A combined variational-topological approach for dispersion-managed solitons in optical fibers

Transcription

1 Z. Angew. Mah. Phy. c 2 Springer Bael AG DOI.7/ Zeichrif für angewande Mahemaik und Phyik ZAMP A combined variaional-opological approach for diperion-managed olion in opical fiber Rober Hakl and Pedro. Torre Abrac. We derive ome ufficien condiion for he exience of diperion-managed olion in a nonlinear Schrödinger equaion wih periodically varying coefficien. The proof relie on a combinaion of he variaional mehod wih he developmen of a novel upper and lower funcion mehod. Mahemaic Subjec Claificaion (2). 35Q55 34C25 78A6. Keyword. Opical olion Schrödinger equaion Singular equaion Periodic oluion Upper and lower funcion.. Inroducion An opical olion can be decribed a an elecromagneic wave which i localized in ime and can propagae along an opical medium wihou ignifican diorion of i hape [3, 6, 4]. In a nonlinear opical medium, hi phyical effec i achieved by mean of a uiable balance beween he chromaic diperion and he nonlinear refracive repone. From a pracical poin of view, he concep of olion i crucial o implemen efficien opical fiber communicaion yem. From a mahemaical poin of view, he propagaion of an opical pule in a fiber cable wih varying diperion i governed by he equaion iψ z 2 β 2(z)Ψ + σ(z) Ψ 2 Ψ=iG(z)Ψ, where Ψ i he complex-valued envelope funcion of he elecric field, z i he longiudinal coordinae of he fiber line and i ime. The funcion β 2,σ,G model repecively he diperion, nonlinear refracive repone and effecive gain or lo along he fiber line. In hi paper, i i aumed ha he opical fiber ha a periodic rucure o ha he coefficien are periodic wih a common minimal period which i normalized o L =. A can be een in he cied reference, hi eem o be he mo inereing cae for pracical purpoe. I i cuomary o remove he righ-hand ide erm of he laer equaion by he ranformaion z Ψ(z,) =A(z,)exp G()d. Then, he equaion under udy i he cubic Schrödinger equaion wih periodic coefficien ia z +d(z)a + c(z) A 2 A =. (.) Now he gain lo power erm i included in he coefficien c. To find olion-like oluion of Eq. (.) i a cenral problem no only in Nonlinear Opic bu alo for a variey of phyical and biological applicaion.

2 R.HaklandP..Torre ZAMP A well-known mehod for he analyical udy of Eq. (.) i he variaional approach decribed in full deail in [3] (ee alo he complee li of reference herein). Equaion (.) i rewrien in he Lagrangian form, wih he acion funcional [ i S = Lddz = ddz 2 (AA z A A z )d(z) A 2 c(z) ] 2 A 4. The following rial funcion i choen A(z,) = Q(/T (z)) T (z) ( exp i M(z) ) T (z) 2 (.2) where he hape of he inpu pule Q i in principle arbirary, being he mo ypical choice a gauian Q(x) =C exp( x 2 /2). Inering hi anaz ino he acion funcional, one obain he yem of ordinary differenial equaion T =4d(z)M M = d(z)c T 3 c(z)c 2 T 2, (.3) wih fixed conan C = Q (x) 2 dx Q(x) 4 dx x2 Q(x) 2 dx, C 2 = 4 x 2 Q(x) 2 dx. (.4) The funcion T (z)andm(z) decribe he opical pule widh and he chirp (ime-dependen phae) of he breahing cenral par of he opical olion. The dynamic of yem (.3), ofen known a TM-equaion in he relaed lieraure, i of key imporance on hi field. Then he problem i reduced o find condiion for he exience of -periodic oluion of yem (.3), ha i, T,M verifying T () = T (),M() = M(). Alhough he variaional approach i approximae, i i recognized a an effecive heoreical mehod o gain inigh on he dynamic of he yem. A hi poin, all he heoreical reul preened in he lieraure aume ha boh coefficien c, d are piecewie conan. Thi aumpion make poible o apply a maching echnique for he repecive phae plane in order o find explici exience condiion. To have piecewie coninuou coefficien i a coheren aumpion in he framework of Nonlinear Opic, bu he main problem i ha for a large number of piece compuaion become oo hard o handle wih. The paper [7] olve explicily he cae of c conan (he o-called lole cae) and d compoed by wo piece, bu for more han wo piece only numerical reul are known []. Our approach i of a differen naure. We propoe he ue of a claical approach like he upper and lower funcion mehod []. Thi echnique i very known in he qualiaive analyi of econd order ODE and ha been applied o equaion wih ingulariie in he recen paper [2]. We ake advanage of he echnique developed here o open a new pah in he udy of DM-olion in opical fiber. The main echnical difficuly i ha in he general cae wih arbirary coefficien, he yem (.3) can no be wrien a a econd-order ODE (a in fac i i done in he paricular cae conidered in [7]). Thi ha forced u o develop a pecific upper and lower funcion mehod for hi framework. I will be hown ha uch mehod i he naural exenion o he fir-order yem of known reul for he econd order calar ODE (ee, e.g., [,2] and reference herein), o in hi ene from a mahemaical poin of view i i inereing by ielf. The rucure of he paper will be a follow. In Sec. 2, we develop a new mehod of upper and lower funcion in cononance wih he problem under conideraion. In Sec. 3, uch a mehod i properly applied o a family of yem which include he TM-equaion.

