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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 20 — Sep. 24, 2012
  • pp: 22675–22682
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Particle-wave duality in quantum tunneling of a bright soliton

Ching-Hao Wang, Tzay-Ming Hong, Ray-Kuang Lee, and Daw-Wei Wang  »View Author Affiliations


Optics Express, Vol. 20, Issue 20, pp. 22675-22682 (2012)
http://dx.doi.org/10.1364/OE.20.022675


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Abstract

One of the most fundamental difference between classical and quantum mechanics is observed in the particle tunneling through a localized potential: the former predicts a discontinuous transmission coefficient (T) as a function in incident velocity between one (complete penetration) and zero (complete reflection); while in the latter T always changes smoothly with a wave nature. Here we report a systematic study of the quantum tunneling property for a bright soliton, which behaves as a classical particle (wave) in the limit of small (large) incident velocity. In the intermediate regime, the classical and quantum properties are combined via a finite (but not full) discontinuity in the tunneling transmission coefficient. We demonstrate that the formation of a localized bound state is essential to describe such inelastic collisions, showing a nontrivial nonlinear effect on the quantum transportation of a bright soliton.

© 2012 OSA

Fig. 1 Schematic plots for different tunneling dynamics through a local potential, V(x). The incident, transmission, and reflection velocities are denoted as vi, vt, and vr, respectively. (a) Classical picture with either a total transmission (T = 1) or total reflection (T = 0). (b) Quantum mechanical picture with a partial transmission (0 < T < 1). The notations, Ψi, Ψt, and Ψr represent the incident, transmitted, and reflected wavefunctions, respectively. (c) Inelastic scattering process of a BS through a potential well, V(x) < 0, where a localized bound state, Ψb, appears after scattering.

Here, we consider the dynamics of a weakly interacting BEC at zero temperature, which can be well-approximated by the Gross-Pitaevskii equation, referred also as the nonlinear Schrödinger equation (NLSE) [23

23. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999). [CrossRef]

],
[122x2+g|Ψ(x,t)|2+V(x)]Ψ(x,t)=itΨ(x,t),
(1)
where the particle mass m and h̄ are both set to 1, Ψ(x, t) represents the condensate wavefunction, g measures the inter-particle interaction, and V(x) = V0δ(x) indicates a defect potential. When the interaction is attractive, g < 0, a stable bright soliton is supported in a uniform system with the solution [5

5. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

]
Ψbs(x,t)=β|g|sech[β(xxcvit)]eiθ(x,t),
(2)
where the center of the wavepacket is denoted by xc, and θ(x, t) ≡ vixEt, with the total energy Evi2/2+μ and the chemical potential μ = −β2/2, respectively. The velocity for a BS is characterized by vi. For 1D solitons, the free parameter is β, which can be set to unit with the normalization condition by taking β = |g|/2, i.e., |Ψbs(x,t)|2dx=2β/|g|=1. In this case, the only two independent parameters left are the normalized potential strength, 0V0/|g|, and the normalized initial velocity, ivi/|g|, which define our parameter space. In the following, we consider the transportation process when such a BS wavepacket is generated at t = 0, centered at xc → −∞, and then propagates along the positive x-axis with an initial velocity vi. This wavepacket then scatters the local defect V (x) at the position x = 0, resulting in possible transmitted, reflected, and localized wave functions after a certain time measured.

First of all, the calculated transmission coefficient (T) as a function of vi and V0 > 0 for a repulsive potential is illustrated in Fig. 2(a) by directly solving Eq. (1) numerically. Here, T is defined as alimt|Ψ(x,t)|2dx with a small value a > 0 to exclude the contribution from any possible localized bound states. As one can see from Fig. 2(a), in the region with a small value of vi and |V0|, there exists a line that characterizes the discontinuity in the transmission coefficient. The existence of such a discontinuity certainly reflects the particle nature of a BS, i.e., totally transmitted (T = 1) or totally reflected (T = 0) as shown in Fig. 1(a). However, this line of border for the particle nature breaks down at a critical point in the parameter space where the incident velocity and corresponding potential strength are denoted by v˜i*~0.8 and |V˜0*|~1.2. Beyond these values, instead of a disrupt change, the contours of transmission coefficient T changes continuously as a regular wave.

