A scalar nonlocal bifurcation of solitary waves for coupled nonlinear Schrödinger systems

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 5 (22) NONLINEARITY PII: S95-775(2)349-4 A scalar nonlocal bifrcation of solitary waes for copled nonlinear Schrödinger systems Alan R Champneys and Jianke Yang 2,3 Department of Engineering Mathematics, Uniersity of Bristol, Bristol BS8 TR, UK 2 Department of Mathematics and Statistics, The Uniersity of Vermont, 6 Colchester Aene, Brlington, VT 54, USA a.r.champneys@bris.ac.k and jyang@emba.m.ed Receied 2 March 22, in final form 25 Jly 22 Pblished 4 October 22 Online at stacks.iop.org/non/5/265 Recommended by P Deift Abstract An eplanation is gien for preios nmerical reslts which sggest a certain bifrcation of ector solitons from scalar (single-component) solitary waes in copled nonlinear Schrödinger (CNLS) systems. The bifrcation in qestion is nonlocal in the sense that the ector soliton does not hae a small-amplitde component, bt instead approaches a solitary wae of one component with two infinitely far-separated waes in the other component. Yet, it is arged that this highly nonlocal eent can be predicted from a prely local analysis of the central solitary wae alone. Specifically, the linearization arond the central wae shold contain asymptotics which grow at precisely the speed of the other-component solitary waes on the two wings. This approimate argment is spported by both a detailed analysis based on matched asymptotic epansions, and nmerical eperiments on two eample systems. The first is the sal CNLS system inoling an arbitrary ratio between the self-phase and cross-phase-modlation terms, and the second is a CNLS system with satrable nonlinearity that has recently been demonstrated to spport stable mlti-peaked solitary waes. The asymptotic analysis frther reeals that when the cres which define the proposed criterion for scalar nonlocal bifrcations intersect with bondaries of certain local bifrcations, the nonlocal bifrcation cold trn from scalar to nonscalar at the intersection. This phenomenon is obsered in the first eample. Lastly, we hae also selectiely tested the linear stability of seeral solitary waes jst born ot of scalar nonlocal bifrcations. We fond 3 Athor to whom correspondence shold be addressed /2/ $3. 22 IOP Pblishing Ltd and LMS Pblishing Ltd Printed in the UK 265

2 266 A R Champneys and J Yang that they are linearly nstable. Howeer, they can lead to stable solitary waes throgh parameter contination. Mathematics Sbject Classification: 35Q55, 74J35, 37G. Introdction Solitary waes play an important role in the soltion dynamics of nonlinear eoltion eqations. If the solitary waes are stable, they often emerge as final states in an initial-ale problem. Een if these waes are nstable, the mere eistence of sch waes has important implications for soltion eoltions. In recent years, it has been discoered that complicated solitary waes cold bifrcate from simple solitary waes. Local bifrcations of wae and daghter waes ( ector solitons ) from singlecomponent waes ( scalar solitons ) hae been stdied in arios forms of copled nonlinear Schrödinger (CNLS) systems [4 6,, 5, 4]. The bifrcation is local, in that the bifrcated solitary wae is infinitesimally close to the original solitary wae (as a graph) at the point of bifrcation. The condition for sch a local bifrcation to occr is based on the linearization arond the single-component plse haing a soltion with prely decaying asymptotics at infinity (see sections 2 and 3). Nonlocal bifrcations are where the bifrcated solitary wae is not infinitesimally close to the original wae at the point of bifrcation. These hae been reported nmerically in the sal CNLS eqations (the generalized Manako system, inoling an arbitrary ratio between the self-phase and cross-phase stdied in section 3, henceforth referred to as the CNLS system) [5, 5, 6], and in second- and third-harmonic generation systems [6, 2, 8]. If the bifrcated wae looks like seeral ector solitons gled together, then nonlocal bifrcation has also been treated analytically by an asymptotic tail-matching method [6] (see section 4 for application of this method in the contet of this paper). General mechanisms hae also been identified that lead to sch ector nonlocal bifrcations in classes of CNLS systems, sch as eigenale degeneracy or the eistence of a local bifrcation [6, 2, 9]. The linear stability of these mltiple-plsed ector solitons for the CNLS system has been stdied in [7], and it has been shown that sch states are linearly nstable. Howeer, if the bifrcated wae is gled together by scalar (i.e. single-component) solitary waes, no analysis has been performed to or best knowledge. In this paper, we focs on this scalar nonlocal bifrcation. We will show that it is closely related to local bifrcations, and can be treated on the same footing. The criterion we propose for a scalar nonlocal bifrcation (in section 2) is that the soltion of the linearized eqation arond the central single-component plse shold hae only prely growing asymptotics instead of prely decaying asymptotics at infinity. In the following sections, we test this criterion against two eample systems. The first, in section 3, is the CNLS system. The second, in section 5, is a CNLS system with satrable nonlinearity. In both cases, good agreement is obtained between or bifrcation condition and the nmerics. Near the bifrcation point, we hae also deeloped a detailed asymptotic analysis based on the aboe-mentioned tail-matching method, which is performed for the CNLS system in section 4. This analysis prodces an eplicit formla for the spacing between scalar solitons being pieced together, and this formla agrees well with the nmerics. An etension of that analytical theory to more general CNLS systems, sch as or second eample stdied in section 5 is straightforward, and will be omitted in this paper. Lastly, we discoer in the corse of or asymptotic analysis that if the cre which defines or proposed criterion for scalar nonlocal

