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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 2657–2667
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Solitons supported by spatially inhomogeneous nonlinear losses

Olga V. Borovkova, Yaroslav V. Kartashov, Victor A. Vysloukh, Valery E. Lobanov, Boris A. Malomed, and Lluis Torner  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 2657-2667 (2012)
http://dx.doi.org/10.1364/OE.20.002657


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Abstract

We uncover that, in contrast to the common belief, stable dissipative solitons exist in media with uniform gain in the presence of nonuniform cubic losses, whose local strength grows with coordinate η (in one dimension) faster than |η| . The spatially-inhomogeneous absorption also supports new types of solitons, that do not exist in uniform dissipative media. In particular, single-well absorption profiles give rise to spontaneous symmetry breaking of fundamental solitons in the presence of uniform focusing nonlinearity, while stable dipoles are supported by double-well absorption landscapes. Dipole solitons also feature symmetry breaking, but under defocusing nonlinearity.

© 2012 OSA

1. Introduction

Spatial solitons exist in a variety of physical settings and nonlinear media. In particular, they appear as self-trapped light beams in optical waveguides. The standard knowledge is that self-focusing and self-defocusing nonlinearities in uniform media give rise to bright and dark solitons, respectively [1

1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).

]. Breaking this rule requires suitable non-uniform media. In particular, periodic transverse modulations of the refractive index support bright gap solitons even in defocusing media [2

2. V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B 18(14), 627 (2004). [CrossRef]

,3

3. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78(1), 179–215 (2006). [CrossRef]

]. Solitons with novel properties have been also predicted and demonstrated in optical and matter-wave systems with periodic modulations of the nonlinearity along the longitudinal or transverse directions [4

4. L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11(10), 1268–1270 (1999). [CrossRef]

10

10. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011). [CrossRef]

]. Still, while nonlinear lattices support stable one-dimensional solitons in focusing media [11

11. H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005). [CrossRef] [PubMed]

22

22. H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010). [CrossRef]

], a common belief is that the existence of bright solitons requires the presence of a linear lattice in the case of defocusing nonlinearities [2

2. V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B 18(14), 627 (2004). [CrossRef]

,3

3. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78(1), 179–215 (2006). [CrossRef]

,23

23. V. Pérez-García and R. Pardo, “Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates,” Physica D 238(15), 1352–1361 (2009). [CrossRef]

]. A linear lattice potential is necessary too for the formation of multipole solitons [24

24. B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003). [CrossRef]

28

28. Y. J. He and H. Z. Wang, “(1+1)-dimensional dipole solitons supported by optical lattice,” Opt. Express 14(21), 9832–9837 (2006). [CrossRef] [PubMed]

], as they do not exist in uniform focusing media. Nevertheless, it was shown recently that stable bright fundamental and higher-order solitons do exist in the absence of any linear potential in one- and multi-dimensional conservative media with a defocusing nonlinearity whose strength grows toward the periphery of the medium faster than rD, where r is the radial coordinate and D the spatial dimension [29

29. O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. 36(16), 3088–3090 (2011). [CrossRef] [PubMed]

31

31. Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. 36(23), 4587–4589 (2011). [CrossRef] [PubMed]

]. On mathematical grounds, the key factor explaining the existence of solitons in such a counterintuitive setting is the non-linearizability of the underlying equations for the decaying soliton tails (the formal linearization would predict, as usual, that bright solitons do not exist in this setting). A similar mechanism explains the existence of bright solitons in a more exotic model, combining a uniform self-defocusing nonlinearity and a spatially-dependent diffraction coefficient vanishing at r [32

32. W.-P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84(4), 043801 (2011). [CrossRef]

].

Solitons exist in dissipative media too. Dissipative solitons have drawn considerable attention in various physical systems in both one- and multi-dimensional settings. Stability is a fundamental issue for such solitons, because losses in the medium must be compensated by gain. However, a spatially uniform gain destabilizes any localized wavepacket by making the background around it unstable. Two solutions to this problem have been elaborated: The use of a nonlinear gain and higher-order stabilizing absorption, such as in the physical settings governed by the complex cubic-quintic Ginzburg-Landau equations [33

33. B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D 29(1-2), 155–172 (1987). [CrossRef]

38

38. P. Marcq, H. Chate, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation,” Physica D 73(4), 305–317 (1994). [CrossRef]

], or various schemes with localized linear gain [39

39. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Spec. Top. 173(1), 233–243 (2009). [CrossRef]

48

48. C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(6), 066609 (2011). [CrossRef]

], as well as with a localized cubic gain, in the absence of higher-order losses [49

49. O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, “Stable nonlinear amplification of solitons without gain saturation,” Europhys. Lett. in press.

