We reveal that even weak inherent discreteness of a nonlinear model can lead to instabilities of the
localized modes it supports. We present the first example of an oscillatory instability of dark solitons,
and analyze how it may occur for dark solitons of the discrete nonlinear Schrödinger and generalized
Ablowitz-Ladik equations. [S0031-9007(98)08088-0

Johansson, Magnus

Kivshar, Yuri

Physical Review Letters

English

The Australian National University

VOLUME 82, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1999ersity,Discreteness-Induced Oscillatory Instabilities of Dark SolitonsMagnus Johansson1 and Yuri S. Kivshar21Department of Physics and Measurement Technology, Linköping University, S-581 83 Linköping, Sweden2Optical Sciences Center, Research School of Physical Sciences and Engineering, The Australian National UnivCanberra ACT 0200, Australia(Received 18 August 1998)We reveal that even weak inherent discreteness of a nonlinear model can lead to instabilities of thelocalized modes it supports. We present the first example of an oscillatory instability of dark solitons,and analyze how it may occur for dark solitons of the discrete nonlinear Schrödinger and generalizedAblowitz-Ladik equations. [S0031-9007(98)08088-0]PACS numbers: 42.65.Tg, 05.45.Yv, 63.20.Pwtoasthehendshes.e-eend)siteWave instabilities are probably the most remarkabnonlinear phenomena that may occur in nature [1]. Oof the first instabilities discovered for nonlinear modewas the modulational instability, which is known to ban effective physical mechanism in fluids [2] and optic[3] for breakup of continuous modes into solitary waveAlso, the solitary waves themselves may become unstaband the analysis of their instabilities is an importanproblem of nonlinear physics. Instabilities are knownoccur for both bright [4] and dark [5] solitary waves odifferentnonintegrablenonlinear models.Recently, a new type of solitary-wave instability,os-cillatory instability, has been found to occur for brighBragg gap solitons in the generalized Thirring model [6Such an instability is characterized by complex eigenvaues, and its scenario is associated with a resonancetween the long-wavelength radiation and soliton internmodes which appear in the soliton spectrum when tmodel becomes nonintegrable [7]. In spite of the fact thoscillatory instabilities appear often in dissipative mode[8], their manifestation in continuous Hamiltonian modeis rare [9], and so farno example has been known for oscillatory instability of dark solitons.The aim of this Letter is twofold. First, we analyzewhat we believe to be the first examples of oscillatoinstabilities of dark solitons, by considering the important cases of the discrete nonlinear Schrödinger (DNLand generalized Ablowitz-Ladik (AL-DNLS) models. Werevealtwo different scenariosfor the dark-soliton oscilla-tory instability, which may occur due to either a resonanbetween radiation modes and the soliton internal mode,a resonance between two soliton internal modes. Secowe demonstrate that even a weak inherent discretenmay drastically modify the dynamics of a nonlinear sytem leading to instabilities which have no analog in thcontinuum limit.First, we consider the well-known DNLS equation,i Ùcn 1 Cscn11 1 cn21d 1 jcnj2cn ­ 0 , (1)where the dot stands for the derivative in time. Stationalocalized solutions of Eq. (1) in the formcnstd ­ fneiLt,whereL ­ 2C cosk 1 sfs0dd2, may exist as dark-soliton0031-9007y99y82(1)y85(4)$15.00lenelsess.le,ttoft].l-be-alheatlsls-ry-S)ceornd,esss-erymodes with the nonvanishing boundary conditionsfn !6fs0deikn (n ! 6`), provided the background wave ismodulationally stable, i.e., forC cosk , 0 [10]. Withoutloss of generality, we can put the background intensityunity, fs0d ­ 1.The structure of the dark-soliton modes of Eq. (1) hbeen discussed earlier [11,12]. Here, we considercaseC . 0, in which the background wave hask ­ pand is “staggered,” as shown in Figs. 1(a) and 1(b). Ttransformationcn ! s21dncn immediately yields the cor-responding “unstaggered” modes (k ­ 0) for negativeC.The dark-soliton modes presented in Figs. 1(a) a1(b) describe two types of stationary “black” soliton[5] in a discrete lattice, the on-site mode (A mode)centered with zero intensity at a lattice site, and tintersite mode (B mode) centered between two siteThese two modes can be uniquely followed from thcontinuous limit (C ! `) to the “anticontinuous” limit(C ­ 0). At C ­ 0, the A mode takes the formfn ­s. . . , 21, 11, 21, 0, 11, 21, 11, . . .d, and it describes asingle “hole” in a background wave with constant amplitude and ap phase shift across the hole. Similarly, thB mode takes the formfn ­ s. . . , 21, 11, 21, 21, 11,21, . . .d, and it describes the lattice oscillation modwith a p phase shift between two neighboring sites ano hole.Linear stability of theA mode for small enoughCfollows from Aubry’s theorem (Theorem 9 in Ref. [13]φn|φn|2(a)φn|φn|2(b)FIG. 1. (a), (b) Two types of “staggered” dark-soliton modein a lattice. Shown are the oscillation amplitude at each s(upper row) and the valuejcnj2 (lower row).