3 Bifrcation of solitary waes for CNLS systems 267 bifrcations intersects with bondaries of certain local bifrcations, the nonlocal bifrcation may trn from scalar to nonscalar at the intersection. This phenomenon indeed occrs in the CNLS system. We hae also stdied the linear stability of solitary waes jst born ot of scalar nonlocal bifrcations. The reslts sggest that these waes are always linearly nstable. Howeer, they can lead to stable solitary waes throgh parameter contination in the second model. 2. A geometric argment Consider a general system of ordinary differential eqations (ODEs) of the form + f (, ) =, (2.) ω 2 + f 2 (, ) =, (2.2) where <ω<. It is assmed that f and f 2 are smooth nonlinear fnctions of their argments which anish as (, ) (, ) and may well depend on other system parameters. Moreoer, they are sch that the problem with or with, which are both inariant sbspaces within the for-dimensional phase space of the ODE system (2.) and (2.2), contain een homoclinic soltions: ((), ()) = ( h (), ), h ( ) = h (), as ±, (2.3) ((), ()) = (, h ()), h ( ) = h (), as ±. (2.4) In what follows we shall se the terms plse homoclinic to the origin, solitary wae and soliton entirely synonymosly. We shall refer to the inariant sbspace homoclinic soltions (2.3) and (2.4) as scalar - and -plses, respectiely. These scalar solitons are contrasted with ector solitons which are homoclinic soltions that hae nonzero - and -components. Note that all homoclinic soltions to the origin are generic, i.e. they persist nder parameter pertrbation, since the system is both reersible and Hamiltonian [3]. Now consider the linearization of (2.) and (2.2) arond the -plse: { } = + f [ h (), ] h () + f [ h (), ], (2.5) = ω 2 + f 2 [ h (), )]. (2.6) Note that the linear eqations decople. So let s look at the specific class of soltions to this linear problem which hae =. Now we hae to simply sole the second eqation, (2.6). The general asymptotics of sch soltions satisfy c ± e ω + c ± 2 eω +o(e ω ) as ± (2.7) for some constants c ± and c± 2. Note that we are able to write the o( )-term by the assmption that <ω<so that the asymptotics of f 2 ((), ) decay more rapidly than ep( ω ). Consider een soltions (2.7) of (2.6), hence c i + = c i := c i, i =, 2. This defines a niqe soltion (p to scale) for the linear initial-ale problem (2.6) for. Now, sppose that at a particlar ale of ω, this soltion had a particlar tail asymptotics (2.7) with c 2 = (see figre (b)). Then the soltion to the linear problem wold be localized. Going back to the flly nonlinear problem, by standard bifrcation theory reslts, we hae satisfied the necessary condition for the local bifrcation of a wae and daghter wae consisting of the mother -plse and a small-amplitde -component. The anishing of c 2 is a codimensionone condition, and hence local bifrcations will lie on lines in a parameter plane. As already

4 268 A R Champneys and J Yang (a) = = (b) = = h h = = h = h = Figre. Sketch figre defining (a) scalar nonlocal and (b) local bifrcations. In each case the top panel depicts schematically the inariant planes { } and { } and the soltion of the linearized problem arond h. The lower two panels depict the corresponding bifrcated ector solitons in phase space and as graphs. mentioned in the introdction, the eistence of sch bifrcations in CNLS systems hae been established by a nmber of athors. Sppose instead that we find a soltion at some parameter ale that satisfies c =. Note that this describes a pre eponentially growing soltion to leading order as ±. See figre (a). Howeer, consider the nonlinear implications of this within the for-dimensional phase space {(,,, ) : = = } of the ODEs (2.) and (2.2). We hae fond an initial condition for an een soltion that is an infinitesimal pertrbation of h (), and which is attracted as towards the inariant plane V := {(,,, ) : = = }. Moreoer, this rate of attraction is faster ( ep()) than the eponential contraction or epansion with that plane ( ep(±ω)) near the origin. Hence, the condition that <ω< ensres that the inariant plane V is normally hyperbolic. Using standard reslts for normally hyperbolic manifolds, the eental behaior of this pertrbed trajectory is goerned by its behaior on V. The fact c = implies that the trajectory is attracted onto the local nstable manifold within V. Bt the local nstable manifold on V is precisely the piece of trajectory that forms the -plse soltion h. Also, since we are talking abot an infinitesimal pertrbation to the nderlying plse h, the time taken to be attracted to h in this way is arbitrarily long. Hence, we hae the scalar nonlocal bifrcation of two h plses at ± as depicted in figre (a). The aboe is only a plasible argment, bt it is highly appealing from an intitie point of iew. In section 4, we will deelop an in-depth asymptotic theory for this scalar nonlocal

5 Bifrcation of solitary waes for CNLS systems 269 bifrcation in the CNLS eqations. The reslts of that theory flly spport the aboe intitie argment. Before proceeding to the eamples, let s make a few short remarks here. (a) First, note the need to assme that <ω<. This was reqired in order to make the = inariant plane normally hyperbolic, or eqialently to assme that the asymptotic attraction onto this plane was o(e ω ). If this condition is iolated, then there is no sense in which the pertrbed trajectory conerges only to the nstable manifold within V and hence the argment fails. (b) Second, it is interesting to note that or analysis sggests that local and scalar nonlocal bifrcations can be treated on an eqal footing. One reqires that c anishes, the other that c 2 anishes. It is perfectly possible to imagine a scenario where, as a parameter is aried, the een soltion () to the linear problem (2.6) generates etra internal oscillations. If it does so in a smooth way, then it is clear that it mst pass repeatedly throgh sccessie zeros of c and c 2. Hence, one wold find each scalar nonlocal bifrcation sandwiched between two sccessie local bifrcations. (c) In fact, in the aboe it is not qite enogh to assme that c anishes for a scalar nonlocal bifrcation. We mst hae that c 2 has the correct sign to attach to the component of the nstable manifold of the nonlinear eqation that contains the pre -plse. In all the eamples below, the pre- eqation is odd and hence both h and h are soltions. Ths, either sign of c 2 will lead to a scalar nonlocal bifrcation. (d) Finally, in systems with odd nonlinearity, the aboe argments can be repeated to find antisymmetric scalar nonlocal bifrcations where the two daghter waes at infinity are h and h. We now trn to two eamples to test the alidity of this approimate reasoning. 3. Eample : the CNLS eqations The sal CNLS eqations may be written in dimensionless form as iu t + U + ( U 2 + β V 2 )U =, (3.) iv t + V + ( V 2 + β U 2 )V =. (3.2) They hae been sed to describe the interaction between wae packets in dispersie conseratie media, and also the interaction between orthogonally polarized components in nonlinear optical fibres (see [5, 5] and references therein). Looking for steady soltions of the form U = e iω2 t (), V = e iω2 2 t (), and performing scaling so that ω = and ω 2 = ω, we arrie at the following set of ODEs: + ( 2 + β 2 ) =, (3.3) ω 2 + ( 2 + β 2 ) =. (3.4) Here, β is a real and positie cross-phase-modlational coefficient, and ω( ) is a propagation constant parameter. It is noted that if [(; ω), (; ω)] is a soltion, then another soltion at propagation constant /ω can be obtained ia the transformation [5] [ ( ) ω ω ; ω, ( ) ] ω ω ; ω. (3.5) Ths, in this paper, we restrict ω sch that ω.

