o speculae ha embedded solions in he TNLS equaion may acually be nonlinearly sable. Unforunaely, his speculaion urns ou o be incorrec. Our numerical simulaions show ha hese solions are sill semi-sable, jus like isolaed embedded solions in oher physical sysems [5, 13, 14, 18]. In oher words, he freedom of arbirary energy of embedded solions in he TNLS equaion is no sufficien o sabilize hese solions. To explore he nonlinear sabiliy of embedded solions in he TNLS equaion, we numerically simulae his equaion, saring wih an embedded solion under perurbaions. Since a whole family of embedded solions have he same sabiliy properies, i is sufficien o pick one solion from each family and es is sabiliy. In numerical simulaions, we adop he coordinaes, which move a he speed v of he embedded solion. Then he TNLS Equaion (2) becomes iφ ivφ + φ + 2 φ = iφ, (25) where has been defined in Equaion (4). In hese coordinaes, an embedded solion is given by (3). Consisen wih he iniial perurbaions we have used in oher wave sysems [5, 14, 18, 23], we use he iniial condiion φ(,) = (1 + α)ψ(), (26) where α is a small real consan. Tha is, our iniial perurbed sae is he original solion, amplified by a facor (1 + α). In his case, he energy of he iniial sae is (1 + α) 2 muliplied by he energy of he unperurbed solion (see Equaion (16)). When α>, he perurbed sae has higher energy han he unperurbed solion, whereas when α<, he perurbed sae has lower energy han he unperurbed solion. In previous sudies on isolaed embedded solions, we have ermed he former perurbaions as energy increasing, and he laer perurbaions as energy decreasing [5, 14, 18]. However, in he presen case, we will refrain from using such erms. The reason is ha embedded solions here are coninuous, so is heir energy. Thus i may be ambiguous or even

13 Coninuous Families of Embedded Solions 371 α=.2 α=.8 α= solid: =3 dashed: =2 solid: =12 dashed: =6 solid: =8 dashed: = Figure 6. Evoluion of he embedded solion of he firs family, shown in Figure 2(c), under perurbaions (26) for various values of α. Firs column: α =.2; second column: α =.8; hird column: α =.9. confusing o say energy-increasing or -decreasing perurbaions. Oher iniial perurbaions, differen from (26) can also be aken, bu hey are no expeced o change he qualiaive conclusions. Because of he gauge ransformaion (9) (11), i makes no difference, which embedded solion ψ() in a soluion family we use in our simulaions. In oher words, for a fixed value of α, wheher he perurbed sae (26) evenually breaks up or persiss is independen of he choice of he embedded solion in a soluion family. We firs consider he firs family of embedded solions. Exensive numerical simulaions have shown ha his family of embedded solions persis when α.81, while hey break up oherwise. The simulaion resuls for hree values of α,.2,.8, and.9, are ploed in Figure 6, where he paricular solion used is as displayed in Figure 2(c) wih (λ, v) = (.8,.3). We see ha for α =.2, he embedded solion sheds coninuous-wave radiaion, whose ampliude seadily increases over ime. As a resul, he solion breaks up. This behavior is ypical of isolaed embedded solions under energy-decreasing perurbaions [5, 14, 18]. For α =.8, he solion also sheds coninuous-wave radiaion, bu he radiaion ail decreases over ime. Meanwhile, he cenral pulse adjuss iself, and approaches a nearby embedded solion wih a higher velociy. As a resul, he solion persiss under his perurbaion. This behavior is somewha similar o isolaed embedded solions under energy-increasing perurbaions [5, 14, 18]. However, he new feaure here is ha he final embedded solion is differen from he unperurbed solion, alhough he final solion clearly sill belongs o he firs family. Wha happens here is ha par