© 1998 The American Physical Society 85VOLUME 82, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1999-eseal-exe-cythenteil-s,lo-ofionm,ic)nrelating the linear stability of multibreathers to the extremof the effective actionwhich is a function of the relativephasesan of the single breathers. For the DNLS equation(1) with a positive nonlinearity,local minimaof the effec-tive action correspond to stable solutions. A perturbativexpression of the effective action to orderC2 was obtainedin Eq. (A10) in [14]. For smallC, it is enough to considerthe phase interactions between nearest and next-neaneighboring sites. Then, the lowest order contribution to the effective action from neighboring excitedsites isPnf2CjLj cossan11 2 and 2 C2 cos2san11 2andg, while the contribution from the next-nearest-neighbor interaction is2C2Pnfin0 cossan11 2 an21d 12C2 cossan011 2 an021d, where n0 is the site withthe “hole.” For the A mode, an11 2 an ­ p,an11 2 an21 ­ 0, n fi n0, an011 2 an021 ­ p, sothat the effective action has a local minimum and thmode is stable for smallC.For theB mode, there are two neighboring sites havinthe same phase at the center, which suggests thatcorresponding extremum of the effective action is a saddNumerical investigation of the eigenvalue problem alsshows that theB mode is unstable for allC.Let us consider the linear stability of the dark-solitonmodes for nonvanishingC. Assumingcnstd ­ ffn 1enstdgeiLt, we obtain the linearized equations foreni Ùen 1 Csen11 1 en21d 1 2jfnj2en1 f2nepn 2 Len ­ 0 .Writing en ­ jn 1 ihn for realfn yieldsddtµjnhn∂­µ0 H12H2 0∂ µjnhn∂; M̂µjnhn∂,where for a system ofN sitesM̂ is a2N 3 2N matrix andH1 and H2 are N 3 N matrices with time-independentcoefficients, H6ij ­ fL 2 s2 7 1df2i gdi,j 2 Csdi,j11 1di,j21d (boundary conditions not explicitly taken intoaccount). Linear stability is then equivalent to the matriM̂ having all its eigenvalues on the imaginary axis.When C ­ 0, all eigenvalues ofM̂ lie at zero except,for the A mode, one complex conjugated pair at6i cor-responding to a mode localized on the hole. WhenC isincreased, the eigenvalues at zero spread on the imaginaxis creatinga phonon band of extended states,which cor-responds to a continuous spectrum for largeN . Assumingen ­ aeiskn2vtd 1 be2iskn2vtd yields the dispersion re-lation, v ­ 6p16C2 cos4sky2d 1 8C cos2sky2d, so thatthe eigenvalues corresponding to the continuous bandbetween 0 (atk ­ p) and 6ip16C2 1 8C (at k ­ 0).On the other hand, the pair of eigenvalues at6i will movetowards zero on the imaginary axis asC is increased. Forsmall C, we can assume that the corresponding eigemode is almost completely localized at the hole, so then011 ø en021 ø 0. Sincefn0 ­ 0, we obtain that thismode oscillates with frequencyL ­ 1 2 2C, so that thecorresponding eigenvalues are6is1 2 2Cd, to the low-86aerest-egthele.oxarylien-atest order inC. Then, equating this result with the expression above for the edge of the continuous band givan estimate to the value ofC where these two eigenval-ues coincide:C ­ 1yp3 2 1y2 ø 0.07735. This agreesquite well with the exact, numerically obtained valuC ; Ccr ø 0.07647. At C ­ Ccr , a Hopf-type bifurca-tion occurs, as two complex conjugated pairs of eigenvues leave the imaginary axis and go out in the complplane (see Fig. 2). Thus,an oscillatory instabilityoccursfor theA mode whenC . Ccr , with the instability growthrate l given by the real part of the unstable eigenvalu(see Fig. 2). For theB mode, all eigenvalues lie at zerowhen C ­ 0. As soon asC is increased, one pair goesout on the real axis and stays there for allC . 0. Thus,theB mode is always unstable.For C . Ccr , the imaginary part of the unstable eigenvalue moves towards zero (i.e., the oscillation frequendecreases) asC is increased. The real part of theeigenvalue vsC is shown in Fig. 2 for two differ-ent system sizes, and compared with estimations ofmaximum growth rate from direct numerical integratioof the DNLS model for chains large enough to eliminathe influence of the boundaries. In all cases the instabity is largest forC ø 0.32 wherel ø 0.120. For largerC, there appear stable “windows” for the finite systemwhere the mode is stabilized by the boundaries. Thecation of these windows depends critically on the size00.020.040.060.080.10.120 0.2 0.4 0.6 0.8 1 1.2 1.4Re λCoupling parameter, C-1-0.500.51-0.001 0 0.001Im λC=0.05-1-0.500.51-0.001 0 0.001Re C=0.076-1-0.500.51-0.001 0 0.001C=0.0771e-061e-050.00010.0010.010.10.5 1 1.5 2 2.5 3 3.5growth rateCλFIG. 