Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schroedinger equation
aa r X i v : . [ n li n . PS ] N ov Bistability and instability of dark-antidark solitonsin the cubic-quintic nonlinear Schr¨odinger equation
M. Crosta , , A. Fratalocchi , ∗ and S. Trillo † Dept. of Physics, Sapienza University of Rome, I-00185, Rome, Italy PRIMALIGHT, Faculty of Electrical Engineering; Applied Mathematics and Computational Science,King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia Dipartimento di Ingegneria, Universit`a di Ferrara, Via Saragat 1, 44122 Ferrara, Italy (Dated: September 24, 2018)We characterize the full family of soliton solutions sitting over a background plane wave and ruledby the cubic-quintic nonlinear Schr¨odinger equation in the regime where a quintic focusing termrepresents a saturation of the cubic defocusing nonlinearity. We discuss existence and propertiesof solitons in terms of catastrophe theory and fully characterize bistability and instabilities of thedark-antidark pairs, revealing new mechanisms of decay of antidark solitons.
PACS numbers: 42.65.Tg,42.65.Pc,42.65.Jx,47.35.Jk
I. INTRODUCTION
Optical spatial solitons are important for their capa-bility to beat diffraction and their potential for engineer-ing a variety of optical reconfigurable structures includ-ing (and not limited to) couplers, deflectors and logicgates. In Kerr media where the paraxial propagation isdescribed by the nonlinear Schr¨odinger (NLS) equation,only one soliton solution exists once the parameters (i.e.,nonlinearity and peak intensity or width) are fixed andthat solution is stable. However, more general nonlinearresponses can result into bistability of solitons (strictlyspeaking solitary waves) and/or their instability againstthe growth of weak perturbations.In this paper we are interested to investigate such fea-tures for solitons sitting on a finite background (i.e., dark-like) in the context of the Cubic-Quintic NLS (CQNLS)with a defocusing cubic and focusing quintic nonlinearresponse. The importance of such model lies in the factthat it constitutes the simplest model for a defocusingsaturable Kerr effect [1, 2], whose parameters can be ef-fectively measured in a relatively simple way by two-wavecoupling or Z-scan [3, 4].As far as bistability is concerned, the case of darksolitons has been investigated with reference to variousmodel including the CQNLS [2, 5–9]. In particular Her-mann has shown that dark solitons exhibit bistability ofthe second kind, i.e. characterized by solutions possess-ing the same full-width-half-maximum (FWHM), albeitpossessing different amplitudes (and generally invariantsof motion). This type of bistability was introduced inRefs. [10, 11] by Gatz and Hermann, to distinguish itfrom the earlier definition [12–14] which implies the ex-istence of different solutions possessing the same valueof one invariant of motion (e.g., the power) for differentvalues of the internal parameter, typically the nonlinear ∗ † [email protected] propagation constant β . The analysis carried out by Her-rmann, however, is limited to stationary solitons, while afull family of moving dark solitons can exist. The analy-sis is further complicated by the fact that the CQNLS inthe regime considered here is known to possess coexistingantidark (or bright on pedestal) solutions [9, 15]. The fullfamily of dark and antidark solitons, once parametrizedby the velocity, exhibits intriguing features which havebeen overlooked, and which we discuss below. Further-more, whether the full family of the dark-antidark mov-ing pairs is stable or not, and which are the instabilitymechanisms is still an open problem. Our systematic in-vestigation of these problems provides two answers: (i) itshows that the criterium demonstrated by Barashenkov[16] for dark solitons provides the correct exhaustive an-swer to the stability problem also for antidark solitons;(ii) it clarifies that the decay of antidark solitons canfollow new scenarios in proper regions of the parameterspace, rather than always blowing-up as