6 27 A R Champneys and J Yang We note that when β =, the partial differential eqations (PDEs) (3.) and (3.2) are called the Manako system, which is integrable [8]. In this case, all solitary waes of the ODE system (3.3) and (3.4) hae closed-form analytical epressions [2]. When β =, the PDEs are two copies of the single NLS eqation which is also integrable [2]. The solitary waes for β = are simply sech plses. When β or, the strctre of solitary waes in this system is mch more complicated. This strctre was partially nraelled in [4,, 2, 5, 5, 6]. It is known that local bifrcations occr along cres in the (β, ω)-plane that are gien by closed-form epressions (see below). These local bifrcations are where wae and daghter-wae strctres are born. In other words, at local bifrcations, a small and localized -component deelops from a pre -plse. It was also obsered nmerically that scalar nonlocal bifrcations, sch as those described in this paper, occr. That is, passing throgh the bifrcation is a singlecomponent plse, for which is zero eerywhere, and is gien by a sech fnction. Bifrcating from this is a soltion for which the -component remains abot the same, bt the -component sddenly deelops two plses which are far-separated from the central -plse. These -plses can be symmetrically or antisymmetrically distribted. Their sizes jmp from zero to a certain finite size across the bifrcation. In this section, we analytically determine the bondaries of these scalar nonlocal bifrcations throgh the criterion deeloped in section 2 and compare them with direct nmerical reslts. 3.. Local and scalar nonlocal bifrcations First, we recall the reslts for local bifrcations in this system [4, 4, 5], assming that a small -component bifrcates from a pre -plse. Ths, at the bifrcation point, the -component is infinitesimally small. Ths the -component is simply goerned by the eqation + 3 =, (3.6) whose homoclinic soltion is () = 2 sech. (3.7) According to standard reslts, a necessary condition of a local bifrcation of a homoclinic soltion with a small-amplitde -component from the -plse (3.7) is that there is a nontriial localized soltion to the linearized problem of the -component. This takes the form of a linear Schrödinger eqation: ω 2 +2β sech 2 =, (3.8) and for local bifrcation we reqire as ±. This eqation can be soled eactly [7], as we now eplain. With the ariable transformation = sech s ψ, ξ = sinh 2, (3.9) where +8β s =, (3.) 2 the Schrödinger eqation (3.8) becomes ξ(+ξ)φ + [ ( s)ξ + 2] φ + 4 (s2 ω 2 )φ =, (3.) which is a hyper-geometric eqation. Its een and odd soltions are φ = F ( 2 ω 2 s, 2 ω 2 s, 2, ξ), (3.2) φ 2 = ξf ( 2 ω 2 s + 2, 2 ω 2 s + 2, 3 2, ξ), (3.3)

7 Bifrcation of solitary waes for CNLS systems 27 where F is the hyper-geometric fnction. In order for the soltion = sech s φ to decay to zero as goes to infinity, one mst hae 2 ω 2 s = n, (3.4) where n is a non-negatie integer, and n < 2 s. Then, the soltion is n = sech s ( n ) k ( (/2)ω (/2)s) k ( ξ) k, (3.5) (/2) k= k k! which decays to zero as ξ ω/2 (i.e. e ω ). In the soltion (3.5), (a) k is defined as { a(a +)(a +2)...(a+ k ), k >, (a) k (3.6), k =. In order for soltion 2 = sech s φ 2 to decay to zero as goes to infinity, one mst hae 2 ω 2 s + 2 = n 2, (3.7) where n 2 is a non-negatie integer, and n 2 < 2 (s ). Then, the 2 soltion is n 2 2 = sech s ( n 2 ) k ( (/2)ω (/2)s +/2) k ( ξ) k sinh, (3.8) (3/2) k= k k! which also decays to zero as ξ ω/2 (i.e. e ω ). When conditions (3.4) and (3.7) are combined, we find that the bondaries for local bifrcations are ω = ωn LB (β) = s n, (3.9) where s is gien by (3.), n is a non-negatie integer, and n<s. The first bondary (n = ) eists for any β ; the second bondary (n = ) eists only for β ; the third bondary (n = 2) eists only for β 3, etc. The first three bondaries ω,,2 LB are plotted in figre 2 as dashed lines for illstration. Note from the aboe constrction that een n corresponds to the eistence of symmetric bifrcating waes (een in both and ) whereas odd n corresponds to antisymmetric bifrcation (een in, odd in ). Now, how can we define the scalar nonlocal bifrcations? It can be noted that on the aboe local bifrcation bondaries, the appropriate soltion or 2 has the following asymptotic behaior: () sgn n ()(c n e ω + d n e ω +o(e ω )), as, (3.2).8 n= n= n=2 ω n= n= n= β Figre 2. Local and scalar nonlocal bifrcation bondaries in the CNLS system (3.3) and (3.4). Scalar nonlocal bifrcation bondaries are solid lines, and are gien by eqation (3.29) (with n = ) and (3.22) (with n =, 2). Local bifrcation bondaries are dashed lines, and are gien by eqation (3.9).

8 272 A R Champneys and J Yang where c n =, and d n is a nonzero constant. In other words, this soltion is localized. Condition c n = in the asymptotics (3.2) is the condition for local bifrcations in this problem. Now, following the argments laid ot in section 2, a scalar nonlocal bifrcation occrs when the -component of the linearized eqation arond the -plse satisfies conditions at infinity that it has a prely growing component. That is, scalar nonlocal bifrcations occr when one of the and 2 soltions of the Schrödinger eqation (3.8) has the following asymptotic behaior: () sgn j ()(α e ω + γ e ω +o(e ω )) as, (3.2) with γ = bt some nonzero α, and j = or is an integer indicating the symmetry of the soltion. Now, a remarkable thing happens. Becase it is easy to see sing the soltions of eqation (3.8) obtained aboe, that condition γ = in the asymptotic eqation (3.2) is eactly satisfied on the bondary cres, ω = ωn NLB (β) := n s, <n s<, (3.22) where n is a non-negatie integer and s(β) was defined in eqation (3.). In fact, on these bondaries, the fnction has an nbonded soltion, n ˆ = sech s ( n ) k ((/2)ω (/2)s) k ( ξ) k, (3.23) (/2) k k! k= when n = 2n is een, and an nbonded soltion, n 2 ˆ 2 = sech s ( n 2 ) k ((/2)ω (/2)s +/2) k ( ξ) k sinh, (3.24) (3/2) k k! k= when n = 2n 2 + is odd. The asymptotic behaiors of these soltions are where ˆ() sgn n ()(g n e ω + h n e (ω 2) ),, (3.25) g n = 2 s ((/2)ω (/2)s) n 4 n (/2)n, n = 2n, 2 s ((/2)ω (/2)s +/2) n2 4 n 2 (3/2)n2, n = 2n 2 +, (3.26) and h n is another constant which can be easily calclated. When ω<, the second term in (3.25) decays faster than e ω. Ths, the coefficient γ in the asymptotics (3.2) is zero on these bondaries. The bondary cre for scalar nonlocal bifrcations (3.22) can be written alternatiely as β = βn NLB (ω) = 8 {(2n 2ω +)2 }. (3.27) These bondaries are plotted in figre 2 as solid lines for comparison with bondaries of local bifrcations (dashed lines), which are gien according to (3.9) by β = βn LB (ω) = 8 {(2n +2ω +)2 }. (3.28) Hence, by constrction, the cres of nonlocal bifrcations simply represent the contination of local bifrcation cres throgh ω = (and mapped back p ia ω ω, since only ω 2 appears in the eqations). In particlar, at the singlar ale ω = we hae that ω NLB n = ω LB n.