2. Real part of the unstable eigenvalues as a functof C in the model (1) forN ­ 61 (dotted) andN ­ 121(dashed) sites, and the growth rate (solid) for an infinite systecalculated for lattices of up to 15 000 sites. Inset: Asymptotfit of the growth rate (points) with stretched exponential (line(see text). Below: Instability scenario shown as the evolutioof the eigenvalues of̂M for N ­ 121 sites.VOLUME 82, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1999en isthateun-ndectic.mroti-elsofheFory,itsn-eekn--ityesa-us-t ofer-atesthe system and the boundary conditions (in Fig. 2, peodic boundary conditionscN11 ­ c1 are used).Similar “reentrant instabilities” have been found fodiscrete breathers of the finite-width Klein-Gordon chain[15]. Qualitatively, the explanation is that for small systems, the eigenvalues corresponding to the “continuouspectrum are rather sparsely distributed, and the corsponding eigenvectors are localized over the size of tsystem. Thus, as the eigenvalue corresponding to thestable mode approaches the imaginary axis when increing C, it may find a hole in the spectrum and join theimaginary axis for a while. Then, a further increase oC causes a collision with the next imaginary eigenvaluand a new instability occurs. This procedure repeats itsuntil the eigenvalue has passed the phonon band eigvalue which is closest to zero. After this the statestable for all larger values ofC (e.g., in Fig. 2 the stateis stable for all C . 1.11 when N ­ 61, and for allC . 1.38 when N ­ 121). Increasing the system sizeimplies that the eigenvalue distribution on the imaginaraxis will be more dense, so that the stable windows wbe smaller andcompletely disappear in the limit of theinfinite system, where the unstable eigenvalue never jointhe imaginary axis.For an infinite DNLS chain, the instability growth ratedecreases in an exponential-like way to zero for largeC,and thus indicates that the dark mode is unstable forC . Ccr . A quite good asymptotic fit is obtained witha stretched exponential, , exps2bCgd, with g ø 0.7(see inset in Fig. 2). We also investigated the casevaryingfs0d to keep thecomplementary norm[5] constantwhen varyingC. In that case, we found the asymptoticdecay of the instability growth rate withC to be fasterthan purely exponential.Figure 3 shows the time evolution for two differ-ent values ofC when the initial state is a slightly per-turbed dark A mode. A small perturbation has beenC=0.30250 500 750 10030000.51nt|ψn|2C=0.75250 500 750 10030000.51nt|ψn|200.510 100 200 300 400|ψn|2tn0n0+1n0+2n0+300.510 50 100|ψn|2tn0n0-1n0-2n0-3FIG. 3. Time evolution ofjcnj2 with slightly perturbed darkA modes as initial conditions forC ­ 0.30 (left) andC ­ 0.75(right). Lower figures show the detailed dynamics for a fewsites around the center.ri-rs-s”re-heun-as-fe,elfen-isyillsallofchosen to be approximately in the direction of thunstable eigenvector. If instead a random perturbatiochosen, the qualitative behavior is the same, exceptthere will be an initial small amount of radiation beforthe unstable internal eigenmode gets excited. Thestable eigenvector is always spatially symmetric arouthe central site, and therefore antisymmetric with respto the dark mode itself, since the latter is antisymmetrThus, the main effect of the instability is a transition froa black (zero intensity at the middle) to grey (nonzeintensity at the middle) soliton, as can be clearly idenfied at least forC . 0.3. This is similar to the instabil-ity scenario of dark solitons in some continuous mod[5], except the oscillatory dynamics. The directionthe resulting grey soliton is determined by the sign of tperturbation projected on the unstable eigenmode.large C, the grey soliton is almost black, moves slowland the radiation is small, while for smallerC the mini-mum value of the soliton intensity increases as well asvelocity and the amount of radiation. For small and itermediate values of the coupling parameterC, the result-ing grey soliton decays continuously into radiation (sFig. 3).It is important to study the oscillatory instability of darsolitons for other types of nonlinear lattices. Here, we cosider the AL-DNLS equationi Ùcn 1 Cscn11 1 cn21d 1mjcnj2cn 112 sm 2 1d jcnj2scn11 1 cn21d ­ 0, where0 # m # 1. The casem ­ 0 corresponds to the integrable AL model, whereas the casem ­ 1, to the DNLSmodel analyzed above.With the general formcnstd ­ fneiLt, and boundaryconditions fn ! 