conjectured inthe previous literature, though collapse is in general al-lowed even in 1+1D because of the high power of thefocusing term.Besides being important per se , the knowledge of thedynamics of the whole soliton family of the CQNLS isalso important in view of recent studies which extendthe investigation of competing nonlinearities to the non-paraxial [17] and nonlocal [18, 19] regimes. Moreover, thefull characterization of the soliton solutions and their in-stabilities constitute the starting ground for describingthe feature of dispersive shock waves (DSW, involvingmultiple solitons in the weakly dispersive regime) [20],an active area of research where succesful experimentshave been recently performed in non-Kerr media underdifferent excitation conditions [21–23]. In this respect,here we provide the first prediction of a dispersive shockwave produced directly by the decay of a solitary waveof the CQNLS model. II. DARK-ANTIDARK SOLUTIONS
We start from the (dimensionless) CQNLS equation: i ∂u∂z + 12 ∂ u∂x − | u | u + α | u | u = 0 , (1)which describes a saturable Kerr-like nonlinearitythrough its truncated expansion at second-order in thenormalized intensity | u | , with α being an external freeparameter that weights the quintic nonlinear response(the smaller α , the weaker the saturation effect). Soli-tons of such system have been reported before. How-ever, we reformulate the full problem from the begin-ning, giving novel analytical formulas which prove con-venient for the purpose of our analysis. Solitons cor-respond to translationally invariant solutions of the theform u ( z, x ) = p ρ ( θ ) exp [ iφ ( θ ) + iβz ], where θ = x − vz and β = g ( ρ ) is the nonlinear phase shift experienced bya plane-wave background with intensity ρ ≡ | u | , in themedium where the nonlinear refractive index varies withintensity ρ = | u | according to the law g ( ρ ) = − ρ + αρ / internal parameters whichwe choose, in analogy to general dark soliton solutionsof the defocusing NLS equation, as ρ (intensity back-ground, which fixes also β ) and v (soliton velocity, whichfixes also the darkness or brightness of the soliton). Notethat, here, the quintic term prevents the simple rescalingto ρ = 1 without rescaling α , and the velocity compli-cates further the scenario, so we keep the three param-eters free. The modulus ρ plays the role of equivalent”position”, and obeys the standard Hamiltonian dynam-ics with ”momentum” p = ˙ ρ ≡ dρ/dθ ,˙ p = − ∂ H ∂ρ , ˙ ρ = ∂ H ∂p ; H = p V ( ρ ) , (2) V = 2 ρ (cid:20) α ρ − ρ ) − ( ρ − ρ ) + 2 k ( ρ − ρ ) − c ρ (cid:21) . Here c = vρ and k = v + ρ − α ρ . Once ρ ( θ ) isobtained by solving Eqs. (2), the phase profile φ ( θ ) canbe found by integrating the following equation:˙ φ = v (cid:18) − ρ ρ (cid:19) . (3) A. Soliton solutions
Soliton solutions sitting on the plane-wave background ρ = ρ correspond to homoclinic separatrix trajectoriesof Eqs. (2), characterized by the energy level H = E with E = V ( ρ ) = − c . Such separatrices emanate from thesaddle point ( ρ, p ) = ( ρ ,
0) of the Hamiltonian H ( ρ, p ).For α = 0, the potential V ( ρ ) − E is a double well cor-responding to a double-loop separatrix as shown in Fig.1(a,b). Therefore one has, in general, a coexisting pair of dark and antidark solitons that corresponds to the mo-tion along the left well ρ m ≤ ρ ≤ ρ (dark solitons), andthe right well ρ ≤ ρ ≤ ρ a (antidark solitons), respec-tively. Here ρ m ≤ ρ ≤ ρ a are the roots of V ( ρ ) − E (explicit expressions of ρ m , ρ a are reported in AppendixA). In terms of such roots, we derive (see Appendix A)the following explicit solutions for dark solitons: ρ d ( θ ) = ρ m + rρ a tanh [ w ( θ − θ )]1 + r tanh [ w ( θ − θ )] , (4)where r = ( ρ − ρ m ) / ( ρ a − ρ ) and w = p α ( ρ a − ρ )( ρ − ρ m ) / ρ = ρ m . Similarly for anti-dark solitons, we obtain ρ a ( θ ) = ρ a + r ρ m tanh [ w ( θ − θ )]1 + r tanh [ w ( θ − θ )] , (5)where r and w are the same as for dark solitons. FIG. 1. (Color online) Dark-antidark pair sitting on the unitbackground ρ = 1, for α = 0 . v = 0 .