]. However, the important question whether spatial shaping of absorption can be used for generation and stabilization of new types of dissipative solitons was not addressed yet.

In this work we show that, contrary to the above-mentioned belief, stable dissipative fundamental and higher-order solitons can exist in media with the uniform gain, provided that the gain is combined with non-uniform nonlinear absorption, whose local strength grows rapidly enough toward the periphery of the system.

2. The model

We describe the propagation of light in such a one-dimensional medium by the nonlinear Schrödinger/Ginzburg-Landau equation for the dimensionless field amplitude q:
iqξ=122qη2+σrq|q|2+ipiqiγ(η)q|q|2,
(1)
where ξ and η are the propagation distance and transverse coordinate, respectively, pi>0 is the strength of the uniform linear gain, and γ(η)>0 describes the spatial profile of the nonlinear absorption. We consider focusing, zero, and defocusing Kerr-type nonlinearities, that correspond to σr=1,0,+1, respectively. Solitons are sought for as q(η,ξ)=w(η)exp(ibξ), where b is the propagation constant, and w(η)=wr(η)+iwi(η) is a complex function satisfying the stationary equation,

(b+ipi)w=(1/2)d2w/dη2+[σriγ(η)]w|w|2.
(2)

This model can be realized in optics, by combining a spatially uniform gain and inhomogeneous doping with two-photon-absorbing elements. Experimental techniques allowing the creation of appropriate nonuniform dopant concentration profiles have been elaborated in another context [50

50. J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D Appl. Phys. 36(3), R1–R16 (2003). [CrossRef]

]. Another possibility, proposed in Refs [29

29. O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. 36(16), 3088–3090 (2011). [CrossRef] [PubMed]

31

31. Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. 36(23), 4587–4589 (2011). [CrossRef] [PubMed]

], is to induce an effective spatial modulation of the nonlinearity by applying an external nonuniform field which affects the local detuning of uniformly distributed dopants.

We start with the consideration of the simplest setting where the strength of the nonlinear absorption in Eqs. (1) and (2) grows towards the periphery as
γ(η)=σiexp(αη2),
(3)
and γ(η) has a single minimum at the center (a single-well absorption profile). By means of rescaling we fix α=1 and keep the strength of the two-photon absorption, σi, as a free parameter. First, to confirm the existence of localized solitons in this setting, it is relevant to mention that Eqs. (2) and (3) with σr=0 and pi=3/2 admit an exact dipole-soliton solution,
w(η)=σi1/2ηexp[(1i)η2/2],
(4)
with b=3/2. While this solution is unstable (see below), the nonlinearity-modulation profile in Eq. (2) with γ(η)=σiη2exp(η2), instead of (3), gives rise, at pi=1/2, to a stable exact fundamental soliton solution,
w(η)=σi1/2exp[(1i)η2/2],
(5)
with propagation constant b=1/2. These examples demonstrate the above-mentioned crucially important peculiarity of this model, namely, the nonlinearizability of Eqs. (1) and (2) due to the growth of the nonlinear absorption at |η| [i.e., the asymptotic form of stationary solutions cannot be found using the formal linearization of Eq. (2)]. This property explains why the uniform linear gain does not imply the instability of localized solutions in this setting.

3. Numerical results

Systematic numerical solutions of Eqs. (2),(3) reveal that the single-well absorption landscape gives rise to fundamental solitons with symmetric and asymmetric profiles, i.e., to the effect of the spontaneous symmetry breaking (SSB), if the conservative part of cubic nonlinearity is focusing (σr=1). Surprisingly, bright solitons in this setting may exist even in the case of zero or defocusing conservative nonlinearities (σr=0,+1). In that case, the solitons always feature symmetric shapes.