3 2. The mehod of upper and lower funcion 2.. Baic noaion and definiion A combined variaional-opological approach The following noaion i ued hroughou: R i a e of all real number, R + =[, + [; C ([,ω]; R) i a Banach pace of all coninuou funcion u :[,ω] R wih he norm u C = max { u() : [,ω]}; C ( [,ω]; R 2) i a Banach pace of all coninuou vecor-valued funcion (u, v) :[,ω] R 2 wih he norm (u, v) C = u C + v C ; AC ( [,ω]; R 2) i a e of all vecor-valued funcion (u, v) :[,ω] R 2 wih aboluely coninuou componen; L ([,ω]; R) i a Banach pace of all Lebegue inegrable funcion p :[,ω] R wih he norm p L = p() d; L ([,ω]; R + )={p L ([,ω]; R) :p() for a. e. [,ω]}; L ( [,ω]; R 2) i a Banach pace of all vecor-valued Lebegue inegrable funcion (p, q) :[,ω] R 2 wih he norm (p, q) L = p L + q L ; K ([,ω] D; R), where D R, i he Carahéodory cla, i.e., he e of funcion f :[,ω] D R uch ha f(,x):[,ω] R i meaurable for all x D, f(, ) :D R i coninuou for almo all [,ω], and up { f(,x) : x D } L ([,ω]; R + ) for any compac e D D. if x R hen [x] + = max {,x},[x] = max {, x}; for every a [,ω[ and b ],ω] uch ha a b define { ] a, b [ if a<b I(a, b) =, I[a, b) =I(a, b) {a}. [,b[ ] a, ω] if b<a In he developmen of he general mehod of upper and lower funcion we will adop he more claical ue of a he independen variable. Conider a yem of wo differenial equaion wih periodic boundary condiion u = p()v, (2.) v = f(, u) (2.2) u() = u(ω), v() = v(ω), (2.3) where p L ([,ω]; R) andf K ([,ω] D; R). By a oluion o (2., 2.2) i underood a vecor-valued funcion (u, v) AC ( [,ω]; R 2) wih u() D for [,ω] aifying (2., 2.2) almo everywhere on [,ω]. By a oluion o he problem (2.) (2.3) i underood a oluion o (2., 2.2) aifying (2.3). The queion of he exience of a periodic oluion o he wo-dimenional yem of he ype (2., 2.2) wa udied by Kiguradze and Mukhigulahvili in [5]. However, he reul obained by hem are no applicable o our yem becaue he funcion p i aumed o be ign-conan in heir paper. The reul dealing wih general nonlinear wo-dimenional yem one can find, e.g., in [4]. In [8, 9] one can find condiion guaraneeing he exience of a periodic oluion o he n-dimenional linear yem of boh ordinary and funcional differenial equaion.

4 R.HaklandP..Torre ZAMP Definiion 2.. A vecor-valued funcion (γ,γ 2 ) AC ( [,ω]; R 2) i aid o be an upper funcion (rep. a lower funcion) o he problem (2.) (2.3) ifγ () D for [,ω], γ () =p()γ 2 () for a. e. [,ω], γ 2() f(, γ ()) (rep. γ 2() f(, γ ())) for a. e. [,ω], and he boundary condiion γ () = γ (ω), γ 2 () γ 2 (ω) (rep. γ 2 () γ 2 (ω)) hold. Definiion 2.2. A vecor-valued funcion (p,p 2 ) L ( [,ω]; R 2) i aid o verify a propery (P + )ifevery vecor-valued funcion (u, v) AC ( [,ω]; R 2) aifying admi he inequaliy u () =p ()v() for a. e. [,ω], (2.4) v () p 2 ()u() for a. e. [,ω], (2.5) u() = u(ω), v() v(ω) (2.6) u() for [,ω]. (2.7) Definiion 2.3. A vecor-valued funcion (p,p 2 ) L ( [,ω]; R 2) i aid o verify a propery (P )ifevery vecor-valued funcion (u, v) AC ( [,ω]; R 2) aifying (2.4) (2.6) admi he inequaliy u() for [,ω]. (2.8) Remark 2.. Noe ha (p,p 2 ) verifie a propery (P + )iff( p, p 2 ) verifie a propery (P ). Indeed, le (u, v) AC ( [,ω]; R 2) aify (2.4) (2.6). Pu w() = u() for [,ω]. Then (w, v) AC ( [,ω]; R 2) aifie w () = p ()v() for a. e. [,ω], (2.9) v () p 2 ()w() for a. e. [,ω], (2.) w() = w(ω), v() v(ω), (2.) and vice vera, if (w, v) AC ( [,ω]; R 2) aifie (2.9) (2.) hen, having defined u() = w() for [,ω] we obain ha (u, v) AC ( [,ω]; R 2) aifie (2.4) (2.6) On he properie (P + ) and (P ) Theorem 2.. Le p L ([,ω]; R) and le ϕ L ([,ω]; R + ) aify If eiher he inequaliie. (2.2) [p()] + d<, (2.3) [p()] +d [p()] +d [p()] d (2.4)

5 hold or he inequaliie A combined variaional-opological approach [p()] d [p()] d [p()] d<, (2.5) [p()] + d 2+2 [p()] + d (2.6) [p()] d, (2.7) are fulfilled hen a vecor-valued funcion (p, ϕ) verifie a propery (P + ). Theorem 2.2. Le p L ([,ω]; R) and le ϕ L ([,ω]; R + ) aify (2.2). If eiher he inequaliie hold or he inequaliie [p()] d<, [p()] d [p()] d [p()] + d<, [p()] +d [p()] +d [p()] d [p()] d 2+2 [p()] + d are fulfilled hen a vecor-valued funcion (p, ϕ) verifie a propery (P ). [p()] + d, 2.3. Exience heorem Theorem 2.3. Le (β,β 2 ) AC ( [,ω]; R 2) and (α,α 2 ) AC ( [,ω]; R 2) be an upper and a lower funcion o (2.) (2.3), repecively, wih β () α () for [,ω]. (2.8) Le, moreover, here exi ϕ L ([,ω]; R + ) uch ha f(, β ()) + ϕ()β () f(, x)+ϕ()x f(, α ()) + ϕ()α () for a. e. [,ω], β () x α () (2.9) and a vecor-valued funcion (p, ϕ) verifie a propery (P + ). If, in addiion, p()d, (2.2)