3: (a) potential V ( ρ ) − E vs. ρ ; (b) phase-space picture (contour lines of H ); (c-d)Relative intensity (thick black solid line) and phase (thin redsolid line) profiles of dark (c) and antidark (d) solitons. The expressions in Eqs. (4)-(5) allow us to obtain,by integrating Eq. (3), the nonlinear phase associated tothe two soliton families. We obtain for dark and antidarksolitons, respectively φ d = − vw s tan − (cid:18)r rρ a ρ m tanh[ w ( θ − θ )] (cid:19) + φ , (6) φ a = vw s tan − (cid:18)r ρ m rρ a tanh[ w ( θ − θ )] (cid:19) + φ , (7)where s ≡ q ( ρ a − ρ )( ρ − ρ m ) ρ a ρ m . From Eqs. (6-7) one caneasily calculate the phase jump ∆Φ = Φ(+ ∞ ) − Φ( −∞ )across the soliton.Interestingly, the existence domain of the soliton pairscan be described with the aid of catastrophe theory −1 −0.5 0 0.5 1−2.5−2−1.5−1−0.50 b a ρ =1, v=0 ρ =1, v=0.5 ρ =3, v=0 ρ =3, v=0.5 ρ =4, v=0 ρ =1, v=0.7 FIG. 2. (Color online) Cusp catastrophe picture for dark-antidark soliton pairs. Each curve shows the evolution ofthe parameter a and b of the normal form potential V ( y ) = y / ay / by , calculated for a dark-antidark soliton pairwith fixed internal parameters v and ρ , and α changing fromzero up to its critical value α c , where all the curves arrivetangentially on the cusp curve (Eq. (8), red solid line) thatbounds the soliton existence domain.FIG. 3. (Color online) Potential and phase-space picture for v = 0, ρ = 1, and different values of α : (a-b) α = 0 (idealKerr case); (c-d) α = 0 .
5; (e-f) α = 0 . α c = 1, where the right branch of the separatrix becomesvanishingly small). [24], already applied to characterize dark-antidark soli-ton pairs in a different context (gap soliton theory [25]).In fact, the quartic potential V in Eqs. (2) belongsto the A + family, and gives rise to the so called cusp catastrophe. According to this picture the potential V ( ρ ) in Eq. (2), which is of the general form V ( ρ ) = c ρ + c ρ + c ρ + c ρ , can be cast into the canonicalform [24] V ( y ) = y / ay / by , by means of thechange of variable ρ = (4 c ) − / y − c / (4 c ). Then, inthe control parameter plane ( a, b ) (explicit expressionsof a and b as a function of α, ρ , v are cumbersome butcan be easily derived), solitons exist in the inner region bounded by the curve (so called bifurcation set [24]) (cid:16) a (cid:17) + (cid:18) b (cid:19) = 0 , (8)shown in Fig. 2. Such curve marks the values where thecritical points ( ∂ y V = 0) of the potential become doublydegenerate ( ∂ y V = 0), and exhibits the characteristicshape of a cusp in the origin (three-fold degenerate point, ∂ y V = 0). In terms of the original parameter ρ , v, α , theexistence condition requires α ≤ α c , with the followingcritical value of the quintic coefficient α c : α c = 1 ρ (cid:18) − v ρ (cid:19) . (9)Taking fixed internal parameters ρ and v , while chang-ing α continuosly, makes the control parameters a and b calculated for the soliton to span a smooth curve in thecontrol parameter plane ( a, b ), until at α = α c , the curvehits (arriving tangentially) the boundary set by the cuspcurve [Eq. (8)]. Different values of ρ and v result intodifferent control curves, as displayed in Fig. 2. We pointout that a similar behavior occurs by varying ρ or v ,keeping the other two parameters fixed. In particular,in the latter case, the existence domain turns out to be − v c ≤ v ≤ v c , with the cut-off velocity v c = p ρ − αρ ,obtained by expressing Eq. (9) in terms of v = v c . d i p i n t en s i t y velocity v (a) α =00.30.60.80.9 0 0.5 110 pea k i n t en s i t y velocity v (b) α =0.010.10.30.60.9 FIG. 4. (Color online) (a) dip intensity (darkness) of darksolitons and (b) maximum intensity of antidark solitons as afunction of velocity v for different values of α . Here the back-ground is ρ = 1. The dashed curves represent the existencethreshold set by Eq. (9). Note the vertical log scale in (b). As an example, we show in Fig. 3 how the typicalphase plane (potential) changes when α is varied be-tween zero and the critical value α c . In this case wechoose ρ = 1 and still solitons, viz. v = 0, yielding ρ m = 0 which means that the dark soliton is black re-gardless of the value of α , while the antidark is char-acterized by a peak intensity ρ a = 3 /α − ρ . In thelimit α = 0 shown in Fig. 3(a,b), which represents theideal Kerr case, the potential is cubic, and the separa-trix has only one branch corresponding to the well-knownblack soliton solution ( ρ = tanh ( x )) of the NLS equa-tion, whereas for ρ > ρ > V ( θ = ∞ ) → −∞ )] and wide (since ρ a → ∞ ).Viceversa, as α grows from zero, the behavior of the po-tential at θ = ∞ is inverted, and the right well becomesfinite, allowing for a eight-shaped separatrix correspond-ing to the dark-antidark pair [see Fig. 3(c-d)]. For smallvalues of α the antidark soliton has high peak intensity ρ a above the background ρ , which, however, decreases asthe saturation parameter α increases. For α approachingits critical value α c antidark solitons become shallow [seeFig. 3(e-f)], until they reduce to the plane wave exactlyat α = α c , where ρ a → ρ . The behavior of dark-antidark FIG. 5. (Color online) Color level plot of FWHM [ θ F WHM inEq. (10)] in the parameter plane ( α, ρ ) for v = 0 .
