American Institute of Mathematical Sciences

Advanced
June  2017, 22(4): 1207-1229. doi: 10.3934/dcdsb.2017059

Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation

Georgy L. Alfimov 1, , Pavel P. Kizin 1, and  Dmitry A. Zezyulin 2,

1. 

Moscow Institute of Electronic Engineering, Zelenograd, Moscow, 124498, Russia
2. 

Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Ed. C8, Lisboa P-1749-016, Portugal

Received  May 2016 Revised  September 2016 Published  February 2017

Fund Project: The third author is supported by FCT (Portugal) through the grant No. UID/FIS/00618/2013.

The paper is devoted to nonlinear localized modes (“gap solitons”) for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with a periodic potential and repulsive interparticle interactions. It has been recently shown (G. L. Alfimov, A. I. Avramenko, Physica D, 254, 29 (2013)) that under certain conditions all the stationary modes for the 1D GPE can be coded by bi-infinite sequences of symbols of some finite alphabet (called “codes” of the solutions). We present and justify a numerical method which allows to reconstruct the profile of a localized mode by its code. As an example, the method is applied to compute the profiles of gap solitons for 1D GPE with a cosine potential.

Keywords: Gross-Pitaevskii equation, gap solitons, nonlinear modes, coding.
Mathematics Subject Classification: 35Q55, 35C08, 34C41, 37B10, 65P99.
Citation: Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059
References:
[1]

T. J. Alexander, E. A. Ostrovskaya and Yu. S. Kivshar, Self-Trapped Nonlinear Matter Waves in Periodic Potentials Phys. Rev. Lett. 96 (2006), 040401. doi: 10.1103/PhysRevLett.96.040401.  Google Scholar

[2]

G. L. Alfimov, V. V. Konotop and M. Salerno, Matter solitons in Bose-Einstein Condensates with optical lattices, Europhys. Lett., 58 (2002), 7-13.  doi: 10.1209/epl/i2002-00599-0.  Google Scholar

[3]

G. L. Alfimov and A. I. Avramenko, Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential, Physica D, 254 (2013), 29-45.  doi: 10.1016/j.physd.2013.03.009.  Google Scholar

[4]

G. L. Alfimov, V. A. Brazhnyi and V. V. Konotop, On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation, Physica D, 194 (2004), 127-150.  doi: 10.1016/j.physd.2004.02.001.  Google Scholar

[5]

G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop and M. Salerno, Wannier functions analysis of the nonlinear Schr′odinger equation with a periodic potential Phys. Rev. E 66 (2002), 046608, 6pp. doi: 10.1103/PhysRevE.66.046608.  Google Scholar

[6]

G. L. Alfimov and P. P. Kizin, On initial data for Cauchy problem for equation $u_{xx}+Q(x)u-P(u)=0$ having no singularities on a given interval, Ufa Mathematical Journal 2016, accepted. Google Scholar

[7]

G. L. Alfimov and M. E. Lebedev, On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$, Ufa Mathematical Journal, 7 (2015), 3-16.  doi: 10.13108/2015-7-2-3.  Google Scholar

[8]

V. I. Arnold, Mathematical Methods of Classical Mechanics Springer-Verlag, 1989.  Google Scholar

[9]

I. V. Barashenkov, D. E. Pelinovsky and E. V. Zemlyanaya, Vibrations and Oscillatory Instabilities of Gap Solitons, Phys. Rev. Lett., 80 (1998), 5117-5120.  doi: 10.1103/PhysRevLett.80.5117.  Google Scholar

[10]

I. V. Barashenkov and E. V. Zemlyanaya, Oscillatory instabilities of gap solitons: A numerical study, Comp. Phys. Comm., 126 (2000), 22-27.  doi: 10.1016/S0010-4655(99)00241-6.  Google Scholar

[11]

F. A. Berezin and M. A. Shubin, The Shrödinder Equation Kluwer, Dordrech, 1991. Google Scholar

[12]

R. Fukuizumi and A. Sacchetti, Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials, J. Stat. Phys., 156 (2014), 707-738.  doi: 10.1007/s10955-014-1023-x.  Google Scholar

[13]

P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, A. R. Bishop, H. Nistazakis and R. Carretero-González, Domain walls of single-component Bose-Einstein condensates in external potentials, Mathematics and Computers in Simulation, 69 (2005), 334-345.  doi: 10.1016/j.matcom.2005.01.016.  Google Scholar

[14]

P. P. Kizin, D. A. Zezyulin and G. L. Alfimov, Oscillatory instabilities of gap solitons in a repulsive Bose-Einstein condensate, Physica D, 337 (2016), 58-66.  doi: 10.1016/j.physd.2016.07.007.  Google Scholar

[15]