6fs0deikn (n ! 6`), the frequencyL is determined byL ­ 2C cosk 1 sfs0dd2fm 1 sm 21d coskg. In particular, for the black mode (k ­ p) therelation becomesL ­ 22C 1 sfs0dd2, just as for theDNLS model. As above, we putfs0d ­ 1 without lossof generality.In the AL-DNLS model, there is a lower limit ofCfor the existence of dark solitons, due to the instabilof the background when the effective coupling changsign (see, e.g., Ref. [16]). This occurs whenC 1 sm 21d sfs0dd2y2 ­ 0, so that, whenfs0d ­ 1, the dark modesexist only forC . s1 2 mdy2 (Fig. 4, dashed line).Consideringcnstd ­ ffn 1 enstdgeiLt as above, weobtain the linearized equations for the small perturbtion en, and the dispersion relation for the continuospectrum (for the staggered mode with limjnj!` f2n ­1): v ­ 62pnskd f1 1 nskdg, where nskd ; s2C 1m 2 1d cos2sky2d. Therefore, the eigenvalues corresponding to the continuous band lie between 0 (atk ­p) and 6i2ps2C 1 m 2 1d s2C 1 md (at k ­ 0) forC . s1 2 mdy2.As above, an approximate expression for the onseinstability can be obtained by assuming that the intnal mode is completely localized at the hole, so then011 ø en021 ø 0. Then, the corresponding eigenvalu87VOLUME 82, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1999heto.tondatlstythealtial,dp..in,t,s-00.10.20.30.40.50 0.2 0.4 0.6 0.8 1Coupling parameter,Cµ-0.200.2-0.02 0 0.02Im λRe λC=0.4012A-0.200.2-0.02 0 0.02Im λRe λC=0.4013BFIG. 4. Solid line: Numerically obtained instability thresholdCcr smd for the AL-DNLS model. Dashed line: Minimum valueof C for the existence of the dark localized modes. InsetExamples of the eigenvalues atm ­ 0.2, below (A) and above(B) the threshold curveCcr smd.are 6is1 2 2Cd, to the lowest order inC. The col-lision with the phonon band edge is then expectedoccur for C ø s1 2 4mdy6 1pm2 1 m 1 1y3. Com-paring this result with the numerically obtained values oC for the bifurcation (see Fig. 4) shows that the instability for smaller m occurs for smallerC than expectedfrom this analytical estimate. The reason why the bfurcation occurs earlier is that it in fact results not froma collision with the band edge phonon, but with a second localized mode that has bifurcated from the banedge slightly before (see the insets in Fig. 4). As a mater of fact, studying the bifurcation for the DNLS modevery closely shows that also in this case the collisioprobably occurs with a second localized mode cominfrom the phonon band. However, this localized modonly occurs extremely close to the bifurcation whenmis close to 1 (form ­ 1 we observe it forC . 0.076 464,and the bifurcation occurs atC ­ 0.076 468 . . .). Andsince it will be very weakly localized, we are not ableto say for sure whether it will be localized or not for theinfinite system. But for smallerm, this localized modeundoubtedly exists, and has the same spatial symmetrythe hole mode. It describes an internal degree of freedoof the dark soliton, i.e., itsinternal mode.Otherwise, the qualitative scenario with stable anunstable windows for finite systems and instability athe way up to the continuum limit for the infinite systemremains the same for all AL-DNLS lattices with0 , m #1. The exception is the limit of the AL model (m ­ 0)which is known to beexactly integrable. At m ­ 0, darksolitons are always stable for allC and their spectrum has88s:tof-i--dt-lngeasmdlla pair of eigenvalues exactly at zero corresponding to texact translational invariance.In conclusion, we have described, for the first timeour knowledge, the oscillatory instability of dark solitonsThis new type of dark-soliton instability appears dueinherent discreteness of a nonlinear lattice model, athe universality of the instability scenario suggests thit should be also observed in other nonlinear modesupporting dark solitons. We remark that the instabiliobserved here may be regarded as an extension ofinstabilities existing for small lattices [17].M. J. acknowledges support from the Swedish NaturScience Research Council. Yu. K. acknowledges a parsupport from the Australian Research Council.[1] E. Infeld and G. 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Discreteness-induced oscillatory instabilities of dark solitons

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Discreteness-induced oscillatory instabilities of dark solitons

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