1, dark (a)and antidark (b) solitons. (c,d) dark solitons with v = 0 . v = 0 . α c ( ρ , v ). solitons with velocity exhibits intriguing features, whichcan be gathered by plotting the minimum (dip) intensity ρ m of dark solitons (the larger ρ m , the lower the dark-ness) and the peak intensity ρ a of antidark solitons (thehigher ρ a , the brighter the antidark) versus v at constant ρ , for different values of α , as displayed in Fig. 4. Fornon-zero but small velocities, the picture remains quali-tatively unchanged with respect to the case v = 0, in thesense that the peak intensity ρ a of antidark solitons de-creases continuously from infinity (at α = 0) to ρ a = ρ at the critical value α c , such that the solitons becomeinfinitely shallow (i.e. they reduce to a pure plane wave).In this case, however, the dip intensity of dark solitonsis no longer zero, i.e. they becomes gray solitons withdarkness ρ − ρ m . The darkness decreases for growingvelocities v up to a minimum value at the bound veloc-ity v c = p ρ − αρ (obtained by solving Eq. (9) withrespect to v for fixed α ). This is clearly shown in Fig.4, where we summarize the result for a fixed background ρ = 1. Note from Fig. 4(a) that Kerr (NLS) dark soli-tons ( α = 0) are always darker than the corresponding CQNLS solitons of the same velocity (the curve ρ m ( v )for α = 0 is always below the other curves relative to α = 0), and have also a larger phase jump ∆Φ than theirCQNLS counterparts.Interestingly, however, as the velocity grows largeenough (above v = 0 . ρ a → ρ and hence dark-ness tends to zero). Conversely, under the same condi-tions, antidark solitons cease to become infinitely shal-low, rather reaching a finite minimum peak intensity atthe cut-off condition for their existence [see Fig. 4(b)].From Fig. 4(b) it is also clear that, for small α the bright-ness of antidark solitons is nearly independent on thevelocity. This change of behavior at the cut-off condi-tion for the existence (from infinitely shallow antidark toinfinitely shallow dark solitons) depends the background ρ . It can be shown to occur at the value of α = 3 / (4 ρ ),in correspondence of the velocity v = √ ρ / α = 0 .
75 and v = 0 . ρ and v ) and dif-ferent renormalized invariants M, H, P (see Appendix Afor their definition). However these soliton families canbe bistable also according to the definition by Gatz andHerrmann, i.e. for fixed α different solutions of the samewidth can exist although they sit on a different back-ground ρ . This type of bistability was investigated forstill ( v = 0) dark solitons [7]. In order to generalize thisresult to the full family (any v , and antidark case), wehave calculated the FWHM as θ F W HM = 2 w tanh − p f ( ρ , ρ m , ρ a ) , (10)where f ( ρ , ρ m , ρ a ) = ρ a − ρ ρ a − ρ − ρ m for dark solitons(FWHM taken at half the intensity between ρ and thedip ρ m ) and f ( ρ , ρ m , ρ a ) = ρ − ρ m ρ a + ρ − ρ m for antidark soli-tons (FWHM at half the intensity between ρ and thepeak ρ a ), respectively. The results obtained by mapping θ F W HM in the plane ( α, ρ ) are summarized in Fig. 5.As shown, dark solitons exhibit bistability for sufficientlylarge α , regardless of their velocity. However the rangeof values of α where bistability occurs is greatly reducedfor large velocities [see Fig. 5(c)]. Conversely, as shownin Fig. 5(d), antidark solitons are never bistable in thesame sense (no folding of the level curves of θ F W HM isever observed at any velocity).