V. V. Konotop and M. Salerno, Modulational instability in cigar-shaped Bose-Einstein condensates in optical lattices Phys. Rev. A 65 (2002), 021602. doi: 10.1103/PhysRevA.65.021602.  Google Scholar

[16]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage and Yu. S. Kivshar, Bose-Einstein condensates in optical lattices: Band-gap structure and solitons Phys. Rev. A 67 (2003), 013602. doi: 10.1103/PhysRevA.67.013602.  Google Scholar

[17]

B. A. Malomed and R. S. Tasgal, Vibration modes of a gap soliton in a nonlinear optical medium Phys. Rev. E 49 (1994), 5787. doi: 10.1103/PhysRevE.49.5787.  Google Scholar

[18]

T. Mayteevarunyoo and B. A. Malomed, Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice Phys. Rev. A 74} (2006), 033616. doi: 10.1103/PhysRevA.74.033616.  Google Scholar

[19]

D. E. Pelinovsky, A. A. Sukhorukov and Yu. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials Phys. Rev. E 70 (2004), 036618, 17pp. doi: 10.1103/PhysRevE.70.036618.  Google Scholar

[20]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, (2003).   Google Scholar

[21]

B. Wu and Q. Niu, Landau and dynamical instabilities of the superflow of Bose-Einstein Condensates in optical lattices Phys. Rev. A 64 (2001), 061603(R). doi: 10.1103/PhysRevA.64.061603.  Google Scholar

[22]

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898719680.  Google Scholar

[23]

Yo. Zhang, Zh. Liang and B. Wu, Gap solitons and Bloch waves in nonlinear periodic systems Phys. Rev. A 80 (2009), 063815. doi: 10.1103/PhysRevA.80.063815.  Google Scholar

[24]

Yo. Zhang and B. Wu, Composition relation between gap solitons and bloch waves in nonlinear periodic systems Phys. Rev. Lett. 102 (2009), 093905. doi: 10.1103/PhysRevLett.102.093905.  Google Scholar

show all references

References:
[1]

T. J. Alexander, E. A. Ostrovskaya and Yu. S. Kivshar, Self-Trapped Nonlinear Matter Waves in Periodic Potentials Phys. Rev. Lett. 96 (2006), 040401. doi: 10.1103/PhysRevLett.96.040401.  Google Scholar

[2]

G. L. Alfimov, V. V. Konotop and M. Salerno, Matter solitons in Bose-Einstein Condensates with optical lattices, Europhys. Lett., 58 (2002), 7-13.  doi: 10.1209/epl/i2002-00599-0.  Google Scholar

[3]

G. L. Alfimov and A. I. Avramenko, Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential, Physica D, 254 (2013), 29-45.  doi: 10.1016/j.physd.2013.03.009.  Google Scholar

[4]

G. L. Alfimov, V. A. Brazhnyi and V. V. Konotop, On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation, Physica D, 194 (2004), 127-150.  doi: 10.1016/j.physd.2004.02.001.  Google Scholar

[5]

G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop and M. Salerno, Wannier functions analysis of the nonlinear Schr′odinger equation with a periodic potential Phys. Rev. E 66 (2002), 046608, 6pp. doi: 10.1103/PhysRevE.66.046608.  Google Scholar

[6]

G. L. Alfimov and P. P. Kizin, On initial data for Cauchy problem for equation $u_{xx}+Q(x)u-P(u)=0$ having no singularities on a given interval, Ufa Mathematical Journal 2016, accepted. Google Scholar

[7]

G. L. Alfimov and M. E. Lebedev, On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$, Ufa Mathematical Journal, 7 (2015), 3-16.  doi: 10.13108/2015-7-2-3.  Google Scholar

[8]

V. I. Arnold, Mathematical Methods of Classical Mechanics Springer-Verlag, 1989.  Google Scholar

[9]

I. V. Barashenkov, D. E. Pelinovsky and E. V. Zemlyanaya, Vibrations and Oscillatory Instabilities of Gap Solitons, Phys. Rev. Lett., 80 (1998), 5117-5120.  doi: 10.1103/PhysRevLett.80.5117.  Google Scholar

[10]

I. V. Barashenkov and E. V. Zemlyanaya, Oscillatory instabilities of gap solitons: A numerical study, Comp. Phys. Comm., 126 (2000), 22-27.  doi: 10.1016/S0010-4655(99)00241-6.  Google Scholar

[11]

F. A. Berezin and M. A. Shubin, The Shrödinder Equation Kluwer, Dordrech, 1991. Google Scholar

[12]

R. Fukuizumi and A. Sacchetti, Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials, J. Stat. Phys., 156 (2014), 707-738.  doi: 10.1007/s10955-014-1023-x.  Google Scholar

[13]

P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, A. R. Bishop, H. Nistazakis and R. Carretero-González, Domain walls of single-component Bose-Einstein condensates in external potentials, Mathematics and Computers in Simulation, 69 (2005), 334-345.  doi: 10.1016/j.matcom.2005.01.016.  Google Scholar