9 Bifrcation of solitary waes for CNLS systems 273 Lastly, we note that the first scalar nonlocal bifrcation cre on the left of figre 2 needs a little special treatment. In fact, this solid cre is gien by eqation { s, 8 ω = β and ω 2, 2 [ + +8β], 8 β and ω 2. (3.29) In other words, the lower branch of this cre is as gien in eqation (3.22) with n =, bt its pper branch is gien by a different fnction. It can be shown that on this pper branch, the soltion of the Schrödinger eqation (3.8) also has the asymptotics (3.2) with γ = j =. This cre is the only scalar nonlocal bifrcation bondary whose fnctional form is partially different from (3.22) Nmerical reslts So we hae fond cres on which or proposed condition for scalar nonlocal bifrcations deried in section 2 is satisfied. It remains to be seen what happens to the flly nonlinear eqations for this eample along sch cres. In section 4 we shall consider this problem ia asymptotic analysis. In this section we trn to nmerical methods. First, let s demonstrate frther properties of the strctre of soltions to the linearized problem (3.8) by comptation of its een and odd soltions as the parameters ary. Figre 3 depicts soltions of the constrained linear bondary ale problems: and () =, () =, X X () 2 d = const., (3.3) () 2 d = const., (3.3) for een and odd soltions, respectiely. Here X is a large positie constant. At the right-hand bondary point we can distingish between soltion components that decay with eponential rate e ω and those which grow with rate e ω by considering the corresponding eigenectors in the (, )-plane. Hence, we can define bondary fnctions w = (X) + ω(x), w 2 = (X) ω(x), (3.32) so that a zero of w defines a soltion with no eponentially growing component whereas zeros of w 2 define soltions with no component that decays like e ω. Hence, according to the aboe definitions, w = can be sed as a nmerical test fnction for local bifrcations and w 2 = as a test fnction for scalar nonlocal bifrcations. Specifically figre 3 depicts the reslts of a nmerical contination of een and odd soltions to (3.8), satisfying (3.3) and (3.3), respectiely, for fied ω as β is increased from zero. It can be seen that an alternating seqence of zeros of fnctions w and w 2 occrs as β increases. At the ales of each of the zeros we plot the mode shape (). Note that each sccessie pair of zeros corresponds to the fnction gaining an etra internal zero. The particlar comptation was carried ot with X =. For this ale, it was fond that the β-ales of the depicted zeros of w and w 2 correspond to those of the analytic formlae (3.27) and (3.28) to within fie decimal places. Increasing X reslted in more accracy, bt an increase in the singlarity of the bondary-ale problem close to each zero of w. For this eample, we hae analytic formlae for the conditions defining local and scalar nonlocal bifrcations. Hence these comptational reslts can be interpreted as deeloping nmerical confidence in or method of detecting them in sitations where analytic formlae do not eist (as in section 5). Also, they proide geometrical insight. Thinking of the phase

10 274 A R Champneys and J Yang (a) A w 2 B C A B C D E F D E -2 F β 2 (b) G w 2 2 H I G - H I J K L J K L 5 5 β Figre 3. Cres of (a) w (β) and (b) w 2 (β) for the linearization (3.8) of the CNLS eample, compted for ω =.5 sing and interal [, ]. Solid lines represent asymmetric soltions and dashed lines symmetric ones. The inserts depict soltions of the linearized eqations at the first three zeros of w, both symmetric and antisymmetric, which define local bifrcations; and of w 2 which define the necessary conditions for scalar nonlocal bifrcations according to or geometric theory. space (, ), the conditions for local and scalar nonlocal bifrcations are that the soltion for large shold lie in one of the two eigendirections. By continos dependence on initial condition reslts, we hae that if there are a sccession of local bifrcations with increasing nmber of internal zeros pon increasing a parameter, then the soltion at time X mst rotate in the phase plane. In so doing, we cannot aoid haing a scalar nonlocal bifrcation sandwiched between each two sccessie local bifrcations, see figre 4. Net, let s nmerically inestigate actal solitary wae bifrcations near the proposed scalar nonlocal bifrcation cre (3.22). First, we consider those cres with n = and 2. The scalar nonlocal bifrcations near these cres hae been nmerically eplored in [5]. The reslts are reprodced in figres 5 and 6. In each figre, solitary waes at three different locations of the parameter plane are shown: one is close to the local bifrcation cre (3.9) (dashed line), another one is in the interior, and the third is close to the theoretical scalar