6 R.HaklandP..Torre ZAMP hen he problem (2.) (2.3) ha a lea one oluion (u, v) uch ha β () u() α () for [,ω]. Theorem 2.4. Le (β,β 2 ) AC ( [,ω]; R 2) and (α,α 2 ) AC ( [,ω]; R 2) be an upper and a lower funcion o (2.) (2.3), repecively, wih α () β () for [,ω]. (2.2) Le, moreover, here exi ϕ L ([,ω]; R + ) uch ha f(, β ()) ϕ()β () f(, x) ϕ()x f(, α ()) ϕ()α () for a. e. [,ω], α () x β () (2.22) and a vecor-valued funcion (p, ϕ) verifie a propery (P ). If, in addiion, (2.2) hold hen he problem (2.) (2.3) ha a lea one oluion (u, v) uch ha α () u() β () for [,ω]. Remark 2.2. Noe ha here exi ϕ aifying (2.9), rep. (2.22), e.g. if here exi a parial derivaive f x which belong o K ([,ω] D; R). Then we can pu { } f ϕ() =up (, x) x : γ () x γ 2 () for a. e. [,ω], where γ = min {α,β }, γ 2 = max {α,β } Auxiliary propoiion Lemma 2.. Le (u, v) AC ( [,ω]; R 2) aify u () =p()v() for a. e. [,ω], (2.23) v () ϕ()u() for a. e. [,ω], (2.24) u() = u(ω), v() v(ω) (2.25) wih p L ([,ω]; R),ϕ L ([,ω]; R + ). Le, moreover, u aume boh poiive and negaive value. If (2.3) hold hen v doe no vanih. Proof. Fir we will how ha here exi [,ω] uch ha u( )=, v( ). (2.26) Aume on he conrary ha if u ha a zero a ome poin, hen v ha a zero a he ame poin. Obviouly, according o he aumpion of he lemma, here exi [,ω[, 2 ],ω], 2 uch ha u( )=, u( 2 )=, v( )=, v( 2 )=, (2.27) u() < for I(, 2 ), (2.28) Then, (2.24) inviewof(2.28) yield v () for a. e. I(, 2 ), which ogeher wih (2.25) and (2.27) reul in v() = for I(, 2 ). (2.29) However, (2.23) and (2.29) yield u () = for a. e. I(, 2 ), which ogeher wih (2.27) implie u() =for I(, 2 ). The la equaliy conradic (2.28). Le, herefore, [,ω] be uch ha (2.26) hold. Obviouly, eiher v( ) > (2.3)

7 A combined variaional-opological approach or v( ) <. (2.3) We will how ha v ha no zero. Aume on he conrary ha v ha a zero in he inerval [,ω]. If (2.3) i aified hen in view of (2.25) we can aume wihou lo of generaliy ha ω and, furhermore, here exi ],ω], uch ha v( )=, v() > for I[, ). (2.32) The inegraion of (2.23) overi(,)inviewof(2.25, 2.26), and (2.32) yield u() = p()v()d [p()] + v()d for I(, ). (2.33) I(,) I(,) The inegraion of (2.24) overi(, )inviewof(2.25) and (2.32), reul in v() ϕ()u()d for I[, ). (2.34) Uing (2.33) in(2.34) we obain v() ϕ() I(, ) [p(ξ)] + v(ξ)dξd for I[, ). (2.35) I(, ) I(,) Now le 2 I[, ) be uch ha v( 2 ) = max {v() : I[, )}. (2.36) Then from (2.35) on accoun of (2.32) and (2.36) wege <v( 2 ) v( 2 ) ϕ() [p(ξ)] + dξd v( 2 ) [p()] + d. (2.37) I( 2, ) I(,) However, by uing (2.3) i follow ha v( 2 ) <v( 2 ), a conradicion. Now aume ha (2.3) hold. Pu u() =u(ω ) for [,ω], v() = v(ω ) for [,ω], (2.38) p() =p(ω ) for a. e. [,ω], ϕ() =ϕ(ω ) for a. e. [,ω]. (2.39) Then i can be eaily verified ha (u, v) AC ( [,ω]; R 2) aify and u () =p()v() for a. e. [,ω], (2.4) v () ϕ()u() for a. e. [,ω], (2.4) u() = u(ω), v() v(ω), (2.42) [p()] + d = [p()] + d, Moreover, in view of(2.26, 2.3), and (2.38) we have = u(ω )=, v(ω ) >, and hu he lemma follow from he above-proven.. (2.43) Lemma 2.2. Le (u, v) AC ( [,ω]; R 2) aify (2.23) (2.25) wih p L ([,ω]; R),ϕ L ([,ω]; R + ). Le, moreover, u ake boh poiive and negaive value. If (2.5) and (2.7) hold hen v doe no vanih.

8 R.HaklandP..Torre ZAMP Proof. Aume on he conrary ha v ha a zero. Pu M v = max {v() : [,ω]}, m v = max { v() : [,ω]}, (2.44) M u = max {u() : [,ω]}, m u = max { u() : [,ω]}. (2.45) According o our aumpion, M u >, m u >. (2.46) Moreover, if v hen from (2.23) i follow ha u i a conan funcion, which in view of (2.45) yield M u = m u. However, he laer equaliy conradic (2.46). Therefore, we have M v, m v, M v + m v >. (2.47) Chooe [,ω[, ],ω] uch ha v( )=M v, v( )= m v. (2.48) Obviouly,, and he inegraion of (2.24) overi(, ) on accoun of (2.25, 2.45), and (2.48) yield M v + m v I(, ) ϕ()u()d M u. (2.49) Now chooe 2 [,ω[, 3 ],ω] uch ha u( 2 )= m u, u( 3 )=M u. (2.5) Obviouly, 2 3, and he inegraion of (2.23) overi( 2, 3 ) and over def = [,ω]\ I( 2, 3 ), repecively, in view of (2.25, 2.44), and (2.5), reul in M u + m u = p()v()d M v [p()] + d + m v [p()] d, (2.5) rep. I( 2, 3) I( 2, 3) I( 2, 3) m u M u = p()v()d m v [p()] + d M v [p()] d. (2.52) Nowfrom(2.5) and (2.52), wih repec o (2.46), we obain M u <M v [p()] + d + m v and I( 2, 3) Noe ha from (2.47) and (2.49) i follow ha I( 2, 3) [p()] d, (2.53) M u <m v [p()] + d + M v [p()] d. (2.54) >. (2.55) Thu if we muliply boh ide of (2.53), rep. (2.54), by, on accoun of (2.55) wege M u <M v [p()] + d + m v [p()] d, (2.56) I( 2, 3) I( 2, 3)