14 372 J. Yang and T. R. Akylas of he increased energy in he iniial perurbed sae is absorbed by he original solion and changes i o a nearby solion wih higher energy (velociy), while he res of he increased energy radiaes away. For α =.9, however, he siuaion is differen. Iniially, he perurbed sae appears o adjus iself oward a nearby solion wih higher energy (velociy). Bu laer on, he ail radiaion sars o increase, which evenually breaks up he solion. We have checked all hese simulaions wih higher accuracy and over longer periods of ime, and he resuls remain he same. I is noed ha our numerical resuls above are consisen wih previous numerical simulaions by Calvo and Akylas [1]. The above numerical resuls indicae ha he firs family of embedded solions are semi-sable: wheher he solion persiss or breaks up depends on he iniial perurbaion. Compared o he dynamics of isolaed embedded solions, a new feaure we find here is ha embedded solions are unsable no only for α<, bu also for α above a cerain hreshold value (which is.81 here). A he presen ime, he reason for his new behavior is sill no clear. I could be because when α >.81, he perurbaion becomes oo srong for his double-humped embedded solion. As we know, any solion can be broken up wih a srong enough perurbaion. Bu he fac ha he perurbed sae iniially does adjus iself oward a nearby solion, makes us suspec ha he reason for is evenual breakup may be elsewhere (see Figure 6(c)). A full explanaion for his new behavior may be obained from a deailed inernal-perurbaion calculaion as has been done for isolaed embedded solions in several oher physical sysems [14, 18]. Bu his remains o be seen. Are higher families of embedded solions also semi-sable? To explore his quesion, we have repeaed he above simulaions for an embedded solion in he second family. The iniial perurbaion remains he same as in (26). In his case, we have found ha he second family of embedded solions is sable only when α.18, and breaks up oherwise. Three simulaion resuls wih α =.1,.15, and.2 are ploed in Figure 7, where he paricular embedded solion used is he one shown in Figure 2(d) wih (λ, v) = (.8,.674). These resuls are qualiaively he same as for he firs family, bu he α=.1 α=.15 α= Figure 7. Evoluion of he embedded solion of he second family, shown in Figure 2(d), under perurbaions (26) for various values of α.

15 Coninuous Families of Embedded Solions 373 window of α for solion sabiliy is much narrower. In oher words, embedded solions of he second family are more prone o breakup under perurbaions. For he hird family of embedded solions, our numerical simulaions did no find a window of α for solion persisence. This may be because ha window is oo small and we did no deec i. I is also possible ha such a window acually disappears for he hird (and higher) families of embedded solions. This issue is no pursued furher in his paper. 4. Discussion In his aricle, we have sudied embedded solions and heir sabiliy in he hird-order NLS Equaion (2). We have discovered an infinie number of coninuous families of embedded solions parameerized by heir velociies, or equivalenly by heir energies. We have furher shown ha hese families of embedded solions are all linearly sable. Bu nonlinearly hey are sill semi-sable, jus like isolaed embedded solions in oher physical sysems [5, 14, 18]. In he heory of embedded solions, he following quesion sill remains open: are nonlinearly sable embedded solions in Hamilonian sysems possible or no? This quesion is quie imporan for physical applicaions. Previous work has made i clear ha a necessary condiion for such embedded solions o be possible is ha hey exis as coninuous families, no as isolaed soluions [5, 14, 18]. However, our resuls in his paper indicae ha his condiion is apparenly no sufficien. Wheher oher physical sysems suppor nonlinearly sable embedded solions or no needs furher invesigaion. Acknowledgmens The work of J.Y. was parially suppored by he Naional Science Foundaion under gran DMS , and by a NASA EPSCoR minigran. The work of T.R.A. was parially suppored by he Air Force Office of Scienific Research, Air Force Maerials Command, USAF, under gran F , and by he Naional Science Foundaion under gran DMS References 1. G. P. AGRAWAL, Nonlinear Fiber Opics, Academic Press, San Diego, A. HASEGAWA and Y. KODAMA, Solions in Opical Communicaions, Clarendon Press, Oxford, P. K. A. WAI, H.H.CHEN, and Y. C. LEE, Radiaion by solions a he zero group dispersion wavelengh of single-mode opical fibers, Phys. Rev. A 41:426 (199).

