Embedded soliton solutions : A variational study

Em b edded soliton solutions : A v ariational study Debabrata Pal , Sk. Golam Ali and B. T alukdar ∗ Dep art ment of Physics, Visva-Bhar ati University, Santiniketan 7312 35, Indi a W e use a v ariational method to construct soliton solutions for sy stems c haracterized by opposing disp ersion and competing nonlinearities at fundamental and second harmonics. W e sho w that b oth ordinary and em b edded solitons tend to gain energy when the second harmonic field becomes weak er than the first harmonic field. P ACS n umbers: 42.65.Tg, 05.45.Yv Keyw ords: Em bedded soliton solution ; Second harmonic generation ; Lagrangian based approac h I. INTRO DUCTION Embedded solitons (ES) represent solitary wa ves which reside inside the co n tinuous spec tr um of a no nlinear - wa ve system. This t yp e of solitons w as first r epo rted by Y ang et. al. [1] in optical mo dels characterized by opp osing disp e rsion a nd comp eting nonlinearities at fundamental and second harmonics. Mo r e sp ecifically , optical media with quadratic χ (2) and cubic χ (3) nonlinear s us ceptibilities can s uppor t ES so lutions. The evolution of ES is governed b y the coupled no nlinear pa rtial differ e n tial eq uations [2 ] iu z + 1 2 u 2 t + u ∗ v + γ 1 | u | 2 u + 4 γ 2 | v | 2 u = 0 (1) and iv z − 1 2 δ v 2 t + q v + 1 2 u 2 + 2 γ 2 ( | v | 2 +2 | u | 2 ) v = 0 , (2) where u = u ( z , t ) and v = v ( z , t ) r epresent the fundamental har monic (FH) and second harmonic (SH) fields resp ec- tively . In writing (1) and (2) we ha ve used optical notations s uc h that z and t stand for the pr opagation distance and reduced time. The quantit y − δ is the relative disper sion of SH a nd γ 1 , 2 are the Kerr co efficients.Here q repres en ts the group velo cit y mismatch orig inated by the frequency difference of FH a nd SH fields. Usually , ES’s are s tudied using num erical routines to solve (1) and (2). In view of this one o ften works within the framework of a simplified ph ysical mo del wher e | v | 2 ≪ | u | 2 and neglects the cro ss-phase modulatio n (XPM) term (fifth term) in comparison with self-phas e modula tion (SPM) (four th term) in (1) . The SPM term in (2) is also assumed to b e neglig ible in compar ison with its XPM counterpart. Thus we get a trunca ted mo del represe n ted by iu z + 1 2 u 2 t + u ∗ v + γ 1 | u | 2 u = 0 (3) and iv z − 1 2 δ v 2 t + q v + 1 2 u 2 + 4 γ 2 | u | 2 v = 0 . (4) F or stationar y soliton solutions one can use u ( z , t ) = e ikz U ( t ) , v ( z , t ) = e 2 ikz V ( t ) (5) with k , the FH w av e num ber. The partial differential equations of the full mo del and those of the truncated mo del then reduce to ordinar y differential equations given by − k U + 1 2 ¨ U + U V + γ 1 U 3 + 4 γ 2 V 2 U = 0 , (6) ∗ Electronic address: bino y123@bsnl.in 2 − 2 k V − 1 2 δ ¨ V + qV + 1 2 U 2 + 2 γ 2 ( V 2 + 2 U 2 ) V = 0 (7) and − k U + 1 2 ¨ U + U V + γ 1 U 3 = 0 , (8) − 2 k V − 1 2 δ ¨ V + q V + 1 2 U 2 + 4 γ 2 U 2 V = 0 . (9) Here the dots denote differ en tiation with resp e ct to t . Linea rization of the equations in (1) and (2) [F ull mo del] as well as in (3 ) a nd (4) [T r uncated mo del] shows that b oth mo dels supp ort o rdinary so liton so lutions in the reg ions 0 < k < q 2 if δ > 0 , k > max n 0 , q 2 o if δ < 0 (10) and embedded s o liton so liton solutions in the re g ions k > max n 0 , q 2 o if δ > 0 , 0 < k < q 2 if δ > 0 . (11) The ob ject of the present work is to derive a str aightf orward analytical mo del for comparing the prop erties of soliton solutio ns supp orted by the pair o f eq uations repres e n ting the full and truncated mo dels .In doing so we sha ll consider the cases of ordina ry and embedded solitons s eparately . T o a c hieve this we shall e n visage a v ar iational approach to the pro blem, where one beg ins with a Lagra ngian for the sys tem under considera tion a nd constructs the so-called effective Lagra ngian by taking reco urse to the use of trial functions for the