11 Bifrcation of solitary waes for CNLS systems 275 (a) β = β (b) β = β 2 w = w = 2 (c) β = β 3 (d) ( (X), (X) ) β β 2 β 3 Figre 4. Sketch figre illstrating the large- asymptotics of een soltions to (3.8) as the parameter β aries for fied <ω<. The soltion to the bondary ale problem (3.3) is depicted p to = X. Between the two β-ales (a) and (c) at which condition w = for local bifrcations occr, there is a β-ale (b) for which w 2 =. Panel (d) sketches the locs of bondary points ((X), (X)) as a fnction of β. nonlocal bifrcation bondary (3.22). Displays of these solitary waes are meant to show the reader how solitary waes continosly deform from wae and daghter wae strctres as system parameters β and ω ary. As we can see, in both cases, scalar nonlocal bifrcations indeed occr on the theoretical cres (3.22) (see panel (c) in both figres). In addition, the nmerical bifrcation bondaries (circles) fall precisely on the theoretical cres. So, for these cases, or theory is flly spported by nmerics. We hae also fond similar agreement for the case n = in the nonlocal bifrcation bondary (3.22). Howeer, the n = 3 case is more complicated. The bifrcation for this case is shown in figre 7. The solid line in the parameter plane (pper left panel) is the theoretical cre (3.22) for scalar nonlocal bifrcations. Circles are nmerically detected nonlocal bifrcation bondaries. Notice that the nmerical bondary falls onto the theoretical cre (3.22) only in the lower part. There, the nonlocal bifrcation is indeed scalar, consistent with the geometric argment of section 2. This can be confirmed in figre 7(e). Bt in the pper and middle parts, the nmerical bondary deiates from the theoretical cre (3.22). The reason trns ot to be that, in these parts, the actal nonlocal bifrcation is not scalar. Indeed, an inspection of figres 7(c) and (d) shows that the bifrcated solitary waes in these parts are not scalar NLS solitons pieced together. Rather, they are tre ector solitons pieced together. Ths, or analysis for scalar nonlocal bifrcations does not apply here. We note that near the pper part of the nmerical bondary, the centre of the bifrcated wae is a wae and daghter wae strctre with n = (see eqation (3.9)), and it is flanked by two single-hmp ector solitons on the two sides. This piecing together of different ector solitons as a nonlocal conseqence of local bifrcation has been analytically stdied earlier in [6]. It was shown there that the bondary for this type of nonscalar nonlocal bifrcation is precisely the bondary of local bifrcations (3.9) (here with n = ). This is indeed the case. When the local bifrcation bondary (3.9) for n = is plotted as a dashed cre there, it agrees with the nmerical bondary (circles) ery well. The bifrcation in the middle part of the parameter region is also nonscalar. It is clear from figre 7(d) that this bifrcation is somewhere in between the nonscalar bifrcation

12 276 A R Champneys and J Yang (a) c b a.5 ω β 2 2 (b).5 (c) Figre 5. Solitary waes in the parameter region (pper left figre) bonded by the scalar nonlocal bifrcation bondary (3.22) ( ) and local bifrcation bondary (3.9) (- ---) with n =. Circles represent the nmerical scalar nonlocal bifrcation bondary for n = obtained in [5]. Solitary waes at stars marked by the letters a, b, c in the parameter region are shown with corresponding letters in the title. of figre 7(c) and the scalar bifrcation of figre 7(e). In fact, it is appropriate to consider this middle part of the bifrcation bondary as a transition between the nonscalar bifrcation in the pper part and the scalar bifrcation in the lower part. Nmerical searching has reealed that there is no scalar nonlocal bifrcation obsered along the branch corresponding to (3.22) with n = 3 aboe the point at which the nonscalar bifrcations start. Ths, the condition (3.22) can at best be a necessary condition for scalar nonlocal bifrcations. Why does the bifrcation deiate from scalar here, and where eactly does this deiation begin? These qestions cold not be answered by the approimate geometric argment in section 2. Howeer, an answer will be reealed in a matched asymptotic analysis in the net section. We will show that the deiation starts where the cre (3.22) with n = 3 intersects the local bifrcation bondary ω = /s of -plses. 4. Matched asymptotic theory for scalar nonlocal bifrcations in the CNLS eqations To theoretically eplain the scalar nonlocal bifrcation reslts in the preios section, an analytical theory will now be constrcted. This theory has three objecties. The first one is to proe that the bondaries of scalar nonlocal bifrcations are indeed gien by the condition that the soltion of the linear Schrödinger eqation (3.8) has only the prely growing component, i.e. eqation (3.22). The second objectie is to obtain an analytical formla for the spacing

13 Bifrcation of solitary waes for CNLS systems c b a (a).5 ω β (b).5 (c) Figre 6. Solitary waes in the parameter region bonded by the scalar nonlocal bifrcation bondary (3.22) ( ) and local bifrcation bondary (3.9) (- ---) with n = 2. Circles represent the nmerical scalar nonlocal bifrcation bondary for n = 2 obtained in [5]. between the -plses and the central -plse when the parameters are close to the bondary of scalar nonlocal bifrcations. The third objectie is to determine when nonlocal bifrcations can deiate from scalar to nonscalar. The techniqe we will se is similar to the tailmatching method as deeloped in [6] for the constrction of mlti-plse trains, bt important modifications need to be made. Throghot this analysis, we reqire ω<asaboe. Sppose the ODE system (3.3) and (3.4) allows a soltion where the -component is symmetric and has a dominant plse in the centre (at = ), while the -component is symmetric or antisymmetric and has two dominant plses on the two sides of the -plse (at =± ). Or main assmption is that the -plses are well-separated from the central -plse, i.e.. Then, we can diide the soltion into three regions: (I) the left -plse region centred at = ; (II) the central -plse region centred at = ; and (III) the right -plse region centred at =. Below, we will determine the soltions in each of these three regions. Note that midway between regions II and I or III, both the and soltions are ery small. Ths they are approimately goerned by the linear parts of eqations (3.3) and (3.4); hence, these soltions are linear combinations of prely eponentially growing and prely eponentially decaying fnctions to leading order. If these tail asymptotics from two adjacent regions can match each other, then a solitary wae can be fond. This is the essence of the tail-matching method. When the -component is symmetric or antisymmetric, the tail-matching treatment between regions II and III becomes the same as that between regions II and I. Ths, we will focs only on matching between regions I and II.

14 278 A R Champneys and J Yang.8 c b a (a) ω.4.2 d e β 2 2 (b).5 (c) (d).5 (e) Figre 7. Different types of nonlocal bifrcations appearing near the scalar nonlocal bifrcation cre (3.22) with n = 3 ( ) in the CNLS system (3.3) and (3.4). Circles are nmerically obtained nonlocal bifrcation bondaries. The lower part of the nmerical cre lies on the solid line (3.22), and the bifrcation there is indeed scalar (see (e)). The pper part of the nmerical cre lies on the dash-dotted local bifrcation bondary (3.9) with n =, and the nonlocal bifrcation there is nonscalar (see (c)). The nonlocal bifrcation in the middle part of the nmerical cre is somewhere in between scalar and nonscalar (see (d)). The dashed line in the pper left panel is the local bifrcation cre (3.9) with n = 3. Stars are parameter ales where the solitary waes are shown. In region I, the soltion can be written as =ũ I, = + ṽ I, (4.)