Figure 1
Fig. 1 (Color online) Typical profiles of fundamental symmetric (a) and asymmetric (b) solitons in the focusing medium, and of the symmetric one in the defocusing medium (c) with the single-well nonlinear-absorption profile (3) [red curves for γ(η)], at pi=1.5, σi=0.5.
shows generic examples of stationary solitons in the focusing and defocusing media with the single-well nonlinear-absorption profile, depicted by the red curves. In the case of the focusing nonlinearity, the family of asymmetric solitons bifurcates from the symmetric one with the increase of the linear gain, pi, at the SSB (critical) point, pi=picr1. The bifurcation is presented in Figs. 2(a)
Fig. 2 (Color online) The energy flow (a) and propagation constant (b) versus pi for symmetric (“fs”) and asymmetric (“fa”) fundamental solitons in the focusing medium with the single-well nonlinear-absorption profile (3) at σi=0.5. (c), (d): The same for dipole solitons in the double-well nonlinear-absorption landscape (6) at σi=2.0. Subscripts “fs”, “ds”, and “da” denote symmetric dipoles in the focusing and defocusing media, and asymmetric dipoles in the defocusing medium, respectively. Circles in (a),(b) correspond to the solitons in Figs. 1(a) and 1(b), while circles in (c),(d) pertain to the solitons in Figs. 4(a) and (b), respectively.
and 2(b), where the energy flow
U=+|q(η)|2dη,
and propagation constant b are shown versus the gain parameter, pi.

While the SSB is a common feature of double-well systems [51

51. G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A 55(6), 4318–4324 (1997). [CrossRef]

54

54. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction,” Phys. Rev. Lett. 95(1), 010402 (2005). [CrossRef] [PubMed]

], including dissipative ones [55

55. A. Sigler and B. A. Malomed, “Solitary pulses in linearly coupled cubic-quintic Ginzburg-Landau equations,” Physica D 212(3-4), 305–316 (2005). [CrossRef]

], it seems more surprising in the present setting, based on the single-well absorption landscape. In this connection, it is worthy to mention that a similar SSB of solitons was recently reported in a model combining the self-focusing nonlinearity, uniform cubic loss, and a localized region of linear gain [45

45. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A 83(4), 041806 (2011). [CrossRef]

]. Another noteworthy feature of the present setting is that, even in the case of the focusing nonlinearity, the propagation constant b can be negative, while focusing nonlinearity in uniform settings gives rise to solitons with b>0 (bright solitons with b<0 are possible in media featuring a localized linear gain against the background of uniform linear loss [44

44. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Solitons in a medium with linear dissipation and localized gain,” Opt. Lett. 36(7), 1200–1202 (2011). [CrossRef] [PubMed]

]). Actually, symmetric solitons in Fig. 2(b) feature negative b only at small values of the gain, while the asymmetric branch plunges into the region of b<0 with the increase of pi. The above-mentioned nonlinearizability of the underlying Eq. (1) is a fundamental reason for the existence of the solitons with b<0 [29

29. O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. 36(16), 3088–3090 (2011). [CrossRef] [PubMed]

31

31. Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. 36(23), 4587–4589 (2011). [CrossRef] [PubMed]

].

The dependence of the critical value of gain, at which the SSB of the fundamental solitons occurs, on the strength of the nonlinear absorption is shown in Fig. 3(a)
Fig. 3 (Color online) Existence and stability domains in the (σi,pi) plane. (a) The fundamental solitons in the focusing medium with the single-well nonlinear absorption profile (3) (the asymmetric solitons exist and are stable at pi>picr1). (b) and (c): Dipole solitons in the double-well absorption profile (6), in the focusing and defocusing media, respectively.
. The asymmetric solitons, which are always stable, exist above this curve, while the symmetric ones are found in the entire (σi,pi) plane, but they are stable only beneath the curve, being unstable where they coexist with the asymmetric solitons (the stability of solitons was inspected by calculating the respective eigenvalues for small perturbations). The latter property complies with the fact that the bifurcation observed in Figs. 2(a) and 2(b) is of the supercritical type (in other words, it is a phase transition of the second kind; note that the above-mentioned SSB bifurcation of solitons reported in Ref [45

45. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A 83(4), 041806 (2011). [CrossRef]

]. is also of the supercritical type).

The family of symmetric solitons that exists in the medium with the defocusing or zero conservative part of nonlinearity is completely stable, featuring no SSB. The energy flow of such modes increases almost linearly with the gain, while the propagation constant decreases, always staying negative.