III. INSTABILITY SCENARIOS
Having characterized the features of dark-antidark soli-ton pairs, we discuss their stability. We proceed by ap-plying a known stability criterium according to which thestability is related to the derivative of the invariant mo-mentum M (see Appendix A) of the soliton against its m o m en t u m M (a)v c darkantidark 0 0.5 1 1.5 200.20.40.60.81 H a m il t on i an H momentum M (b)darkantidark FIG. 6. (Color online) (a) Renormalized momentum M vs.soliton velocity v , and (b) Hamiltonian H ( v ) vs. M ( v ), fordark and antidark solitons with fixed ρ = 1 and α = 0 . ∂ v M = 0 forantidark solitons. c oe ff i c i en t α (a) ρ =1 α α m c oe ff i c i en t α (b) ρ =2 α α m FIG. 7. (Color online) Stability maps for antidark solitonsin the parameter plane ( v, α ) for (a) ρ = 1; (a) ρ = 2.The light (cyan) and dark (red) shaded domains correspondto unstable and stable solutions, respectively. The dot-dashedline sets the value α below which antidark solitons have finitebrightness at cut-off. The dynamics illustrated in Fig. 8 andFigs. 9-10 are relative to the sampled values marked by bulletsand stars in (a), respectively. velocity v . The marginal condition ∂M∂v = 0 , (11)separates stable solutions ( ∂ v M <
0) from unstable ones( ∂ v M > M ( v ) for dark solitonfamily shows that such function has always a negative slope. Therefore the whole family of dark solitons is sta-ble in its existence domain, a conclusion that is fully sup-ported by our numerical simulation of the propagation.Viceversa instabilities take place for antidark solitons.In particular, for small α , it turns out that they are al-ways unstable in the whole domain of existence since M ( v ) exhibits always positive slope. Conversely at suf-ficiently large α , the momentum M ( v ) changes its slopenear the cut-off value for existence v c . This is shown inFig. 6, where we compare the momentum M ( v ) for darkand antidark solitons at α = 0 . ρ = 1. Note thatthe change of slope of the momentum means that twoantidark solutions with same momentum and differentvelocity exist. These two solutions differ by their Hamil-tonian H (see Appendix for its expression). Indeed, asshown in Fig. 6(b) the Hamiltonian H as a function of M is folded, showing an upper (unstable) branch and alower (stable) branch. FIG. 8. (Color online) Dynamics of antidark solitons with ρ = 1, α = 0 .
7, exhibiting different behavior depending onthe initial velocity v [see bullets in Fig. 7(a)]: (a) v = 0 . v = 0 .
4, decay into a nearby stableantidark soliton; (c) v = 0, decay into a pair of antidarkshallow solitons with opposite velocities. The velocity which gives the marginal stability ( ∂ v M =0) clearly depends on ρ and α . The results of Fig. 6could be repeated for different values of ρ and α , andsummarized by drawing stability maps in the plane v, α at constant ρ . These are displayed in Fig. 7 for twodifferent values of the background ρ . As shown a rela-tively small island of stability is found in the vicinity ofthe boundary for existence, where they have small bright- FIG. 9. (Color online) Decay of an unstable antidark solitons( ρ = 1, v = 0, α = 0 .