15 Bifrcation of solitary waes for CNLS systems 279 where = 2ω sech[ω( + )], ũ I, ṽ I (4.2) (see figres 5(c) and 6(c)). In the new coordinates, ξ = +, (4.3) the small ũ I component satisfies the linear Schrödinger eqation, ũ Iξξ ũ I + β 2 (ξ)ũ I =, (4.4) to leading order. To obtain solitary waes, we demand that ũ I (ξ), ξ. (4.5) At large positie ξ ales, this ũ I soltion mst match the tails of the dominant soltion () = 2 sech, (4.6) in region II. This matching dictates that the asymptotic behaior of ũ I at large ξ ales is ũ I (ξ) 2 2e e ξ, ξ. (4.7) The linear eqation (4.4), together with the bondary conditions (4.5) and (4.7), completely determines the ũ I soltion in region I. Now we determine the small ṽ I component in region I. When eqation (4.) is sbstitted into (3.4), and terms of order ṽi 2, ṽ3 I and ũ2 I ṽi dropped, we find that, to leading order, ṽ I satisfies the following eqation: ṽ Iξξ ω 2 ṽ I +3 2 (ξ)ṽ I = βũ 2 I (ξ). (4.8) We note that it is important to retain the inhomogeneos term in eqation (4.8), as, otherwise, that eqation with the anishing bondary condition at negatie infinity wold always prodce a localized soltion which is impossible to match to the soltion in region II. The bondary conditions for soltion ṽ I are and ṽ I, ξ, (4.9) ṽ I α e ωξ + γ e ωξ 4 2βωe 2 e (2 ω)ξ, ξ, (4.) ω where α and γ are constants. The last term in eqation (4.) is contribted from the inhomogeneos term of eqation (4.8). In deriing it, the asymptotic behaiors of the ũ I and soltions were sed (see eqations (4.2) and (4.7)). Net, we determine the soltions in region II. In this region, the soltions can be written as = + ũ II, =ṽ II, (4.) where () is gien in eqation (4.6), and ũ II, ṽ II (see figres 5(c) and 6(c)). Here, we only need to focs on the ṽ II soltion. This soltion satisfies the eqation ṽ II ω 2 ṽ II + β 2 ()ṽ II = (4.2) to leading order. The leading asymptotic behaior of this soltion at is ṽ II γ(e ω + δ e ω ),, (4.3) where γ and δ are constants. If we only consider solitary waes with symmetric or antisymmetric -components, then the aboe ṽ II soltion wold hae the same symmetry. This symmetry condition wold niqely determine the coefficient δ. The constant γ is selected by the condition that the tail asymptotics of the ṽ II soltion for in region II mst

16 28 A R Champneys and J Yang match the soltion (4.) for ξ in region I. This matching gies γ = (2 2ω + γ)e ω. Recall that ṽ I, and ths γ. As a reslt, to leading order, γ = 2 2ω e ω. (4.4) This matching also gies the relation α = γδe ω = 2 2ωδ e 2ω. (4.5) One may wonder why the third term in the ṽ I asymptotics (4.) is not matched by ṽ II asymptotics (4.3). In fact, there is a smaller term in the ṽ II soltion which is proportional to e (2 ω). This term arises de to the prodct of 2 and the e ω component in the leading ṽ II soltion (4.3) (see eqation (4.2)). One can check that this term will eactly match the third term in the ṽ I asymptotics (4.). So there is no contradiction here. Bt this is a minor isse which is not critical to or analysis. Now we are in a position to derie a formla for the spacing between the -plses and the middle -plse. This formla comes from the solability condition for the ṽ I eqation (4.8) together with the bondary conditions (4.9), (4.) and (4.5). It is noted that eqation (4.8) is self-adjoint, and it has a localized homogeneos soltion ξ de to the spatial translation inariance of the ODE (3.4). Calclating the integrals of prodcts between ξ and the two sides of eqation (4.8) from to y, and integrating by parts, we get y βũ 2 I ξ dξ = (ṽ Iξ ξ ṽ I ξξ ) y. (4.6) When y, sbstitting the bondary conditions (4.9), (4.) and (4.5) into the aboe eqation, we find that y βũ 2 I ξ dξ 6δω 4 e 2ω + 32βω3 e 2 e 2( ω)y, y. (4.7) ω The aboe eqation is the leading two-term epansion for the integral on its left-hand side. When y approaches infinity, this integral dierges. Bt we can separate this diergent part from the rest of the integral. Notice that 32βω 3 e 2 y ω e2( ω)y = 64βω 3 e 2 e 2( ω)ξ dξ. (4.8) Ths eqation (4.7) can be rewritten as [ũ2 ( ) β I 2 ξ + 28ω3 e 2 e 2( ω)ξ] dξ = 32δω 4 e 2ω. (4.9) The integral aboe is no longer diergent. In fact, one can se the asymptotic relations (4.5) and (4.7) to check that the integrand in that integral approaches zero eponentially as goes to infinity. Eqation (4.9) gies a formla for spacing when the system parameters β and ω are specified. This formla can actally be made more eplicit as follows. Recall that the fnction ũ I is determined by eqation (4.4) and bondary conditions (4.5) and (4.7). Under the notation ũ I (ξ) = 2 2e φ(ξ), (4.2) the fnction φ(ξ) is niqely specified by the following eqation and bondary conditions φ ξξ φ + β 2 (ξ)φ =, (4.2) {, ξ, φ(ξ) e ξ, ξ, (4.22)