9 rep. M u <m v A combined variaional-opological approach I( 2, 3) [p()] + d + M v Now (2.49, 2.56), and (2.57) reul in m v [p()] d <M v M v [p()] d <m v I( 2, 3) [p()] d. (2.57) [p()] + d, (2.58) [p()] + d. (2.59) Noe ha (2.58) and (2.59) inviewof(2.5) and (2.47) yield M v >,m v >. Therefore, muliplying he correponding ide of (2.58) and (2.59) we obain [p()] d [p()] d Noe ha < I( 2, 3) I( 2, 3) I( 2, 3) [p()] d [p()] + d [p()] d and, in view of he inequaliy AB 4 (A + B)2, [p()] + d [p()] + d 4 I( 2, 3) Therefore, uing (2.6) and (2.62) in(2.6) we obain [p()] d< 4 Noe alo, ha from (2.58) and (2.59) i follow ha Thu from (2.63), in view of (2.5) and (2.64) wege ω 2+2 [p()] d< which conradic (2.7). [p()] + d. (2.6) [p()] d (2.6) 2 [p()] + d 2. (2.62) 2 [p()] + d 2. (2.63) [p()] + d>2. (2.64) [p()] + d,

10 R.HaklandP..Torre ZAMP Lemma 2.3. Le (u, v) AC ( [,ω]; R 2) aify (2.23) (2.25) wih p L ([,ω]; R),ϕ L ([,ω]; R + ),and le (2.2) hold. Le, moreover, u aume boh poiive and negaive value. If eiher (2.3) and (2.4) hold or (2.5) and (2.6) are fulfilled, hen v ha a zero. Proof. We will prove he lemma in he cae when (2.3) and (2.4) are fulfilled. The cae when (2.5) and (2.6) are aified can be proven analogouly. Le, herefore, (2.3) and (2.4) hold and aume o he conrary ha v ha no zero. Fir aume ha v i a poiive funcion. Pu M v = max {v() : [,ω]}, m v = min {v() : [,ω]}, (2.65) and define number M u and m u by (2.45). Then, according o our aumpion, we have (2.46) and Chooe, 2 [,ω[,, 3 ],ω] uch ha (2.5) hold and M v >, m v >. (2.66) v( )=M v, v( )=m v. (2.67) Now he inegraion of (2.24) overi(, ), on accoun of (2.25, 2.45), and (2.67), yield M v m v I(, ) ϕ()u()d M u. (2.68) Furher, he inegraion of (2.23) overi( 2, 3 ), in view of (2.25, 2.5, 2.65), and (2.66), reul in M u + m u = I( 2, 3) p()v()d M v From (2.69), wih repec o (2.46) and (2.2), we obain Now (2.68) and (2.7) imply M u <M v M v <m v + M v [p()] + d. (2.69) [p()] + d. (2.7) [p()] + d. (2.7) On he oher hand, he inegraion of (2.23) from o ω, in view of(2.25) and (2.65), yield i.e., = p()v()d M v m v [p()] d M v [p()] + d m v [p()] d, [p()] + d. (2.72) Noe ha having aumed u no o be a conan funcion we find p. Therefore, from (2.4) i follow ha

11 A combined variaional-opological approach [p()] d>. (2.73) Thu if we muliply boh ide of (2.7) by [p()] d, hen, in view of (2.66, 2.72), and (2.73) we obain [p()] d< [p()] + d + [p()] + d [p()] d. (2.74) However, (2.74) conradic (2.4). Now aume ha v i a negaive funcion. Define u, v, p, andϕ by (2.38) and (2.39). Then i can be eaily verified ha (u, v) AC ( [,ω]; R 2) aifie (2.4) (2.42). Furhermore, (2.43) i fulfilled and, in addiion, alo [p()] d = [p()] d. Moreover, v i a poiive funcion, and hu he lemma follow from he above-proven. Lemma 2.4. Le a vecor-valued funcion (p, ϕ) verify he propery (P + ) wih p L ([,ω]; R),ϕ L ([,ω]; R + ). Le, moreover, (2.2) hold. Then he problem u () =p()v()+h() for a. e. [,ω], (2.75) v () = ϕ()u()+q() for a. e. [,ω], (2.76) u() = u(ω), v() = v(ω) (2.77) ha a unique oluion (u, v) for each (h, q) L ( [,ω]; R 2). Moreover, here exi ρ > independen on h and q uch ha he eimae (u, v) C ρ (h, q) L (2.78) hold. Proof. According o he well-known Fredholm alernaive principle, i i ufficien o how ha he correponding homogeneou yem u () =p()v() for a. e. [,ω], (2.79) v () = ϕ()u() for a. e. [,ω], (2.8) wih boundary condiion (2.77) ha only he rivial oluion. Le, herefore, (u, v) AC ( [,ω]; R 2) aify (2.77, 2.79), and (2.8). By Definiion 2.2, u i non-negaive, which ogeher wih (2.8) reul in v () for a. e. [,ω]. Then, he periodiciy condiion (2.77) implie ha v i a conan funcion. Having in mind hi fac, he inegraion of (2.79) fromoω, inviewof(2.77) give =v() p()d. (2.8) Now (2.8) on accoun of (2.2) reul in v() = for [,ω]. (2.82) Uing (2.82) in(2.79) we ge ha u i a conan funcion. Therefore, he inegraion of (2.8) fromo ω in view of (2.77) yield =u(). (2.83)