field v ariable s . Unders ta ndably , the trial functions will involv e a num b er of unknown par ameters . As w e sha ll see the effectiv e Lag rangian will provide a natural ba sis to deter mine these par ameters . In the ab ov e context w e no te that (6) a nd (7) , r esulting from the full mo del , follo w from an actio n principle . In contrast to this , (8) and (9) p ertaining to the truncated mo del are non-Lagr angian . But the latter set o f equations are based on physically founded assumptions . This led Kaup a nd Malomed [2] to adapt the v ar iational approach to the seeming ly flaw ed system represented by (8) and (9) . In their metho d one sta rts with the Lagra ngian of the full system and drops the term containing V 4 to construct an expressio n for the effective L a grangia n by us ing the trial functions for U a nd V . F urther , the implementation of the Ritz optimization proc e dure to ev aluate the v aria tional pa r ameters re q uires o ne mo re a pproximation . W e c laim that the r esults in Ref (2) can b e reder iv ed and reexamined without tak ing reco ur se to the use of this tw o - tier approximation .In par ticular , we find that if we work with the effective Lag rangian of the full system , construct e q uations for the v aria tional parameters and then use the approximation V ≪ U ,we automa tically a rrive at the results o f K a up and Malomed . Mor e significantly , the metho d follow ed by us provides a natural basis to e xamine how the results for U and V for the full mo del differ from those of the truncated mo del . O ne of our main ob jectives in this w ork is to co mpare the results of the full and truncated mo dels and ther eb y g ain so me physical weight for the pr o blem . W e b egin section I I w ith the Lagr angian o f the full system and co ns truct the expressio n for the effective Lagr angian using so me trial functions for U and V . W e then apply the Ritz optimization pr o cedur e to obtain equations for the parameters of the trial functions and ex amine how the results of Ref. 2 are o btained fo r V ≪ U . In sectio n I II we compare the r esults of U and V for the full mo del with thos e for truncated mo del. W e represent the results for bo th ordinary and e mbedded s olitons. II. V ARI A T IONAL FORMULA TION OF (6) , (7) , (8) AND (9) A. Lagrangian representation Our analysis for the pro per ties of or dinary and em be dded soliton solutions supp orted by the full and truncated mo dels will inv olve ess en tially a Ritz optimization pro cedure [3] ba sed on the v a riational functional for (6) and (7) . It is easily se e n that these initial- bounda ry v a lue pr oblems can be conv erted to a v ariationa l problem with the Lagra ng ian w r itten a s L = Z     − k U 2 − (2 k − q ) V 2 − 1 2 ˙ U 2 + δ 2 ˙ V 2 + U 2 V + γ 1 2 U 4 + 4 γ 2 U 2 V 2 + γ 2 V 4     dt. (12) 3 0 2 4 6 8 10 12 0 1 2 3 4 5 6 U,V t U T U F V T V F FIG. 1: U and V as a function of t for non - embedded solitons. In the Ritz optimization pro cedure, the firs t v aria tion of the v ariationa l functional is made to v anish within a set of suitable chosen trial functions. W e thus introduce the ansa tz [2] U = Asech ( √ 2 k t ) and V = B sech 2 ( √ 2 k t ) (13) for the time - dep endent parts of the FH and SH fields. Here the amplitudes A a nd B a re v ariationa l parameter s. The inv er se width √ 2 k will, how ever, not b e v aried. Inserting (13) in (12) and carr ying o ut the time integral we obtain h L i = 2 3 √ 2 k     − 4 k A 2 − 2 (2 k − q ) B 2 + 8 5 δ k B 2 + 2 A 2 B + γ 1 A 4 + 32 5 γ 2 A 2 B 2 + 48 35 γ 2 B 4     , (14) the effective Lagr angian for U and V in (1 3) . The Lagr angian in (14) represents a sp ecific function of the para meters only . Optimization with resp ect to parameters will yield a system of equatio ns which when solved will determine U and V within the chosen set of tria l functions and a concomitant appr oximation for the true solutions. This is the route we follow to determine the v alues of the parameter