[14]

P. P. Kizin, D. A. Zezyulin and G. L. Alfimov, Oscillatory instabilities of gap solitons in a repulsive Bose-Einstein condensate, Physica D, 337 (2016), 58-66.  doi: 10.1016/j.physd.2016.07.007.  Google Scholar

[15]

V. V. Konotop and M. Salerno, Modulational instability in cigar-shaped Bose-Einstein condensates in optical lattices Phys. Rev. A 65 (2002), 021602. doi: 10.1103/PhysRevA.65.021602.  Google Scholar

[16]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage and Yu. S. Kivshar, Bose-Einstein condensates in optical lattices: Band-gap structure and solitons Phys. Rev. A 67 (2003), 013602. doi: 10.1103/PhysRevA.67.013602.  Google Scholar

[17]

B. A. Malomed and R. S. Tasgal, Vibration modes of a gap soliton in a nonlinear optical medium Phys. Rev. E 49 (1994), 5787. doi: 10.1103/PhysRevE.49.5787.  Google Scholar

[18]

T. Mayteevarunyoo and B. A. Malomed, Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice Phys. Rev. A 74} (2006), 033616. doi: 10.1103/PhysRevA.74.033616.  Google Scholar

[19]

D. E. Pelinovsky, A. A. Sukhorukov and Yu. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials Phys. Rev. E 70 (2004), 036618, 17pp. doi: 10.1103/PhysRevE.70.036618.  Google Scholar

[20]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, (2003).   Google Scholar

[21]

B. Wu and Q. Niu, Landau and dynamical instabilities of the superflow of Bose-Einstein Condensates in optical lattices Phys. Rev. A 64 (2001), 061603(R). doi: 10.1103/PhysRevA.64.061603.  Google Scholar

[22]

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898719680.  Google Scholar

[23]

Yo. Zhang, Zh. Liang and B. Wu, Gap solitons and Bloch waves in nonlinear periodic systems Phys. Rev. A 80 (2009), 063815. doi: 10.1103/PhysRevA.80.063815.  Google Scholar

[24]

Yo. Zhang and B. Wu, Composition relation between gap solitons and bloch waves in nonlinear periodic systems Phys. Rev. Lett. 102 (2009), 093905. doi: 10.1103/PhysRevLett.102.093905.  Google Scholar