Fig. 2 Contour plot of the transmission coefficient as a function of incident soliton velocity vi and potential strength |V0|. Cases of potential barrier V0 > 0 and potential well V0 < 0 are separately shown in (a) and (b). White dashed lines are plotted from Eq. (4) and Eq. (9) for the corresponding analytical results (see the text).

On the contrary, in Fig. 2(b), the tunneling properties are different for a BS through an attractive potential (V0 < 0, a potential well). Qualitatively speaking, we have a similar “phase diagram” as the case of a potential barrier, but now the phase boundary, the while dashed curve, becomes a nearly linear line. As it would be demonstrated later, at a small value of V0, such a universal border, independent from any additional parameters, comes from the existence of a localized bound state. The formation of this localized bound state screens the potential well and results in extra interactions on the quantum tunneling of a BS.

In order to give a deeper understanding of these numerical results, we consider the limit of a weak nonlinear interaction for the first step. When the interaction energy is much smaller than the kinetic energy and potential energy, the corresponding transmission coefficient should be similar to that in the standard quantum mechanics textbook [7

7. D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005).

], i.e.,
T(vi/V0)21+(vi/V0)2+O(1vi2).
(3)
As expected for a characteristic wave nature, the transmission coefficient T is always continuous and independent of the sign of potential strength, V0. We note that above results are true both for incident waves in the form of a soliton wavepacket and a plane wave. In such a scenario, the incident BS can be easily distorted by the local potential due to that the nonlinearity is too weak to support the original soliton solution, resulting in a lots of dispersive radiations in the transmitted or reflected waves [24

24. J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys. 274, 187–216 (2007) [CrossRef]

].

On the other hand, in the limit of a weak and repulsive potential along with a small velocity, i.e., the strong interaction limit, we can safely assume that the propagating soliton is not affected by the potential. Hence, one can use the center position of a BS, xc(t), to describe the whole transportation process if there is no bound state generated during the scattering process. In this limit, one can rigorously show that the dynamics of xc(t) behaves like a classical particle moving effectively in a conservative potential, Veff(x), which is just a convolution of the local potential with the soliton wavefunction [25

25. J. Holmer and M. Zworski, “Slow soliton interaction with delta impurities,” J. Modern Dynamics 1, 689–718 (2007). [CrossRef]

], i.e., Veff(x)=β22|g|V0sech2(x). Therefore, the corresponding “conservation law” for the total energy is found to be,
v(t)2+β2|g|2V0sech2[xc(t)]=vi2+β2|g|2V0sech2(xi),
(4)
where v(t)1|g|dxc(t)dt is the defined particle velocity for a BS. As a result, the border across the regions with T = 1 and T = 0 can be defined by taking v(t) = 0 and xc = 0 as the boundary condition, along with the initial condition xi → −∞. Then, we obtain the relation V0/|g| = (|g|/β)2(vi/|g|)2 = 4(vi/|g|)2, which is depicted as the white dashed line in Fig. 2(a).

However, we know that above semi-classical approach used for Eq. (4) fails when V0 is larger than a critical value, denoted as V0* due to the failure of taking the BS as a classical particle. This critical value can be estimated as following: as the center of a BS reaches the location of a potential, it gains a local potential energy, V0|Ψ(0)|2, which cannot be larger than the absolute value of the chemical potential, |μ| = β2/2 = |g|2/8, in order to keep the soliton description valid. From Eq. (2), the critical value for the breakdown is V0*/|g|=0.5. In Fig. 2(a), the agreement between the analytical curve defined by Eq. (4) and our direct numerical simulations is good both qualitatively and quantitatively. When V0 is close to V0*, the BS is in the brink of collapsing, then the reflection part as well as the non-soliton radiation become non-negligible. Discrepancies from above analytical results are therefore expected.

Now we come to the potential well, which should have similar results as the potential barrier in the limit of large V0 and vi (the wave nature in the weak interaction limit). However, in the regime of small V0 and vi, the wavepacket description used above fails for the lack in the consideration of possible localized bound states supported in an attractive interaction. The appearance of a localized bound state indicates extra inelastic scatterings. Therefore, the resulting tunneling amplitude changes dramatically, as compared to the case of a potential barrier. The bound state wavefunction for a localized potential has been well-studied in the literature [26

26. V. Hakim, “Nonlinear Schrödinger flow past an obstacle in one dimension,” Phys. Rev. E 55, 2835–2845 (1997). [CrossRef]