The model with single-well absorption landscape can support higher-order (multipole) solitons too, cf. Equation (4). This is an indication of the important fact that the spatial shaping of nonlinear losses can be used for generation of new types of solitons that do not exist in uniform dissipative systems. However, such higher-order solitons in single-well landscapes always turn out to be unstable, irrespectively of the sign of the conservative nonlinearity. The instability grows with the increase of the gain, usually transforming the multipole solitons into stable fundamental ones.

Nevertheless, stable multipoles (e.g., dipoles) can be found in the system with a double-well nonlinear-absorption profile, described, e.g., by function
γ(η)=σi(η2η02)2exp(αη2),
(6)
in Eqs. (1) and (2), see examples in Fig. 4
Fig. 4 (Color online) Examples of symmetric and asymmetric dipole solitons at pi=3.0 (a) and pi=1.9 (b), respectively, in the defocusing medium with the double-well nonlinear-absorption landscape (5) [red curves for γ(η)], at σi=2.0.
. Generic results can be adequately demonstrated for η0=1. We have found that, in the focusing medium with the uniform gain and double-well profile of the nonlinear absorption, all dipole solitons are symmetric (unlike the fundamental solitons in the single-well absorption landscapes, they do not feature the SSB). The energy flow and propagation constant of the dipoles in the focusing medium are shown in Figs. 2(c) and 2(d), by curves Ufs and bfs, as functions of the gain. The dipoles do not exist above the upper threshold value of the gain, piupp, at which tangential lines to the U(pi) and b(pi) curves become vertical [see Fig. 3(b) for domains of stability and existence of such solitons in the (σi,pi) plane]. The dipole solitons are stable within a limited interval of gain parameters, picr1<pi<picr2 (note the difference from the symmetric fundamental solitons, which have no existence boundaries, and are completely stable in the absence of the SSB).

Various scenarios of the evolution of perturbed dipoles in the focusing medium are displayed in Figs. 5(a)
Fig. 5 (Color online) The dynamics of dipole solitons in the focusing (a)-(c) and defocusing (d)-(f) media with the double-well nonlinear-loss profile (5). (a) Spontaneous transformation of an unstable symmetric dipole into a fundamental soliton at pi=0.8. (b) Stable propagation of the symmetric dipole at pi=1.8. (c) The transformation of an unstable symmetric dipole into breather at pi=3.2. (d,e) The stable propagation of the symmetric and asymmetric dipoles at pi=1.5 and pi=2.1, respectively. (f) Spontaneous transformation of an unstable asymmetric dipole into a breather, with the dynamically restored symmetry, at pi=2.23. All the cases are shown for σi=3, with small noise added to the input.
-5(c). At small values of the gain, pi<picr1, the unstable dipoles usually evolve into symmetric fundamental solitons [Fig. 5(a)]. Stable propagation of the dipole soliton is demonstrated in Fig. 5(b) for picr1<pi<picr2. At larger gain levels, pi>picr2, the dipole solitons develop an oscillatory instability and may transform into persistent breathers, without symmetry breaking. The initial stage of this process is shown in Fig. 5(c), in which case the emerging breather propagates stably over an indefinitely long distance.

On the contrary to the fundamental solitons, the dipoles exhibit the SSB in the defocusing medium with the double-well absorption landscape. This finding resembles a well-known fact that, in dual-core conservative systems, the SSB of spatially odd modes occurs under the action of defocusing nonlinearities (in contrast to the spatially even ground state, which undergoes the symmetry breaking under the action of focusing nonlinearities [51

51. G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A 55(6), 4318–4324 (1997). [CrossRef]

54

54. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction,” Phys. Rev. Lett. 95(1), 010402 (2005). [CrossRef] [PubMed]