12 R.HaklandP..Torre ZAMP By (2.2), we have u() = for [,ω]. (2.84) Thu (2.82) and (2.84) enure ha he only oluion o (2.77, 2.79, 2.8) i he rivial one. Now we will how he eimae (2.78). Le Ω : L ( [,ω]; R 2) C ( [,ω]; R 2) be he operaor aigning o every (h, q) L ( [,ω]; R 2) he unique oluion (u, v) of(2.75) (2.77). Thu, Ω i a linear coninuou operaor. Then, (2.78) obviouly hold wih ρ =up{ Ω(x, y) C : (x, y) L =} Proof of main reul Proof of Theorem 2.. Le (u, v) AC ( [,ω]; R 2) aify (2.23) (2.25). According o Definiion 2.2 i i ufficien o how ha (2.7) hold. Aume on he conrary ha here exi [,ω] uch ha u( ) <. (2.85) Fir noe ha if p hen from (2.23) i follow ha u i a conan funcion. Conequenly, he inegraion of (2.24) fromoω in view of (2.25) yield u( ) (2.86) which ogeher wih (2.2) conradic (2.85). Therefore, in wha follow we can aume ha p. (2.87) According o Lemma 2., 2.2, and2.3 i follow ha u doe no aume poiive value. Therefore, u() for [,ω]. (2.88) However, from (2.24, 2.25), and (2.88) i follow ha v i a conan funcion. Thu he inegraion of (2.23) fromoω, on accoun of (2.25) reul in =v() p()d. (2.89) On he oher hand, from (2.2, 2.87) and (2.4), rep. (2.6), i follow ha p()d. Therefore, from (2.89) we obain v. Thu from (2.23) we ge ha u i a conan funcion which i, according o (2.85), negaive. Now he inegraion of (2.24) fromoω, inviewof(2.25), reul in (2.86). However, (2.86) inviewof(2.2) conradic (2.85). Proof of Theorem 2.2. I immediaely follow from Theorem 2. and Remark 2.. Proof of Theorem 2.3. Le ρ be a number appearing in Lemma 2.4. Pu ρ() =up{ f(, x) : β () x α ()} for a. e. [,ω] (2.9) and ρ = ρ ( ϕ L (α,β ) C + ρ L ). (2.9) Le, moreover, U = {y C ([,ω]; R) :β () y() α () for [,ω]}, (2.92) U 2 = {z C ([,ω]; R) : z C ρ }, (2.93)

13 A combined variaional-opological approach and le Ω : U U 2 C ( [,ω]; R 2) be an operaor which o every (y, z) aign he unique oluion of u = p()v (2.94) v = ϕ()u + ϕ()y + f(, y), (2.95) u() = u(ω), v() = v(ω). (2.96) According o Lemma 2.4, he operaor Ω i defined correcly. Alo noe ha Ω(y, z) =Ω(y, ) for any (y, z) U U 2. Furhermore, he operaor Ω i coninuou. We will how ha Ω ranform U U 2 ino ielf. According o Lemma 2.4 we have Ω(y, z) C = (u, v) C ρ (,ϕy+ f(,y)) L. (2.97) From (2.97), in view of (2.9) (2.92) and he incluion y U, we obain v C (u, v) C ρ. (2.98) Conequenly, v U 2. On he oher hand, in view of (2.9, 2.94) (2.96), u () β () =p()(v() β 2 ()) for a. e. [,ω], (2.99) v () β 2() ϕ()(u() β ()) for a. e. [,ω], (2.) u() β () = u(ω) β (ω), v() β 2 () v(ω) β 2 (ω). (2.) However, a pair (p, ϕ) verifie a propery (P + ), and herefore from (2.99) (2.) wege u() β () for [,ω]. (2.2) Analogouly, we find u() α () for [,ω]. (2.3) Now (2.2) and (2.3) imply u U. Thu we have hown ha Ω ranform a e U U 2 ino ielf. Furhermore, on accoun of (2.9, 2.98, 2.2, 2.3), and he incluion y U,from(2.94) and (2.95) i follow u () ρ p() for a. e. [,ω], v () 2 ϕ() (α,β ) C + ρ() for a. e. [,ω]. Conequenly, Ω ranform U U 2 ino i relaively compac ube. According o Schauder fixed poin heorem, here exi (u,v ) U U 2 uch ha (u,v )=Ω(u,v ). (2.4) However, in view of (2.94) (2.96), from (2.4) i follow ha (u,v ) i a oluion o (2.) (2.3). Proof of Theorem 2.4. I immediaely follow from Theorem 2.3 and Remark Applicaion o he opical problem In hi ecion we will conider he yem ogeher wih he boundary condiion u = p()v, (3.) v = h() u 3 + g() u 2, (3.2) u() = u(ω), v() = v(ω). (3.3) Here p, h, g L ([,ω]; R). We eablih condiion for he exience of a oluion (u, v) o(3.) (3.3) wih u() > for [,ω].