s A and B . B. V ariational parameters and truncated model F rom the v anishing co nditio ns of δ h L i δA and δ h L i δB we obtain − 2 k + B + γ 1 A 2 + 16 5 γ 2 B 2 = 0 (15) and A 2 + 32 5 γ 2 A 2 B − 2 (2 k − q ) B + 8 5 δ k B + 96 35 γ 2 B 3 = 0 . (16) Understandably , these equa tions determine the pa rameters of the full mo del. T o go over to the truncated mo del we can cho ose B ≪ A and neglect B 2 and B 3 in (15) a nd (1 6) to get − 2 k + B + γ 1 A 2 = 0 (17) and A 2 + 32 5 γ 2 A 2 B − 2 (2 k − q ) B + 8 5 δ k B = 0 . (18) These equations were obtained by Kaup a nd Ma lomed [2] first b y neglecting the last term in (12) and then a gain neglecting the contribution of the term 4 γ 2 U 2 V 2 while taking v ar iation with resp ect to A . But w e hav e shown that this type of tw o - tier approximation is no t es sent ial to ma k e a tr a nsition fro m the full to the truncated mo del. 4 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 U,V t U T U F V T V F FIG. 2: U and V as a fun ction of t for em b edded solitons. II I. SOLITON SOLUTIONS W e hav e seen that when the wav e num b er k falls int o the regio n (10) b oth full and truncated mo dels hav e ordinar y soliton solutions . T o see how , in this case ,the r esults of U and V for the full mo del differ from those of the truncated mo del w e hav e chosen to work with k = 0 . 25 , γ 1 = − 0 . 05 , γ 2 = − 0 . 025 , δ = 1 , q = 1 . As for the full mo del , we use these v a lues in (15 ) and (16) to get three v alues for B , namely B 1 = 0 . 4824 , B 2 = 5 . 6895 , B 3 = 19 . 437 9 . W e find that A v a lues corr e spo nding to B 1 and B 3 are imaginar y while A b ecomes a real num b er equa l to 7 . 2109( A F 2 ) when calculated by using the v alue of B 2 ( B F 2 ) .The corre s ponding results for the truncated model ar e B T 2 = 6 . 7227 and A T 2 = 1 1 . 1559 . The supers cripts F and T refer to the full and truncated models . W e shall also use similar sup e rscripts on U and V .The or dina ry or no n-em b edded so liton s olutions are shown in Fig. 1 . F ro m this figure it is clear that the cur v es for U T and V T are mor e p eaked compa r e to the curves for U F and V F . It will , therefore be int eresting to examine how the b e ha viour of U F , V F , U T and V T is affected in the ca s e of embedded so litons . In consis ten t with (11) we take k = 0 . 6 963 , γ 1 = − 0 . 05 , γ 2 = − 0 . 025 , δ = 1 , q = 1 for the em bedded soliton . In this case we find B F 2 = 4 . 9450 , A F 2 = 5 . 65 00 , B T 2 = 6 . 38 22 and A T 2 = 9 . 9896 . In Fig. 2 we display the curves for embedded solitons . In this case also the curves for U T and V T are more pea k ed than the curves for U F and V F . But lo o king clos ely into the curves in Figs. 1 and 2 we see that in the cas e of embedded s olitons the curves for U T and V T fall off more r apidly than their non-embedded co un terparts . In the mo del considered in this w ork the energy of the soliton is giv en b y E = R + ∞ −∞  | u | 2 + 2 | v | 2  dt . F rom the results in Figs. 1 a nd 2 it is clear that E T > E F for bo th o r dinary and em b edded solitons. Understandably , E T and E F stand for the solito n energies o btained by using the truncated and full mo dels. It will, therefore , b e a n interesting curiosity to verify how the approximation V ≪ U affects a typical exp eriment. ACKNO WLEDGEMENTS One of the authors (BT) w ould lik e to ackno wledge the fina ncial suppor t of the Universit y Gr ant s Commission, Gov ernment of India (F. No. 32 - 39 / 200 6 (SR)). The authors ar e thankful to Prof. S. N. Roy , Departmen t of Physics, Visv a - Bhara ti, Santinik etan 7 31235, India [1] J Y ang , B A Malomed and D J Kaup ,Em b edded solitons in Second-Harmonic Generating systems , Phys. Rev . Lett. 83 , 1958 (1999) [2] D J Kaup, B A Malomed , Embedded solitons in Lagrangian and semi-Lagrangian systems , Physica D 184 153 (2003) [3] D Anderson , V ariational approach to nonlinear pulse propagation in optical fib ers , Phys. Rev. A 27 3135 (1983)