17 Bifrcation of solitary waes for CNLS systems 28 where (ξ) is gien by eqations (4.2) and (4.3). Under these notations, formla (4.9) simplifies as e 2( ω) = 4ω4 δ βi, (4.23) where I is the integral [ I = φ 2 ( ) 2 ξ +6ω3 e 2( ω)ξ] dξ. (4.24) Recall that the constant δ is defined by eqations (4.2) and (4.3). To be more eplicit, δ is defined by ψ ω 2 ψ + β 2 ()ψ =, (4.25) and ψ() e ω + δ e ω,. (4.26) In other words, δ is the coefficient of the prely decaying component of the Schrödinger eqation (4.25) at =. The bondary condition for the fnction ψ at = is proided by the symmetry of the -component in the solitary wae we are seeking. Since we are focsing on symmetric and antisymmetric -components, fnction ψ wold hae the same symmetry. This symmetry helps to niqely determine the δ-coefficient in the aboe linear problem. Formla (4.23) is the key reslt of this section. It eplicitly gies the epression for the spacing in solitary waes bifrcating from scalar nonlocal bifrcations. Seeral obserations qickly follow from this formla. First, solitary waes from scalar nonlocal bifrcations eist only when parameters δ and I hae the same sign. Second, when δ =, goes to infinity. Ths, this is a bondary of scalar nonlocal bifrcations. This condition is precisely the one proposed in section 2. For the CNLS system, δ = on the cres (3.22). One may notice from formla (4.23) that also goes to infinity when I =. Howeer, I = does not correspond to a bondary of scalar nonlocal bifrcations. The reason is as follows. For the CNLS system, I = on the local bifrcation bondaries of pre -plses: ω = s n, (4.27) where n is an integer and n<s. This is becase on these bondaries, the soltion φ of eqation (4.2) which satisfies the zero bondary condition at ξ = (see eqation (4.22)) is always localized. Ths, in order for it to satisfy the bondary condition (4.22) at ξ =, φ mst be infinitely large. Hence, I =. The aboe fact applies to the soltion ũ I of eqation (4.4) as well: on the local bifrcation bondary (4.27), soltion ũ I satisfying bondary conditions (4.5) and (4.7) is infinitely large. When this happens, or original assmption ũ I for scalar nonlocal bifrcations breaks down. Hence, if there is a nonlocal bifrcation here at all, it wold not be scalar: the plses on the two wings wold be tre ector solitons. Ths, I = does not gie a bondary of scalar nonlocal bifrcations. Conseqently, a scalar nonlocal bifrcation bondary is gien entirely by the condition δ =, which is or preios condition. An interesting and sbtle isse is: what if cres δ = and I = intersect? As we hae discssed aboe, when I =, the nonlocal bifrcation (if there is one) becomes nonscalar. Ths at the locs of I = and δ =, the nonlocal bifrcation cold trn from scalar to nonscalar. Then the bifrcation bondary wold deiate from the scalar bifrcation cre δ = at the intersection. This phenomenon cold, and does, happen. In fact, figre 7 gies a good eample. Let s reprodce the scalar bifrcation cre of figre 7 (i.e. (3.22) with n = 3) and the tre bifrcation bondary in figre 8 (solid line and circles). On top of it, we plot the

18 282 A R Champneys and J Yang local bifrcation cre (4.27) of -plses with n = (dashed line). We see that the intersection between these two cres is precisely where bifrcation trns from scalar to nonscalar; ths, deiation between the nmerical bifrcation bondary and the scalar bifrcation bondary starts there. This eample and the matched asymptotic analysis aboe tell s that the preios condition for scalar nonlocal bifrcations is a necessary bt not sfficient condition. When local bifrcation cres of -plses intersect with these necessary-condition cres, nonscalar bifrcations cold start; ths, scalar bifrcations on part of the necessary-condition cres will not materialize. The analytical epressions for δ and I may be possible to obtain, as the linear Schrödinger eqations (4.2) and (4.25) can be soled sing hyper-geometric fnctions (see [7] and section 3). Bt sch epressions wold be ery comple. For practical prposes, it is preferable to determine them nmerically. For illstration prposes, we select ω =.6, and show the nmerical ales of these qantities in figre 9 at arios β-ales ranging from to7. In figre 9(a), the integral I is shown. This integral is independent of the symmetry of the soltion. In figres 9(b) and (c), the coefficients δ for antisymmetric and symmetric soltions, respectiely, are shown. This figre is helpfl in reealing on which side of the scalar nonlocal bifrcation bondary solitary waes can be epected. For instance, at ω =.6, δ = when β =.28 and the ()component is antisymmetric. On the left-hand side of this β-ale, δ> and I>; ths, solitary waes with symmetric -component and antisymmetric -component can be epected. Bt on the right-hand side of this β-ale, δ<and I>, therefore, no sch waes can be fond. These predictions completely agree with the nmerics shown in the preios section. Similar agreement is fond near other scalar nonlocal bifrcation cres as well. As ω ωn NLB (β), δ ; ths, according to formla (4.23). Below, we derie the asymptotic formla for when ω ωn NLB (β), or eqialently, β βn NLB (ω) (see eqation (3.27)). The latter limit will be adopted in the following deriation as it is a little more conenient. When β βn NLB (ω), the integral I approaches a finite constant ale I(ω,βn NLB ), while δ goes to zero. Obiosly, the asymptotic formla for crcially depends on the asymptotic formla of δ. We determine the asymptotic formla for δ by reglar pertrbation methods.8.6 ω.4.2 ω=/s β Figre 8. Intersection between the dashed cre ω = /s (see eqation (3.)) defining local bifrcations of daghter- soltions from -plses (with n = ) and the solid cre defining the necessary condition (3.22) for scalar nonlocal bifrcations (with n = 3). Circles show the nmerical reslts for nonlocal bifrcations reprodced from figre 7.

19 Bifrcation of solitary waes for CNLS systems 283 (a) integral I 2 ω =.6 (b) δ for antisymmetric () 5 5 ω= β β (c) δ for symmetric () 5 5 ω= β Figre 9. Parameter ales δ and I at ω =.6 and arios β-ales: (a) the integral I ales; (b) coefficient δ for antisymmetric -components; (c) δ for symmetric -components. below. Eqation (4.25) can be rewritten as ψ ω 2 ψ + βn NLB 2 ψ = ɛ2 ψ, (4.28) where ɛ = β βn NLB. When ɛ =, this eqation has an nbonded soltion: ˆ (), n = 2n, g n ψ n () = (4.29) ˆ 2 (), n = 2n 2 +, g n where fnctions ˆ,2 () and constant g n are gien by eqations (3.23), (3.24) and (3.26). The asymptotic behaior of this soltion is: (see eqation (3.25)). pertrbation series ψ n () e ω +o(e ω ), (4.3) When ɛ is small, the soltion ψ can be epanded into a reglar ψ(,β,ω) = ψ n (, βn NLB,ω)+ ɛψ n (, βn NLB,ω)+O(ɛ 2 ). (4.3) When this epansion is sbstitted into eqation (4.25), at order ɛ, we find that the fnction ψ n satisfies the eqation ψ n ω 2 ψ n + β NLB n 2 ψ n = 2 ψ n. (4.32)