14 R.HaklandP..Torre ZAMP For he ake of breviy we will ue he following noaion: H + = G + = [h()] + d, H = [g()] + d, G = The main reul of hi ecion are he following one. Theorem 3.. Le H + > ( 9 3 8) H,G >G +,andle ( p() d) 2 4 p()d [h()] d, [g()] d. p()d, (3.4) (G H + 98 ) G +H 8 (H 8 + ( ) ) H 9 9 (G ( 8 9 ) 2 G+ ) 3. (3.5) Then he problem (3.) (3.3) ha a lea one oluion (u, v) wih u() > for [,ω]. Theorem 3.2. Le (3.4) hold, H > ( 9 3 8) H+,G + >G,andle ( p() d) 2 4 p()d (G + H 98 ) 8 (H 8 ( ) ) G 8 H+ H + 9 (G 9 + ( ) ) (3.6) 9 G Then he problem (3.) (3.3) ha a lea one oluion (u, v) wih u() > for [,ω]. Such reul are direcly applicable o he TM equaion (.3), giving ome inereing corollarie. Corollary 3.. Le u conider he anaz (.2) wih a general inpu pule profile Q(x). Aume ha c, d L ([, ]; R) verify [d()] + d> ( ) 3 9 [d()] d, 8 [c()] + d> [c()] d (3.7) Then, here exi a conan K K(d, C,C 2 )(where C,C 2 are defined by (.4)), uch ha he TMequaion (.3) have a -periodic oluion provided ha [c()] + d<k(d, C,C 2 ). Corollary 3.2. Le u conider he anaz (.2) wih a gauian inpu pule profile Q(x) =C exp( x 2 /2). Aume ha c, d L ([, ]; R) verify (3.7). Then, here exi a conan H H(c, d) uch ha he TM-equaion (.3) have a -periodic oluion provided ha C <H(c, d). Boh corollarie are direc applicaion of Theorem 3., aking ino accoun in he econd corollary ha for a gauian profile Q(x) =C exp( x 2 /2), we can compue C =,C 2 = C2 2. Of coure, explici 2 expreion of conan K, H are eaily derived. We omi furher deail. The following wo corollarie can be direcly derived from Theorem 3.2.

15 A combined variaional-opological approach Corollary 3.3. Le u conider he anaz (.2) wih a general inpu pule profile Q(x). Aume ha c, d L ([, ]; R) verify [d()] d> ( ) 3 9 [d()] + d, 8 [c()] d> [c()] + d (3.8) Then, here exi a conan K K(d, C,C 2 )(where C,C 2 are defined by (.4)), uch ha he TMequaion (.3) have a -periodic oluion provided ha [c()] d<k(d, C,C 2 ). Corollary 3.4. Le u conider he anaz (.2) wih a gauian inpu pule profile Q(x) =C exp( x 2 /2). Aume ha c, d L ([, ]; R) verify (3.8). Then, here exi a conan H H(c, d) uch ha he TM-equaion (.3) have a -periodic oluion provided ha C <H(c, d). The proof of Theorem 3. and 3.2 require he conrucion of adequae upper and lower funcion. For breviy, we will only give he complee proof of Theorem 3., ince Theorem 3.2 can be proven analogouly. Some lemma are needed. Lemma 3.. Le p, q L ([,ω]; R) be uch ha Then every oluion o he problem i given by u() =c v() = + p()d p()d p()d q() q() p()d, q()d =. (3.9) u = p()v, (3.) v = q(), (3.) u() = u(ω), v() = v(ω) (3.2) p()d q() p(ξ)dξd p(ξ)dξd for [,ω], (3.3) p(ξ)dξd q() p(ξ)dξd for [,ω], (3.4) where c R. Proof. If u and v are given by (3.3) and (3.4) hen, obviouly, hey aify (3.) and (3.), and in view of (3.9), alo (3.2) i fulfilled. Le (u, v) AC ( [,ω]; R 2) aify (3.) (3.2). Then he inegraion of (3.) fromoand from o ω, repecively, give u() =u() + p()v()d (3.5)

16 R.HaklandP..Torre ZAMP and u() =u(ω) p()v()d. (3.6) The inegraion by par of (3.5) and (3.6), in view of (3.) and he periodiciy of u, reul in u() = u() + v() u() =u() v() If we muliply boh ide of (3.7) by u() u() p()d = u() p()d = u() p()d + v() p()d v() p()d p()d q() q() p(ξ)dξd, (3.7) p(ξ)dξd. (3.8) p()d and boh ide of (3.8) by p()d, we arrive a p()d p()d p()d p()d p()d p()d q() q() p(ξ)dξd, (3.9) p(ξ)dξd. (3.2) Now if we um he correponding ide of (3.9) and (3.2) we find ha (3.3) hold rue. The equaliy (3.4) follow from (3.7) and (3.8). Lemma 3.2. Le p, q L ([,ω]; R) aify (3.9) and le u, v AC ( [,ω]; R 2) aify (3.) (3.2). Then ( M u m u p() d) 2 4 p()d [q()] d, (3.2) where M u = max {u() : [,ω]}, m u = min {u() : [,ω]}. (3.22) Proof. According o (3.2) we can exend funcion u, v, p, andq o he inerval [, 3ω] periodically. Then, obviouly, (3.) and (3.) hold for almo every [, 3ω], and u() =u( + ω), v() =v( + ω) for [, 2ω]. Chooe m [,ω[ and M ] m, m + ω[ uch ha u( m )=m u, u( M )=M u. (3.23) According o Lemma 3., he funcion u ha a repreenaion u() =u( m ) m+ω m p()d m+ω p()d m q() p(ξ)dξd + m m p()d m+ω q() m+ω p(ξ)dξd for [ m, m + ω]. (3.24) Nowfrom(3.24), for = M,inviewof(3.23) and he periodiciy of p i follow ha m+ω M M m+ω m+ω M u m u = p()d p()d q() p(ξ)dξd + p()d q() p(ξ)dξd. M m m m M (3.25)