1) into two symmetric dispersive shockfans (trains of dark solitons): (a) color level plot of the in-tensity; (b) snapshot of intensity (black solid line) and phase(red solid line) at z = 5. For comparison the input intensity(renormalized to the maximum of the plot) is shown (solidblue line).FIG. 10. (Color online) As in Fig. 7 for v = 0 . ness. In particular stability requires α > α m , where α m corresponds to the vertex of the stability island [see Fig.7]. In order to test the validity of the marginal stabilitycondition we have made extensive simulations performedby means of the well-known split-step method. Such sim-ulations confirm indeed that ∂ v M = 0 gives the thresholdfor stability also for antidark solitons. The numerics alsoreveal two basic mechanisms of instability, which are il-lustrated below by means of numerical runs performedby launching the exact soliton profile, while the pertur- bation arises from intrinsic roundoff and discretizationerrors.The first scenario is valid for relatively large α , suchthat a stable range of velocities exists. As an exam-ple for illustration purpose, we have chosen α = 0 . ρ = 1, which yield a range of stable velocities v =(0 . , . v , marked by bullets in the map of Fig.7(a), is displayed in Fig. 8. As shown in Fig. 8(a), for v = 0 . v is decreased just below the thresholdfor instability [ v = 0 . v = 0 in such a way that theinitial zero momentum is conserved, otherwise they ap-pear to be asymmetric (case not shown). The scenarioillustrated in Fig. 8 holds also for different values of α ,providing α > α m , i.e. a stable range of velocities ex-ists. It is interesting to note that the value ρ = 3 / (4 ρ ),which discriminates the fact that antidark solitons canbe or not infinitely shallow at cut-off, lies in the regionof stability for any ρ (see dot-dashed lines in Fig. 7).Therefore antidark solitons with arbitrarily small bright-ness that exist for α > α around cut-off, are alwaysstable, whereas antidark solitons which have finite bright-ness at cut-off can propagate stably only for α > α m .The scenario discussed above change qualitativelywhen α < α m , where the island of stability shrinks tozero. In this case the decay instability towards other an-tidark solitons is forbidden because no stable solutionsexist. In this case, the only stable solutions are dark soli-tons and therefore the decay instability occurs towardsthese solutions, even though their shape differ dramati-cally (indeed being ”opposite”) from the input antidarkshape. Importantly when α is small (weak saturation)the antidark solitons possess large amplitude and largepower P a (or number of particles, see Appendix for itsdefinition). Under these conditions we have found thatthe decay instability of the antidark leads to a DSW(see Refs. [20–23] and references therein), i.e. an ex-panding region filled with fast oscillations which behaveasympotically as solitons. In the present case, startingwith a zero-velocity antidark soliton, the decay instabil-ity leads to two symmetric DSW fans, as displayed inFig. 9 for α = 0 .
1. In each of the two fans, the inneredge is set by the darkest and slowest soliton, whereason the outer edge the fan is linked to the plane wavethrough a train of solitons with progressively decreasingdarkness and increasing velocity, which become denser asthe background is approached. Although this behavior isreminiscent of that ruled by the integrable NLS equation(limit α = 0) in the semiclassical regime (i.e., nonlin-earity much stronger than diffraction/dispersion) underexcitation of e.g. a gaussian on pedestal [21], it mustbe emphasized that this is, to the best of our knowl-edge, the first example where the dispersive shock wavesoccurs directly through the decay of an unstable solitarywave of the system. Further characterization of the shockfan need to develop the Whitham modulational theoryfor the system, which is beyond the scope of this paper.However, simple arguments based on the features of soli-tons as shown in Fig. 4, let us predict that the fan ruledby Eq. (1) is narrower as compared to the one ruled bythe integrable NLS equation under the same excitation.This is due to the fact that the soliton with vanishinglysmall darkness, constituting the outer edge of the fan,correspond to progressively reduced velocity as the quin-tic nonlinearity grows (i.e., α increases), as clearly shownin Fig. 4(a).The decay scenario shown in Fig. 9 does not changewhen starting with an antidark soliton with non-zero ve-locity ( v, M = 0), except that the two shock fans becomeasymmetric (the higher the velocity, the higher the asym-metry), in both the number of solitons and the velocityof the darkest soliton (inner edge of the fan), as shownin Fig. 10. This asymmetry is definitely expected, basedon the fact that a symmetric configuration would have M = 0 and hence would lead to violation of momentumconservation. Moreover the asymmetric development ofthe shock is analogous to DSW generated in the inte-grable NLS limit when starting from a gray beam withnon-zero velocity and momentum [23].Finally we point out that, for larger α , yet with α < α c ,the scenario remains qualitatively unchanged though thenumber of