20 284 A R Champneys and J Yang Recalling that the bondary conditions for ψ() and ψ n () are gien by eqations (4.26) and (4.3), the bondary condition for the fnction ψ n is ψ n () δ ɛ eω,. (4.33) The linear operator on the left-hand side of eqation (4.32) is self-adjoint. In addition, ψ n () is a homogeneos soltion. Calclating the inner prodct between ψ n and the inhomogeneos term of eqation (4.32), we readily find that 2 ψ 2 n d = (ψ nψ n ψ n ψ n ). (4.34) Sbstitting the bondary conditions (4.3) and (4.33) into the aboe relation, we find that for both ψ() symmetric and antisymmetric, the constant δ is gien by the asymptotic formla δ = β βnlb n 4ω 2 ψ n 2 d +O(( β βn NLB ) 2 ). (4.35) We hae compared this formla with the nmerical ales of δ as displayed in figres 9(b) and (c). The slope 2 ψ n 2 d/4ω predicted by this formla is in ecellent agreement with the nmerical slope at β = βn NLB. When this formla is sbstitted into eqation (4.23), we finally obtain the leading two-term asymptotic epansion for the spacing fnction as { ( ) ( )} = ln β β NLB n +lnk +O β β NLB n, (4.36) 2(ω ) where the constant K is K = K n (ω) = ω 3 βn NLB I(ω,β NLB n ) 2 ψ n 2 d. (4.37) Net, we make qantitatie comparisons between the spacing formla (4.23), its leading twoterm epansion (4.36) and nmerics near the scalar nonlocal bifrcation bondaries (3.22) with n = and2atω =.6 and arios β-ales (see figre 2). We remind the reader that ω = s is a bondary for antisymmetric -components, and ω = 2 s is a bondary for symmetric -components. At ω =.6, the bondary point is at β NLB =.28 for the former case, and is at β2 NLB =.68 for the latter case. The analytical spacings from formla (4.23) and its two-term asymptotic epansion (4.36) are shown as dashed and dash-dotted lines in figres (a) and (b) for these two cases, respectiely. We hae also nmerically determined the spacings between the -plses and the central -plse in the eact solitary waes. These nmerical ales are shown as solid lines in figres (a) and (b). We see that, when the separation is large, the formla (4.23) and its asymptotic form (4.36) agree with the nmerical ales perfectly. 5. Eample 2: satrable nonlinearity As a second eample we take the CNLS system stdied by Ostroskaya and Kishar []: U 2 + V 2 iu t + U + U =, (5.) +s( U 2 + V 2 ) U 2 + V 2 iv t + V + V =. (5.2) +s( U 2 + V 2 ) This dimensionless model arises after scaling of the model for the incoherent interaction between two linearly polarized optical beams in a biased photorefractie medim. Here, s ( < s < ) is an effectie satration parameter, representing the photorefractie effects.

21 Bifrcation of solitary waes for CNLS systems 285 (a) for antisymmetric ω=.6, n= solid: nmerical dashed: formla (4.23) dash dotted: formla (4.36) β (b) for symmetric ω=.6, n=2 solid: nmerical dashed: formla (4.23) dash dotted: formla (4.36) β Figre. Comparison between the analytical formla (4.23), its leading two-term asymptotic epansion (4.36) and nmerical ales for spacing at ω =.6 and arios β-ales: (a) comparison near the scalar nonlocal bifrcation bondary (3.22) with n = ; (b) comparison near the scalar nonlocal bifrcation bondary (3.22) with n = 2. This system is significant becase it was shown in [] that mlti-hmped stationary plses may be stable soltions, a reslt that eplains the eperimental obserations of [9]. In the limit s, this system redces to the Manako eqations. The solitons in this system are of the form U(,t) = e it (), V (, t) = e iω2t (), (5.3) where and are real fnctions satisfying the following ODEs: =, (5.4) +s( ) ω =. (5.5) +s( ) Looking for single-component plses ( =, or = ) we obtain simple planar eqations, which can be shown by phase plane techniqes to possess symmetric homoclinic orbits ± h, ± h. Unlike the preios eample, we know of no closed-form epressions for these soltions other than at s =. So we trn straight away to nmerical methods. Once again we restrict to <ω<and look for local and scalar nonlocal bifrcations from the plse h. Figre shows the analoge of figre 2 for this eample where the bifrcation bondaries were obtained by nmerically imposing the bondary conditions for local bifrcation and for scalar nonlocal bifrcation on the linearized eqation ω h () +s 2 =. (5.6) h () Figre 2 depicts the corresponding graphs of w (s) and w 2 (s) for fied ω =.5, where w and w 2 were defined in (3.32). Note that we hae qalitatiely the same strctre as in figre 3 in that the mode shapes () gain increasingly many internal zeros as the parameter increases to some limit. Moreoer, between each pair of local bifrcations of a gien symmetry type there is a scalar nonlocal bifrcation, and ice ersa. Here, it wold seem that the limit s plays the same qalitatie role as β in the preios model, and we conjectre that there are infinitely many local and scalar nonlocal bifrcations as we approach this limit. This is borne ot by the reslts in figre which show that the cres in the (s, ω)-plane defined by

22 286 A R Champneys and J Yang (a) (b) ω ω s. s Figre. Cres of local (- ---)andscalar nonlocal ( ) bifrcations from the pre -plse h of (5.4), (5.5) obtained by nmerically soling the linearized eqation (5.6) and reqiring the soltion to hae prely decaying or growing asymptotics at infinity: (a) the -component is een; (b) the -component is odd. Only the first three cres of each type are presented. zeros of w = and w 2 = become increasingly steep as s. It wold be fair to conjectre from the nmerics that all cres (apart from the first symmetric scalar nonlocal bifrcation) approach the point (ω, s) = (, ). There are some nmerical difficlties in compting the nonlocal bifrcation cres p to this point, becase the plse h becomes infinitely broad as s, and hence the soltion () of the linearized problem which grows eponentially at infinity mst be continally rescaled to aoid the soltion becoming arbitrarily large. Now the presence of local bifrcations has been fond before in this system; see figre in [] where local bifrcations are fond ia zeros of a certain integral as a fnction of ω for seeral different ales of s. Here we hae presented a simple procedre for atomatically detecting these local bifrcations as a fnction of all system parameters. As far as we know these cres of local bifrcations are not gien by closed-form analytic epressions as they were in the preios eample. The qestion remains whether the cres which hae been ptatiely called scalar nonlocal bifrcations really represent that for the fll eqations. Figre 3 depicts nmerical soltions of the flly copled nonlinear eqations for fied ω =.5 in a parameter region between the second local and first scalar nonlocal symmetric bifrcations. A one-parameter family