17 A combined variaional-opological approach On he oher hand, according o Lemma 3. again, he funcion u ha alo a repreenaion M +ω u() =u( M ) M +ω p()d q() p(ξ)dξd M p()d M M M +ω M +ω + p()d q() p(ξ)dξd for [ M, M + ω]. (3.26) M M m Noe ha m +ω ] M, M +ω[. Therefore, from (3.26) for = m +ω in view of (3.23) and he periodiciy of p and q i follow ha M m+ω m+ω M M M u m u = p()d p()d q() p(ξ)dξd + p()d q() p(ξ)dξd. (3.27) m M M M m Now if we um he correponding ide of (3.25) and (3.27) we find M m+ω m+ω 2(M u m u )= p()d p()d q() p(ξ)dξ p(ξ)dξ d m M M m+ω M M + p()d q() p(ξ)dξ p(ξ)dξ d. (3.28) Noe ha and M p(ξ)dξ m+ω M p(ξ)dξ p(ξ)dξ m m+ω m M p(ξ) dξ for [ M, m + ω], (3.29) M p(ξ)dξ p(ξ) dξ for [ m, M ]. (3.3) m Therefore, from (3.28), in view of (3.29), (3.3) and he periodiciy of q we obain M 2(M u m u ) p()d p() d m m+ω M p() d q() d. (3.3) Now uing he inequaliy AB 4 (A + B)2 in (3.3) and he periodiciy of p, we arrive a Noe ha (3.9) implie M u m u ( p() d) 2 8 p()d q() d =2 and conequenly, (3.32) and (3.33) reul in (3.2). q() d. (3.32) [q()] d, (3.33)

18 R.HaklandP..Torre ZAMP Lemma 3.3. Le p L ([,ω]; R),ϕ L ([,ω]; R + ), (2.2) hold and Then (p, ϕ) verifie he propery (P + ). Proof. In he cae where p()d > we pu x = while if p()d < hen we pu x = [p()] d, y = [p()] + d, y = 4 p()d ( p() d) 2. (3.34) [p()] + d, (3.35) [p()] d. (3.36) In boh cae (3.35) and (3.36), from (3.34) i follow ha (y + x) 2 4(y x). (3.37) Nowfrom(3.37) i follow ha x 2, (3.38) y 2 x +2 2x 2+2 x, (3.39) and, uing he inequaliy 4AB (A + B) 2, alo yx y x. (3.4) Therefore, if p()d > hen in view of (3.35) from(3.38) (3.4) we obain he inequaliie (2.5) (2.7). If p()d < hen in view of (3.36) from(3.38) and (3.4) we ge (2.3) and (2.4). Thu he concluion of he lemma follow from Theorem 2.. Proof of Theorem 3.. A fir we conruc an upper funcion. Pu ( ) 2 ( ) A = G G +, D = H + H, (3.4) 9 8 ( ) 3 ( ) 2 ( ) A 2 B = A, C = D, x = D 3. (3.42) Then, obviouly, AH + BH + CG + DG =, and, herefore, according o Lemma 3., he problem w = p()w 2 (3.43) w 2 = A[h()] + B[h()] + C[g()] + D[g()] (3.44) w () = w (ω), w 2 () = w 2 (ω) (3.45) ha a oluion (w,w 2 ) AC ( [,ω]; R 2).Pu m w = min {w () : [,ω]}, (3.46) M w = max {w () : [,ω]}. (3.47)

19 A combined variaional-opological approach Then, according o Lemma 3.2, in view of(3.4) and (3.42) we have Pu Then, obviouly, and, on accoun of (3.46), M w m w β () = ( p() d) 2 4 p()d (G H + 98 G +H ). (3.48) ( ) /2 + x (w () m w ) for [,ω], (3.49) xd β 2 () =xw 2 () for [,ω]. (3.5) Furhermore, from (3.48) in view of(3.5, 3.4), and (3.42) we obain β () > for [,ω] (3.5) ( ) /2 β () for [,ω]. (3.52) xd x (M w m w ) Now from(3.49) wih repec o (3.53) i follow ha β () On he oher hand, in view of (3.4) and (3.42) wehave ( ) /3 ( ) /2. (3.53) xa xd ( ) /3 for [,ω]. (3.54) xa ( ) /3 ( ) /2 =, xb xd ( ) /2 ( ) /3 =. (3.55) xc xa Therefore, (3.52, 3.54), and (3.55) reul in xa β 3 xb () for [,ω], (3.56) xc β 2 xd () for [,ω]. (3.57) Thu, from (3.49) and (3.5), in view of (3.43) (3.45, 3.56), and (3.57) we obain β () =p()β 2 () for a. e. [,ω], β 2() h() g() β 3 + () β 2() for a. e. [,ω],

20 β () = β (ω), R.HaklandP..Torre β 2 () = β 2 (ω). Therefore, (β,β 2 ) i an upper funcion o (3.) (3.3). Now we conruc a lower funcion. Le y ],G /G + [ be uch ha ( p() d) 2 and pu 4 p()d G yh ( ) (G y G + ) 3 y /2 ZAMP (3.58) y = G y G +. H + (3.59) Obviouly, y H + + y G + G = and, herefore, according o Lemma 3., he problem z = p()z 2 (3.6) z 2 = y [h()] + + y [g()] + [g()] (3.6) z () = z (ω), z 2 () = z 2 (ω) (3.62) ha a oluion (z,z 2 ) AC ( [,ω]; R 2).Pu m z = min {z () : [,ω]}, (3.63) M z = max {z () : [,ω]}. (3.64) Then, according o Lemma 3.2, wehave ( M z m z 2 4 p()d G. (3.65) Pu α () = y + y2 y y 3 (z () m z ) for [,ω], (3.66) Then, obviouly, and, on accoun of (3.63), α 2 () = y2 y 3 z 2 () for [,ω]. (3.67) α () > for [,ω] (3.68) y α () for [,ω]. (3.69) y Furhermore, from (3.65) inviewof(3.58) and (3.59) we obain y 2 y 3 (M z m z ) y ( ) y /2. (3.7) y Nowfrom(3.66) wih repec o (3.64) and (3.7) i follow ha α () y3/2 y for [,ω]. (3.7)