]). Examples of such symmetric and asymmetric dipoles are displayed in Fig. 4, while the symmetry-breaking bifurcation isshown by the respective U(pi) and b(pi) curves in Figs. 2(c) and 2(d). The emerging asymmetric dipoles are stable, while the symmetric ones are unstable past the bifurcation point [in compliance with the supercritical character of the bifurcation in Figs. 2(c) and 2(d)]. As shown in Fig. 3(c), the symmetric dipoles in the two-well absorption landscapes, with the defocusing nonlinearity, are stable in the domain of picr1<pi<picr2, which also implies that they can be stable only if the nonlinear-absorption strength, σi, exceeds a certain minimal value. The upper stability border, pi=picr2, is actually determined by the symmetry-breaking bifurcation. The asymmetric dipoles that emerge above the picr2 curve are also stable only within a limited range of the gain parameter,picr2<pi<picr3. Figures 5(d)-5(f) illustrate the stable and unstable evolution of the dipoles in these areas. In particular, at pi>picr3, the asymmetric dipoles are subject to an oscillatory instability, which transforms them into breathers, leading to an effective dynamical restoration of the symmetry [Fig. 5(f)].

4. Analytical results

The numerical findings reported here can be explained in an analytical form by means of the Thomas-Fermi (TF) approximation, which, at the lowest order, neglects the diffraction term and conservative nonlinearity, i.e., all real terms, in Eq. (2) (the TF approximation was recently applied to a dissipative system with the combination of the localized linear gain and uniform linear loss in Ref [40

40. D. A. Zezyulin, G. L. Alfimov, and V. V. Konotop, “Nonlinear modes in a complex parabolic potential,” Phys. Rev. A 81(1), 013606 (2010). [CrossRef]

], but in an essentially more complex form than here). This approximation yields a real stationary wave function,

wTF(η)=(pi/σi)1/2exp(η2/2).
(7)

Strictly speaking, the TF approximation applies under the condition that σi/σr and pi are large parameters, but, in reality, the accuracy of the approximation is reasonable even when these quantities take values 1. In particular, the expression for the energy flow, UTF=π1/2pi/σi, following from Eq. (7), approximates the U(pi) dependence in Fig. 2(a) quite accurately, up to the bifurcation point: for σi=0.5, the slope of the linear dependence, given by this expression, is UTF/pi3.54, while its numerically found counterpart is 3.52.

Note that the steep anti-Gaussian modulation of the nonlinear absorption in Eq. (3) is not necessary for the existence of stable solitons in the presence of the uniform linear gain. Thus, Eq. (2) with exponential (rather than anti-Gaussian) absorption landscape, γ(η)=a+cosh2(η), gives rise to an exact chirped fundamental soliton,
q(η,ξ)=(3μ/2a)1/2exp(ibξ)sech1+iμ(η),μ=(3σr/2a)+sgn(a)[(3σr/2a)2+2]1/2,
with propagation constant b=(1μ2)/2 and pi=μ[1+(3/2a)]. This soliton resembles the well-known Pereira-Stenflo solution [56

56. L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R. Soc. Lond. A Math. Phys. Sci. 326(1566), 289–313 (1972). [CrossRef]

58

58. J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246(5), 412–422 (1998). [CrossRef]

], but is stable, unlike it. In this absorption landscape, an exact dipole soliton solution can be constructed too, q(η,ξ)=exp(5iξ/2)sech1i6(η)sinh(η), for parameters a=σr=0 and pi=(27/2)1/2.

Finally, using the TF approximation, it is easy to find that the mildest nonlinear-absorption profile supporting solutions with a convergent energy flow (norm) is γ(η)|η|1+ε with any ε>0. Furthermore, in the D-dimensional version of Eq. (2) the same is true for γ(r)|r|D+ε.

5. Conclusions

Our analysis has revealed that fundamental and multipole one-dimensional solitons, propagating in both focusing and defocusing nonlinear dissipative media with the uniform linear gain, can be stabilized with the help of an inhomogeneous nonlinear absorption, whose strength grows with the coordinate faster than |η|. Some soliton solutions were found in the exact analytical form. In the single-well absorption landscape, only fundamental solitons are stable, featuring the spontaneous symmetry breaking in the focusing medium. A double-well absorption landscape is required for the stabilization of dipoles, in which case the symmetry breaking occurs under the defocusing nonlinearity. Basic results for the symmetric fundamental solitons can be obtained in quite an accurate analytical form by means of the TF approximation.