dark solitons generated in the decay of theantidark soliton decreases.Though the aim of this paper was the full charac-terization of solitary waves sitting on a finite back-ground, before concluding, we point out that such soli-tons can also coexist with bright solitary waves withzero pedestal. Seeking for bright solitons of the form u ( x, z ) = p ρ ( x ) exp( iβz ), a simple calculation showsthat the peak intensity of these solutions turns out tobe ρ a = 3(1 + p βα/ / (2 α ). The existence domainof such solutions include arbitrarily small α , and hencethey coexist with dark-antidark pairs. Given the defo-cusing nature of the leading order (cubic) nonlinearitythis might appear surprising. However, it is not difficultto understand that such bright solitons are sustained en-tirely by the quintic focusing nonlinearity since ρ a di-verges in the limit α →
0, where the quintic nonlinearityvanishes indeed. Without deepening the study of thebright case, already investigated in the framework of theCQNLS equation, we expect them to be unstable at leastin the limit of small α , where they do not affect the decaydynamics of antidark solitons or the stable dynamics ofdark solitons discussed above. IV. CONCLUSIONS
In summary we have discussed the main properties ofsolitary solutions with finite background of the CQNLSequation with focusing quintic term. Bistability and in-stabilities have been studied for the full family of solu-tions obtained in new analytical form, and parametrizedby the intensity of the background wave and the velocity.We have found that the solutions exhibit a non-trivialbehavior against the velocity such that, depending onthe value of the quintic nonlinearity, either dark or an-tidark solitons, become infinitely shallow at their boundfor existence. Furthermore dark solitons are bistable forany velocity, whereas antidark are never bistable. Finallydark solitons are stable against weak perturbations whileantidark are mostly unstable, exhibiting different mecha-nisms of instability. In particular, a novel mechanism in-volving the decay of an antidark soliton into a dispersiveshock wave has been characterized. Further work will bedevoted to assess how the quintic nonlinearity affects theformation and dynamics of dispersive shock waves whichdevelop from more general (non-solitary) inputs. Poten-tially also the collapse, i.e. blow-up at a finite distance(which is known to occur even in the 1+1D case that wedealt with, owing to the high order of the focusing term[26, 27]) could play a substantial role that needs furtherinvestigation.From the experimental point of view, the most naturalsetting for testing these results is the study of parax-ial beam evolution in nonlinear optics of centrosymmet-ric media, where the quintic term accounts for the sat-uration of the Kerr nonlinear index n ( n <
0, defo-cusing media), usually quantified in terms of the high-order nonlinear index n [7], the overall index changebeing ∆ n = −| n | I + n I , where I is the optical in-tensity. While the nonlinear indexes n , n (or equiv-alently the nonlinear susceptibilities) depends solely onmaterial properties and can be accurately characterizedby means of consolidated techniques [3, 4], the normal-ized coefficient α used throughout the paper turns out todepend on the input intensity as well, and hence the im-pact of the quintic nonlinearity can be tuned by changingthe optical power [17]. An other area where the predic-tions based on the present model can be relevant andcan lead to experimental test, is the dynamics of ultra-cold atoms (Bose-Einstein condensates), where the quin-tic term arises from higher-order (three-body) atom in-teractions [28], and tuning of the nonlinearities can beachieved by means of Feshback resonances. V. APPENDIX A
Analytical solutions can be worked out from the equa-tion which follows directly from Eq. (2)˙ ρ = r α p Q ( ρ ) , (12)where Q ( ρ ) = α [ E − V ( ρ )]. The polynomial Q ( ρ ) canbe expressed in terms of its ordered roots ρ m , ρ (doubleroot), ρ a , with ordering ρ m ≤ ρ ≤ ρ a , as Q ( ρ ) = ( ρ − ρ a )( ρ − ρ m )( ρ − ρ ) . From Eq. (12) one obtains thefollowing quadrature integral, Z ρ ( θ ) ρ ( θ ) dρ p Q ( ρ ) = r α Z θθ dθ, (13)where ρ ( θ ) = ρ m or ρ ( θ ) = ρ a for dark or antidarksolitons, respectively. The integral (13) gives, upon in-version, the solutions given in the text. Explicitly, theextremal roots ρ m , ρ a are expressed by: ρ m = 3 − αρ − p (3 − αρ ) − v α α ,ρ a = 3 − αρ + p (3 − αρ ) − v α α . (14)Note that, in the limit v = 0, the smaller root ρ m van-ishes, and as a consequence the dark soliton becomes ablack soliton. Once the soliton solutions are known one can easily cal-culate the renormalized invariants (momentum, Hamilto-nian) which are defined as follows M = i Z + ∞−∞ ( u ∗ x u − u x u ∗ ) (cid:18) − ρ ρ (cid:19) dx, (15) H = Z + ∞−∞ " | u x | Z ρ ( x ) ρ [ g ( ρ ) − g ( ρ )] dρ dx, (16)and the power or number of particles, P a (antidark) and P d (dark): P a = Z + ∞−∞ | u | − ρ dx ; P d = Z + ∞−∞ ρ − | u | dx (17)The quantities M and H are those employed in the textto assess the stability of soliton solutions. VI. ACKNOWLEDGEMENTS
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