16 374 J. Yang and T. R. Akylas 4. T. R. AKYLAS and T. J. KUNG, On nonlinear wave envelopes of permanen form near a causic, J. Fluid Mech. 214:489 (199). 5. J. YANG, B. A. MALOMED, and D. J. KAUP, Embedded solions in second-harmonic-generaing sysems, Phys. Rev. Le. 83:1958 (1999). 6. P. S. JANG and D. J. BENNEY, Dynamics Technology, Inc, Technical Repor No. DT (1981), unpublished. 7. H. A. HAUS, J. D. MOORES, and L. E. NELSON, Effec of hird-order dispersion on passive mode locking, Op. Le. 18:51 (1993). 8. M. KLAUDER, E. W. LAEDKE, K. H. SPATSCHEK, and S. K. TURITSYN, Pulse propagaion in opical fibers near he zero dispersion poin, Phys. Rev. E 47:R3844 (1993). 9. D. C. CALVO and T. R. AKYLAS, On he formaion of bound saes by ineracing nonlocal soliary waves, Physica D 11:27 (1997). 1. D. C. CALVO and T. R. AKYLAS, Sabiliy of bound saes near he zero-dispersion wavelengh in opical fibers, Phys. Rev. E 56:4757 (1997). 11. S. KICHENASSAMY and P. J. OLVER, Exisence and nonexisence of soliary wave soluions o higher-order model evoluion equaions, SIAM J. Mah. Anal. 23:1141 (1992). 12. A. R. CHAMPNEYS and M. D. GROVES, A global invesigaion of soliary wave soluions o a wo-parameer model for waer waves, J. Fluid Mech. 342:199 (1997). 13. J. YANG, Dynamics of embedded solions in he exended KdV equaions, Sud. Appl. Mah. 16:337 (21). 14. Y. TAN, J. YANG, and D. E. PELINOVSKY, Semi-sabiliy of embedded solions in he general fifh-order KdV equaion, Wave Moion 36:241 (22). 15. A. V. BURYAK, Saionary solion bound saes exising in resonance wih linear waves, Phys. Rev. E 52:1156 (1995). 16. J. FUJIOKA and A. ESPINOSA, Solion-like soluion of an exended NLS equaion exising in resonance wih linear dispersive waves, J. Phys. Soc. Japan 66:261 (1997). 17. R. GRIMSHAW and P. COOK, Soliary waves wih oscillaory ails, in A. T. Chwang, J. H. W. Lee, and D. Y. C. Leung, eds. Proceedings of he Second Inernaional Conference on Hydrodynamics, Hong Kong (1996), D. E. PELINOVSKY and J. YANG, A normal form for nonlinear resonance of embedded solions, Proc. R. Soc. Lond. A 458: (22). 19. A. R. CHAMPNEYS, B. A. MALOMED, and M. J. FRIEDMAN, Thirring solions in he presence of dispersion, Phys. Rev. Le. 8:4169 (1998). 2. A. R. CHAMPNEYS and B. A. MALOMED, Moving embedded solions, J. Phys. A 32:L547 (1999). 21. A. R. CHAMPNEYS and B. A. MALOMED, Embedded solions in a hree-wave sysem, Phys. Rev. E 61:886 (1999). 22. A. R. CHAMPNEYS, B. A. MALOMED, J. YANG, and D. J. KAUP, Embedded solions: Soliary waves in resonance wih he linear specrum, Physica D 152:34 (21). 23. J. YANG, B. A. MALOMED, D. J. KAUP, and A. R. CHAMPNEYS, Embedded solions: A new ype of soliary waves, Mah. Compu. Simul. 56:585 (21). 24. T. UEDA and W. L. KATH, Dynamics of coupled solions in nonlinear opical fibers, Phys. Rev. A 42:563 (199). 25. T. R. AKYLAS, Envelope solions wih saionary cress, Phys. Fluids A 5:789 (1993). 26. R. GRIMSHAW, B. MALOMED, and E. BENILOV, Soliary waves wih damped oscillaory ails: An analysis of he fifh-order Koreweg de Vries equaion, Physica D 77:473 (1994).

17 Coninuous Families of Embedded Solions T. S. YANG and T. R. AKYLAS, On asymmeric graviy capillary soliary waves, J. Fluid Mech. 33:215 (1997). 28. D. E. PELINOVSKY, Insabiliies of dispersion-managed solions in he normal dispersion regime, Phys. Rev. E 62:4283 (2). 29. D. J. KAUP, Perurbaion heory for solions in opical fibers, Phys. Rev. A 42:5689 (199). 3. J. YANG, Vecor solions and heir inernal oscillaions in birefringen nonlinear opical fibers, Sud. Appl. Mah. 98:61 97 (1997). UNIVERSITY OF VERMONT MIT, CAMBRIDGE (Received January 17, 23)