, 27

27. N. Pavloff, “Breakdown of superfluidity of an atom laser past an obstacle,” Phys. Rev. A 66, 013610(1–8) (2002). [CrossRef]

, 28

28. B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A 71, 033609(1–10) (2005). [CrossRef]

, 29

29. D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen. 38, 1777–1792 (2005). [CrossRef]

], and its analytic form can be written as following:
Ψb(x)=βb|g|sech(βb|x|+xb),
(5)
where βb measures the slope (the inverse of soliton width) and amplitude of the bound state, and xb ≡ tanh−1(|V0|/βb) is the shift of effective peak position from the potential center. It is easy to see that the bound state wavefunction is composed of two soliton-like solutions, but with different center positions and β. By matching the discontinuity in the wavefunction slopes with the potential strength, for a given renormalization of the bound state, i.e., βb is fixed, such bound states exist only in a weak potential limit and disappears when |V0| > βb. Such an anti-intuitive result originates from the fact that the maximum slope of a BS is limited by its renormalization due to the nonlinear (interaction) effect. Although the localized bound state also exists in a repulsive potential defect, it cannot be easily produced in the tunneling process due to the mismatch in the boundary conditions. Therefore it does not affect the tunneling property as we discussed above.

Inspired by the numerical simulations of the tunneling process (not shown here), we consider the following simplified picture of tunneling dynamics in the presence of a bound state, i.e., inelastic scattering. To derive an analytical formula for the border when a soliton scatters by a potential well, we assume: (i) the reflected wavefunction is negligible, (ii) the bound state appears after the scattering, and (iii) the transmitted wave also has a soliton profile. Since both the soliton solution and the localized bound state are governed by two parameters, β and v, as shown in Eq. (2), the relevant parameters to describe a soliton tunneling are therefore: (βi, vi) for the incident soliton, (βt, vt) for the transmitted one, and (βb, vb = 0) for the localized bound state. Since vi is given and βi ≡ |g|/2 is required for the initial unit normalization, now we only have three parameters to be determined: βb, βt, and vt.

Instead of matching the boundary of wavefunctions during the scattering process, we use the conservation laws to extract these three unknown parameters in a more general method. Based on the above three assumptions, we can write down three equations:
2βi|g|=1=2|g|(βb|V0|)+T,
(6)
βi22+vi22=βb22(1T)+(βt22+vt22)T,
(7)
vi=2βt|g|vt=Tvt
(8)
where we have expressed the transmission coefficient 2βt/|g| as a function of T via the assumption (iii). Equations (6) and (7) represent the conservations of total probability and total energy by including both the localized bound state and the transmitted soliton. Equation (8) can be understood as the conservation of current density due to the change of soliton amplitude. We note that the condition |V0| < βb is required, in order to have T > 0 in Eq. (6). By eliminating the other two variables, one can obtain T as a function of the normalized potential strength 0 ≡ |V0/g| and the normalized initial velocity ivi/|g|,
T=V˜0(V˜0+1)V˜02(V˜0+1)2(4v˜i2V˜0+3v˜i2)2V˜0+3/2,
(9)
where the “+” solution is physically invalid. From Eq. (9), we find several interesting properties in the tunneling of a BS through the potential well. First of all, above solution for the transmission coefficient is real only when |V˜0|2(|V˜0|+1)2(4v˜i2|V˜0|+3v˜i2)0, or when vi is smaller than a critical velocity, vc(|V0|), i.e.,
vc|g||V0/g|(|V0/g|+1)3+4|V0/g|,
(10)
where V0 and vi has to be bounded by requiring T < 1. In Fig. 2(b), the border for the discontinuity in the transmission coefficient is compared with our analytical formula and direct numerical simulations, which results in very good agreement. More importantly, we find that the critical velocity defined in Eq. (10), i.e., the curve for the discontinuous transmission coefficient, shows a rather straight line (although is not exact), instead of a parabolic one for the potential barrier. Now the major contributions come from the existence of a localized bound state in this inelastic scattering process.