It may be quite interesting to extend the analysis for two-dimensional solutions. In particular, the two-dimensional version of Eq. (2) with σr=0, pi=2, and the nonlinear-loss modulation function σiexp(r2) (here r and θ are the two-dimensional polar coordinates), gives rise to an exact solution for a solitary vortex, q=σi1/2exp[2iξ+iθ(1i)(r2/2)], cf. Equation (4). Further, for the modulation function σir2exp(r2)and σr=0, pi=1, an exact solution for a fundamental soliton is available, q=σi1/2exp[iξ(1i)(r2/2)], cf. Equation (5).

Acknowledgments

The work of O.V. Borovkova was supported by the Ministry of Science and Innovation of Spain, grant FIS2009-09928.

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2.

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D. L. Machacek, E. A. Foreman, Q. E. Hoq, P. G. Kevrekidis, A. Saxena, D. J. Frantzeskakis, and A. R. Bishop, “Statics and dynamics of an inhomogeneously nonlinear lattice,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036602 (2006). [CrossRef] [PubMed]

15.

Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. 31(19), 2888–2890 (2006). [CrossRef] [PubMed]

16.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007). [CrossRef] [PubMed]

17.

P. Niarchou, G. Theocharis, P. G. Kevrekidis, P. Schmelcher, and D. J. Frantzeskakis, “Soliton oscillations in collisionally inhomogeneous attractive Bose-Einstein condensates,” Phys. Rev. A 76(2), 023615 (2007). [CrossRef]

18.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008). [CrossRef] [PubMed]

19.

Y. Sivan, G. Fibich, and B. Ilan, “Drift instability and tunneling of lattice solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(4), 045601 (2008). [CrossRef] [PubMed]

20.

F. K. Abdullaev, A. Gammal, M. Salerno, and L. Tomio, “Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices,” Phys. Rev. A 77(2), 023615 (2008). [CrossRef]

21.

H. A. Cruz, V. A. Brazhnyi, and V. V. Konotop, “Partial delocalization of two-component condensates in optical lattices,” J. Phys. B 41(3), 035304 (2008). [CrossRef]

22.

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010). [CrossRef]

23.

V. Pérez-García and R. Pardo, “Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates,” Physica D 238(15), 1352–1361 (2009). [CrossRef]

24.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003). [CrossRef]

25.

D. Neshev, A. Sukhorukov, Y. Kivshar, E. Ostrovskaya, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]

26.

J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003). [CrossRef] [PubMed]

27.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12(13), 2831–2837 (2004). [CrossRef] [PubMed]

28.

Y. J. He and H. Z. Wang, “(1+1)-dimensional dipole solitons supported by optical lattice,” Opt. Express 14(21), 9832–9837 (2006). [CrossRef] [PubMed]

29.

O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. 36(16), 3088–3090 (2011). [CrossRef] [PubMed]

30.

O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, “Bright solitons from defocusing nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(3), 035602 (2011). [CrossRef] [PubMed]

31.

Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. 36(23), 4587–4589 (2011). [CrossRef] [PubMed]

32.

W.-P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84(4), 043801 (2011). [CrossRef]

33.

B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D 29(1-2), 155–172 (1987). [CrossRef]

34.

O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. France 49(11), 1829–1833 (1988). [CrossRef]

35.

W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64(7), 749–752 (1990). [CrossRef] [PubMed]

36.

V. Hakim, P. Jakobsen, and Y. Pomeau, “Fronts vs solitary waves in non equilibrium systems,” Europhys. Lett. 11(1), 19–24 (1990). [CrossRef]

37.

B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg-Landau equation,” Phys. Rev. A 42(10), 6009–6014 (1990). [CrossRef] [PubMed]

38.

P. Marcq, H. Chate, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation,” Physica D 73(4), 305–317 (1994). [CrossRef]

39.

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Spec. Top. 173(1), 233–243 (2009). [CrossRef]

40.

D. A. Zezyulin, G. L. Alfimov, and V. V. Konotop, “Nonlinear modes in a complex parabolic potential,” Phys. Rev. A 81(1), 013606 (2010). [CrossRef]

41.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59(1), 81–89 (2010). [CrossRef]

42.

F. Kh. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(5), 056606 (2010). [CrossRef] [PubMed]

43.

D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, “Dissipative double-well potential: nonlinear stationary and pulsating modes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(5), 056213 (2010). [CrossRef] [PubMed]

44.

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Solitons in a medium with linear dissipation and localized gain,” Opt. Lett. 36(7), 1200–1202 (2011). [CrossRef] [PubMed]

45.