21 A combined variaional-opological approach Therefore, (3.69) and (3.7) reul in y 2 y 3 ( ) 2 α 2() y y for [,ω], (3.72) ( ) 3 α 3() y y for [,ω]. (3.73) Thu, from (3.66) and (3.67), in view of (3.6) (3.62, 3.72), and (3.73) we obain α () =p()α 2 () for a. e. [,ω], α 2() h() g() α 3 + () α 2() for a. e. [,ω], α () = α (ω), α 2 () = α 2 (ω). Therefore, (α,α 2 ) i a lower funcion o (3.) (3.3). Moreover, according o (3.4, 3.42, 3.54, 3.59, 3.69), and y >, we have β () α () for [,ω]. Noe ha funcion ψ (, y) = 3 β 4()y + y 3, ψ 2(, y) = 2 β 3()y + y 2 are non-decreaing for every [,ω] in he econd argumen whenever y β (). Therefore, if we pu ϕ() = 3[h()] + β 4 () + 2[g()] + β 3 () for a. e. [,ω], (3.74) f(, x) = h() x 3 + g() x 2 for a. e. [,ω], x >, hen one can eaily verified ha (2.9) i fulfilled. Moreover, in view of (3.4, 3.42, 3.52, 3.55), and (3.74), we have 3H +(xd) 2 +2G +xb = ( 9 8 ) 9 (G ( ) ) ( G 9 + ( H + ( ) ) H 8 ( ) 3 9 G H ( ) ) 3 9 G +H 8 8 G+H+. (3.75) Now, in view of he inequaliy H + > ( 9 3 8) H we have 8 G +H + < ( ) 3 9 G + H. (3.76) 8 8 Uing (3.76) in(3.75) wege 3 ( 9 8 Nowfrom(3.77), in view of (3.5), we obain ) 2 (G ( ) ) G+ ( 9 H + ( 9 8 ) 3 H ) 4 ( G H G +H ). (3.77) p()d ( p() d) 2. Thu, according o Lemma 3.3 i follow ha (p, ϕ) verifie he propery (P + ). Now he concluion follow from Theorem 2.3.

22 R.HaklandP..Torre ZAMP Acknowledgmen Rober Hakl uppored by he Academy of Science of he Czech Republic, Iniuional Reearch Plan No. AVZ953 and by una de Andalucía, Spain, Projec FQM226; Pedro. Torre uppored by Minierio de Educación y Ciencia, Spain, projec MTM28-252, and by una de Andalucía, Spain, Projec FQM226 Reference. De Coer, C., Habe, P. : Upper and lower oluion in he heory of ODE boundary value problem: claical and recen reul. In: Zanolin, F. (ed.) Nonlinear Analyi and Boundary Value Problem for Ordinary Differenial Equaion CISM ICMS, pp. 78. Springer, New York (996) 2. Hakl, R., Torre, P..: On periodic oluion of econd-order differenial equaion wih aracive-repulive ingulariie.. Differ. Equ. 248, 26 (2) 3. Haegawa, A., Kodama, Y.: Solion in Opical Communicaion. vol. 37, Clarendon Pre, Oxford (995) 4. Kiguradze, I.T., Mukhigulahvili, S.V.: On nonlinear boundary value problem for wo dimenional differenial yem. Differ. Equ. 4(6), pp (24). [Tranlaed from Differenialnye Uravneniya, vol. 4, No. 6, 24, ] 5. Kiguradze, I., Mukhigulahvili, S.: On periodic oluion of wo dimenional nonauonomou differenial yem. Nonlinear Anal. 6(2), (25) 6. Kivhar, Y.S., Argrawal, G.P.: Opical Solion. Adademic Pre, London (23) 7. Kunze, M.: Periodic oluion of a ingular Lagrangian yem relaed o diperion-managed fiber communicaion device. Nonlinear Dyn. Sy. Theory, (2) 8. Mukhigulahvili, S.: On a periodic boundary value problem for cyclic feedback ype linear funcional differenial yem. Arch. Mah. (Bael) 87(3), (26) 9. Mukhigulahvili, S., Gryay, I.: An opimal condiion for he uniquene of a periodic oluion for linear funcional differenial yem. EQTDE, No. 59, 2 (29). hp:// Rachůnková, I., Saněk, S., Tvrdý, M.: Singulariie and laplacian in boundary value problem for nonlinear ordinary differenial equaion. In: Handbook of Differenial Equaion (Ordinary Differenial Equaion), vol. 3. Elevier (26). Schwarz, O.Y., Turiyn, S.K.: Muiple period diperion managed olion. Phy. Rev. A 76, 4389 (27) 2. Torre, P.., Zhang, M.: A monoone ieraive cheme for a nonlinear econd order equaion baed on a generalized ani maximum principle. Mah. Nachr. 25, 7 (23). doi:.2/mana Turiyn, S.K., Gabiov, I., Laedke, E.W., Mezenev, V.K., Muher, S.L., Shapiro, E.G., Schäfer, T., Spachek, K.H.: Variaional approach o opical pule propagaion indiperion compenaed ranmiion yem. Op. Commun. 5, 7 35 (998) 4. Turiyn, S.K., Shapiro, E.G., Medvedev, S.B., Fedoruk, M.P., Mezenev, V.K.: Phyic and mahemaic of diperion managed opical olion. C. R. Phy. 4, 45 6 (23) Rober Hakl Iniue of Mahemaic AS CR Žižkova Brno Czech Republic hakl@ipm.cz Pedro. Torre Deparameno de Maemáica Aplicada Faculad de Ciencia Univeridad de Granada Campu de Fuenenueva /n 87 Granada Spain porre@ugr.e (Received: anuary 4, 2; Revied: April 24, 2)