The validity of our assumptions used above is totally based on the ”soliton-in” and ”soliton-out” picture, alone with the condition that the formed bound state also has a soliton profile. Although our simplified theory does predict the locations of transition, in Fig. 3 the transmission coefficient T we obtained in Eq. (9) is slightly less than the values obtained by direct numerical simulations. The discrepancy between analytical and numerical data is not surprising because we have neglected the radiation parts (non-soliton waves) in the transmitted waves, which cannot be captured in such a simple theory. It should be remarked that the discontinuity in the transmission coefficient, the spikes, comes from our numerical errors due to the choice of different grid sizes in simulations. Last but not least, we restate that the estimations we made here for the tunneling dynamics of a potential well cannot be applied to the potential barrier, because the bound state wavefunction supported by a potential barrier has to be a double-humped one in the profile due to a mismatching boundary condition. On the side of wave nature, the transmission coefficient is identical both for potential well and potential barrier when one considers a regular (noninteracting) wave tunneling, but on the side of particle nature, it turns out to be very different in the classical particle picture. This also reflects the non-trivial effect of interaction (nonlinearity) in the soliton scattering problem.

Fig. 3 A comparison of transmission coefficient between numerical simulations (solid lines) and analytical results from Eq. (9) (dashed points). Note that the parameters are the same as those used in Fig. 2(b).

In conclusion, we have systematically studied the tunneling of a bright soliton subjecting to a localized potential defect. By performing direct numerical simulations, we obtain a full phase diagram of the transmission coefficient in terms of the incident velocity and potential strength. Our results show a fundamentally important transport property, which can be easily observed in ultracold atoms, nonlinear optics, or even soft-matter systems.

Acknowledgments

This research is supported by the National Science Council of Taiwan.

References and links

1.

T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett. 98, 210602(1–4) (2007). [CrossRef]

2.

L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007). [CrossRef] [PubMed]

3.

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, 2008). [CrossRef]

4.

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005). [CrossRef] [PubMed]

5.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

6.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002). [CrossRef] [PubMed]

7.

D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005).

8.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989). [CrossRef]

9.

H. Sakaguchi and M. Tamura, “Scattering of solitons and dark solitons by potential walls in the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 74, 292–298 (2005). [CrossRef]

10.

C. Lee and J. Brand, “Enhanced quantum reflection of matter-wave solitons,” Europhys. Lett. 73, 321–327 (2006). [CrossRef]

11.

T. Ernst and J. Brand, “Resonant trapping in the transport of a matter-wave soliton through a quantum well,” Phys. Rev. A 81, 033614 (2010). [CrossRef]

12.

J. A. González, A. Bellorín, and L. E. Guerrero, “Soliton tunneling with sub-barrier kinetic energies,” Phys. Rev E 60, R37–R40 (2009). [CrossRef]

13.

G. Kälbermann, “Soliton interacting as a particle,” Phys. Lett. A 252, 37–42 (1999). [CrossRef]

14.

G. Kälbermann, “Soliton tunneling,” Phys. Rev. E 55, R6360–R6362 (1999).

15.

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008). [CrossRef]

16.

J. L. Helm, T. P. Billam, and S. A. Gardine, “Bright matter-wave soliton collisions at narrow barriers,” Phys. Rev. A 85, 053621 (2012). [CrossRef]

17.

B. Gertjerenken, T. P. Billam, L. Khaykovich, and C. Weiss, “Scattering bright solitons: quantum versus mean-field behavior,” arXiv:1208.2941v2 (2012).

18.

X. D. Cao and B. A. Malomed, “Soliton-defect collisions in the nonlinear Schrödinger equation,” Phys. Lett. A 206, 177–182 (1995). [CrossRef]

19.

S. Burtsev, D. J. Kaup, and B. A. Malomed, “Interaction of solitons with a strong inhomogeneity in a nonlinear optical fiber,” Phys. Rev. E 52, 4474–4481 (1995). [CrossRef]

20.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Strong NLS soliton defect interactions,” Physica D 192, 215–248 (2004). [CrossRef]

21.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007). [CrossRef]

22.

C. P. Jisha, A. Alberucci, R.-K. Lee, and G. Assanto, “Optical solitons and wave-particle duality,” Opt. Lett. 36, 1848–1850 (2011). [CrossRef] [PubMed]

23.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999). [CrossRef]

24.

J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys. 274, 187–216 (2007) [CrossRef]

25.

J. Holmer and M. Zworski, “Slow soliton interaction with delta impurities,” J. Modern Dynamics 1, 689–718 (2007). [CrossRef]

26.

V. Hakim, “Nonlinear Schrödinger flow past an obstacle in one dimension,” Phys. Rev. E 55, 2835–2845 (1997). [CrossRef]

27.