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A 83(4), 041806 (2011). [CrossRef]

46.

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Dissipative surface solitons in periodic structures,” Europhys. Lett. 91(3), 34003 (2010). [CrossRef]

47.

V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable radially symmetric and azimuthally modulated vortex solitons supported by localized gain,” Opt. Lett. 36(1), 85–87 (2011). [CrossRef] [PubMed]

48.

C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(6), 066609 (2011). [CrossRef]

49.

O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, “Stable nonlinear amplification of solitons without gain saturation,” Europhys. Lett. in press.

50.

J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D Appl. Phys. 36(3), R1–R16 (2003). [CrossRef]

51.

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A 55(6), 4318–4324 (1997). [CrossRef]

52.

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, “Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates,” Phys. Rev. Lett. 79(25), 4950–4953 (1997). [CrossRef]

53.

S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, “Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping,” Phys. Rev. A 59(1), 620–633 (1999). [CrossRef]

54.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction,” Phys. Rev. Lett. 95(1), 010402 (2005). [CrossRef] [PubMed]

55.

A. Sigler and B. A. Malomed, “Solitary pulses in linearly coupled cubic-quintic Ginzburg-Landau equations,” Physica D 212(3-4), 305–316 (2005). [CrossRef]

56.

L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R. Soc. Lond. A Math. Phys. Sci. 326(1566), 289–313 (1972). [CrossRef]

57.

N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20(10), 1733 (1977). [CrossRef]

58.

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246(5), 412–422 (1998). [CrossRef]

OCIS Codes
(190.5940) Nonlinear optics : Self-action effects
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: December 20, 2011
Revised Manuscript: January 12, 2012
Manuscript Accepted: January 12, 2012
Published: January 20, 2012

Citation
Olga V. Borovkova, Yaroslav V. Kartashov, Victor A. Vysloukh, Valery E. Lobanov, Boris A. Malomed, and Lluis Torner, "Solitons supported by spatially inhomogeneous nonlinear losses," Opt. Express 20, 2657-2667 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2657