N. Pavloff, “Breakdown of superfluidity of an atom laser past an obstacle,” Phys. Rev. A 66, 013610(1–8) (2002). [CrossRef]

28.

B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A 71, 033609(1–10) (2005). [CrossRef]

29.

D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen. 38, 1777–1792 (2005). [CrossRef]

OCIS Codes
(020.1335) Atomic and molecular physics : Atom optics
(020.1475) Atomic and molecular physics : Bose-Einstein condensates
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: June 21, 2012
Revised Manuscript: August 17, 2012
Manuscript Accepted: September 4, 2012
Published: September 19, 2012

Citation
Ching-Hao Wang, Tzay-Ming Hong, Ray-Kuang Lee, and Daw-Wei Wang, "Particle-wave duality in quantum tunneling of a bright soliton," Opt. Express 20, 22675-22682 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22675


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References

  1. T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett.98, 210602(1–4) (2007). [CrossRef]
  2. L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett.98, 130404(1–4) (2007). [CrossRef] [PubMed]
  3. C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, 2008). [CrossRef]
  4. C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett.95, 170410(1–4) (2005). [CrossRef] [PubMed]
  5. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).
  6. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science296, 1290–1293 (2002). [CrossRef] [PubMed]
  7. D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005).
  8. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys.61, 763–915 (1989). [CrossRef]
  9. H. Sakaguchi and M. Tamura, “Scattering of solitons and dark solitons by potential walls in the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn.74, 292–298 (2005). [CrossRef]
  10. C. Lee and J. Brand, “Enhanced quantum reflection of matter-wave solitons,” Europhys. Lett.73, 321–327 (2006). [CrossRef]
  11. T. Ernst and J. Brand, “Resonant trapping in the transport of a matter-wave soliton through a quantum well,” Phys. Rev. A81, 033614 (2010). [CrossRef]
  12. J. A. González, A. Bellorín, and L. E. Guerrero, “Soliton tunneling with sub-barrier kinetic energies,” Phys. Rev E60, R37–R40 (2009). [CrossRef]
  13. G. Kälbermann, “Soliton interacting as a particle,” Phys. Lett. A252, 37–42 (1999). [CrossRef]
  14. G. Kälbermann, “Soliton tunneling,” Phys. Rev. E55, R6360–R6362 (1999).
  15. A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett.100, 153901(1–4) (2008). [CrossRef]
  16. J. L. Helm, T. P. Billam, and S. A. Gardine, “Bright matter-wave soliton collisions at narrow barriers,” Phys. Rev. A85, 053621 (2012). [CrossRef]
  17. B. Gertjerenken, T. P. Billam, L. Khaykovich, and C. Weiss, “Scattering bright solitons: quantum versus mean-field behavior,” arXiv:1208.2941v2 (2012).
  18. X. D. Cao and B. A. Malomed, “Soliton-defect collisions in the nonlinear Schrödinger equation,” Phys. Lett. A206, 177–182 (1995). [CrossRef]
  19. S. Burtsev, D. J. Kaup, and B. A. Malomed, “Interaction of solitons with a strong inhomogeneity in a nonlinear optical fiber,” Phys. Rev. E52, 4474–4481 (1995). [CrossRef]
  20. R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Strong NLS soliton defect interactions,” Physica D192, 215–248 (2004). [CrossRef]
  21. Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett.99, 133901(1–4) (2007). [CrossRef]
  22. C. P. Jisha, A. Alberucci, R.-K. Lee, and G. Assanto, “Optical solitons and wave-particle duality,” Opt. Lett.36, 1848–1850 (2011). [CrossRef] [PubMed]
  23. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys.71, 463–512 (1999). [CrossRef]
  24. J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys.274, 187–216 (2007) [CrossRef]
  25. J. Holmer and M. Zworski, “Slow soliton interaction with delta impurities,” J. Modern Dynamics1, 689–718 (2007). [CrossRef]
  26. V. Hakim, “Nonlinear Schrödinger flow past an obstacle in one dimension,” Phys. Rev. E55, 2835–2845 (1997). [CrossRef]
  27. N. Pavloff, “Breakdown of superfluidity of an atom laser past an obstacle,” Phys. Rev. A66, 013610(1–8) (2002). [CrossRef]
  28. B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A71, 033609(1–10) (2005). [CrossRef]
  29. D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen.38, 1777–1792 (2005). [CrossRef]

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