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References

  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).
  2. V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B 18(14), 627 (2004). [CrossRef]
  3. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78(1), 179–215 (2006). [CrossRef]
  4. L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11(10), 1268–1270 (1999). [CrossRef]
  5. I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. 19(3), 537 (2002). [CrossRef]
  6. F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003). [CrossRef]
  7. H. Saito and M. Ueda, “Dynamically stabilized bright solitons in a two-dimensional bose-einstein condensate,” Phys. Rev. Lett. 90(4), 040403 (2003). [CrossRef] [PubMed]
  8. M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97(3), 033903 (2006). [CrossRef] [PubMed]
  9. B. A. Malomed, Soliton Management in Periodic Systems (Springer, New York, 2006).
  10. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011). [CrossRef]
  11. H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005). [CrossRef] [PubMed]
  12. G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217(1), 31–57 (2006). [CrossRef]
  13. J. Garnier and F. K. Abdullaev, “Transmission of matter wave solitons through nonlinear traps and barriers,” Phys. Rev. A 74(1), 013604 (2006). [CrossRef]
  14. D. L. Machacek, E. A. Foreman, Q. E. Hoq, P. G. Kevrekidis, A. Saxena, D. J. Frantzeskakis, and A. R. Bishop, “Statics and dynamics of an inhomogeneously nonlinear lattice,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036602 (2006). [CrossRef] [PubMed]
  15. Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. 31(19), 2888–2890 (2006). [CrossRef] [PubMed]
  16. J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007). [CrossRef] [PubMed]
  17. P. Niarchou, G. Theocharis, P. G. Kevrekidis, P. Schmelcher, and D. J. Frantzeskakis, “Soliton oscillations in collisionally inhomogeneous attractive Bose-Einstein condensates,” Phys. Rev. A 76(2), 023615 (2007). [CrossRef]
  18. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008). [CrossRef] [PubMed]
  19. Y. Sivan, G. Fibich, and B. Ilan, “Drift instability and tunneling of lattice solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(4), 045601 (2008). [CrossRef] [PubMed]
  20. F. K. Abdullaev, A. Gammal, M. Salerno, and L. Tomio, “Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices,” Phys. Rev. A 77(2), 023615 (2008). [CrossRef]
  21. H. A. Cruz, V. A. Brazhnyi, and V. V. Konotop, “Partial delocalization of two-component condensates in optical lattices,” J. Phys. B 41(3), 035304 (2008). [CrossRef]
  22. H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010). [CrossRef]
  23. V. Pérez-García and R. Pardo, “Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates,” Physica D 238(15), 1352–1361 (2009). [CrossRef]
  24. B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003). [CrossRef]
  25. D. Neshev, A. Sukhorukov, Y. Kivshar, E. Ostrovskaya, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]
  26. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003). [CrossRef] [PubMed]
  27. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12(13), 2831–2837 (2004). [CrossRef] [PubMed]
  28. Y. J. He and H. Z. Wang, “(1+1)-dimensional dipole solitons supported by optical lattice,” Opt. Express 14(21), 9832–9837 (2006). [CrossRef] [PubMed]
  29. O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. 36(16), 3088–3090 (2011). [CrossRef] [PubMed]
  30. O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, “Bright solitons from defocusing nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(3), 035602 (2011). [CrossRef] [PubMed]
  31. Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. 36(23), 4587–4589 (2011). [CrossRef] [PubMed]
  32. W.-P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84(4), 043801 (2011). [CrossRef]
  33. B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D 29(1-2), 155–172 (1987). [CrossRef]
  34. O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. France 49(11), 1829–1833 (1988). [CrossRef]
  35. W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64(7), 749–752 (1990). [CrossRef] [PubMed]
  36. V. Hakim, P. Jakobsen, and Y. Pomeau, “Fronts vs solitary waves in non equilibrium systems,” Europhys. Lett. 11(1), 19–24 (1990). [CrossRef]
  37. B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg-Landau equation,” Phys. Rev. A 42(10), 6009–6014 (1990). [CrossRef] [PubMed]
  38. P. Marcq, H. Chate, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation,” Physica D 73(4), 305–317 (1994). [CrossRef]
  39. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Spec. Top. 173(1), 233–243 (2009). [CrossRef]
  40. D. A. Zezyulin, G. L. Alfimov, and V. V. Konotop, “Nonlinear modes in a complex parabolic potential,” Phys. Rev. A 81(1), 013606 (2010). [CrossRef]
  41. C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59(1), 81–89 (2010). [CrossRef]
  42. F. Kh. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(5), 056606 (2010). [CrossRef] [PubMed]
  43. D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, “Dissipative double-well potential: nonlinear stationary and pulsating modes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(5), 056213 (2010). [CrossRef] [PubMed]
  44. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Solitons in a medium with linear dissipation and localized gain,” Opt. Lett. 36(7), 1200–1202 (2011). [CrossRef] [PubMed]
  45. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A 83(4), 041806 (2011). [CrossRef]
  46. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Dissipative surface solitons in periodic structures,” Europhys. Lett. 91(3), 34003 (2010). [CrossRef]
  47. V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable radially symmetric and azimuthally modulated vortex solitons supported by localized gain,” Opt. Lett. 36(1), 85–87 (2011). [CrossRef] [PubMed]
  48. C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(6), 066609 (2011). [CrossRef]
  49. O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, “Stable nonlinear amplification of solitons without gain saturation,” Europhys. Lett.in press.
  50. J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D Appl. Phys. 36(3), R1–R16 (2003). [CrossRef]
  51. G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A 55(6), 4318–4324 (1997). [CrossRef]
  52. A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, “Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates,” Phys. Rev. Lett. 79(25), 4950–4953 (1997). [CrossRef]
  53. S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, “Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping,” Phys. Rev. A 59(1), 620–633 (1999). [CrossRef]
  54. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction,” Phys. Rev. Lett. 95(1), 010402 (2005). [CrossRef] [PubMed]
  55. A. Sigler and B. A. Malomed, “Solitary pulses in linearly coupled cubic-quintic Ginzburg-Landau equations,” Physica D 212(3-4), 305–316 (2005). [CrossRef]
  56. L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R. Soc. Lond. A Math. Phys. Sci. 326(1566), 289–313 (1972). [CrossRef]
  57. N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20(10), 1733 (1977). [CrossRef]
  58. J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246(5), 412–422 (1998). [CrossRef]

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