3] the coding of bounded solutions for the nonlinear equation (4) is possible are shown in the first and the second gaps: if $\omega$ and $A$ belong to Region 1, then the bounded solutions can be coded using an alphabet of $N=3$ symbols; in Region 2 an alphabet of $N=5$ symbols is necessary.">Figure 1.  The diagram of gaps and bands of the Matheiu equation $\psi_{xx}+(\omega-A\cos 2x)\psi=0$ . The regions where according to [3] the coding of bounded solutions for the nonlinear equation (4) is possible are shown in the first and the second gaps: if $\omega$ and $A$ belong to Region 1, then the bounded solutions can be coded using an alphabet of $N=3$ symbols; in Region 2 an alphabet of $N=5$ symbols is necessary.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Action of the map $T$ on $\Delta_0$ which consists of three connected components, $D_{-1}$, $D_{0}$ and $D_{1}$. The cosine potential (6) was used for (7). Parameters are $\omega=1$ and $A=-3$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  An island $D$ with $v$-curve $\beta$, $v$-strip $V$, $h$-curve $\alpha$ and $h$-strip $H$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 1, in this case $\Delta_0$ consists of three connected components, $D_{-1}$, $D_0$ and $D_{1}$; Panel (b): parametric point $(\omega,A)=(4,-10)$ from Region 2 in Figure 1, in this case $\Delta_0$ consists of five connected components, from $D_{-2}$ to $D_2$.">Figure 2.  The sets ${\mathcal{U}}_\pi^\pm$ and $\Delta_0$ for (7) with the potential (6). Panel (a): parametric point $(\omega,A)=(1,-3)$ from Region 1 in Figure 1, in this case $\Delta_0$ consists of three connected components, $D_{-1}$, $D_0$ and $D_{1}$; Panel (b): parametric point $(\omega,A)=(4,-10)$ from Region 2 in Figure 1, in this case $\Delta_0$ consists of five connected components, from $D_{-2}$ to $D_2$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Ordering of $h$-strips in the island $D_0$ in the case $L=1$. The island $D_0$ contains three $h$-strips with multi-indices of length two: $H_{0(-1)}$, $H_{00}$, $H_{01}$. Each of these $h$-strips contains three $h$-strips with multi-indices of the length three. Within each of the $h$-strips the ordering of embedded $h$-strips inherits the pattern determined by the "global" arrow (in bold) and arrows over each of the islands.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5 visualized as a ternary tree ($2L+1=3$). As an example, red path (shown by thick lines in the grayscale version of the figure) indicates the position of $h$-strip $H_{01(-1)(-1)}$. As the graph shows, this $h$-strip is situated inside $H_{01(-1)}$, to the right from $H_{01(-1)0}$.">Figure 6.  The same ordering pattern as in Figure 5 visualized as a ternary tree ($2L+1=3$). As an example, red path (shown by thick lines in the grayscale version of the figure) indicates the position of $h$-strip $H_{01(-1)(-1)}$. As the graph shows, this $h$-strip is situated inside $H_{01(-1)}$, to the right from $H_{01(-1)0}$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The scheme of the island $D_0$, positions of $h$-strips and the points $h_{0}^\pm$, $h_{0i_1}^\pm$ for the case of coding using the alphabet of three symbols (the symbols are ''$-1$'', ''$0$'' and ''$1$''). The point $p_{-M}$ is situated in the strip $H_{0\times M}$ between the points $h_{0\times M}^-$ and $h_{0\times M}^+$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  The gap solitons from Region 1, the first gap. Each panel shows the spatial profile of the soliton. The corresponding codes are: (a) $\{\ldots,0,1,0,\ldots\}$; (b) $\{\ldots,0,1,1,0,\ldots\}$; (c) $\{\ldots,0,1,1,1,0,\ldots\}$; (d) $\{\ldots,0,1,0,1,0,\ldots\}$; (e) $\{\ldots,0,1,-1,0,\ldots\}$; (f) $\{\ldots,0,1,1,-1,0,\ldots\}$; (g) $\{\ldots,0,1,-1,1,0,\ldots\}$; (h) $\{\ldots,0,1,0,-1,0,\ldots\}$. For all shown solutions $\omega = 1$ and $A = -3$. The vertical dashed line in each panel indicates the position of the point $p_0$, such that $Tp_0\notin D_0$, $T^{-n} p_0\in D_0$, $n\geq 0$, see the explanation in subsection 4.2. Blue thin lines show schematically the cosine potential (6).
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8), and blue thin lines show schematically the cosine potential (6).">Figure 9.  The gap solitons from Region 2, the second gap. The corresponding codes are: (a) $\{\ldots,0,2,0,\ldots\}$; (b) $\{\ldots,0,2,2,0,\ldots\}$; (c) $\{\ldots,0,2,-2,0,\ldots\}$; (d) $\{\ldots,0,2,-1,0,\ldots\}$; (e) $\{\ldots,0,-1,0,\ldots\}$; (f) $\{\ldots,0,-1,-1,0,\ldots\}$; (g) $\{\ldots,0,-1,1,0,\ldots\}$; (h) $\{\ldots,0,2,1,0,\ldots\}$. For all shown solutions $\omega = 4$, $A = -10$. The vertical line indicates the position of the point $p_0$ (see caption of Figure 8), and blue thin lines show schematically the cosine potential (6).
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  (a) Soliton from the second gap ($\omega=4$, $A=-10$) with the code $\{\ldots,0, 1, 2, -2, 2, -1, 1, 2, -2, 2, -1, 0,\ldots\}$. (b) Soliton from the third gap ($\omega=10$, $A=-20$) with the code $\{\ldots, 0, -1, 2, 3, -3, 3, -3, 3, -3, 2, 1, 0,\ldots\}$. For each soliton the length of the code is $m=10$.
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  The relation between the entries of the code (18), the entries of the orbit $\mathbf{s}=\{\ldots p_{-1},p_0,p_1,\ldots\}$ and the values of $\widehat\psi(x)$ of (7), $i_1\ne 0$, $i_m\ne 0$
Code $a$ $\ldots$ $0$ $\ldots$ $0$ $i_1$ $\ldots$ $i_m$ $0$ $\ldots$
Orbit $\bf s$ $\ldots$ $p_{-M}$ $\ldots$ $p_0$ $p_1$ $\ldots$ $p_m$ $p_{m+1}$ $\ldots$
Island $\ldots$ $D_0$ $\ldots$ $D_0$ $D_{i_1}$ $\ldots$ $D_{i_m}$ $D_0$ $\ldots$
$\widehat\psi(x)$ $\ldots$ $\ldots$ $\widehat{\psi}(-M\pi)$ $\ldots$ $\widehat{\psi}(0)$ $\widehat{\psi}(\pi)$ $\widehat{\psi}(m\pi)$ $\widehat{\psi}((m+1)\pi)$ $\ldots$
Table Options

Download as excel

Code $a$ $\ldots$ $0$ $\ldots$ $0$ $i_1$ $\ldots$ $i_m$ $0$ $\ldots$
Orbit $\bf s$ $\ldots$ $p_{-M}$ $\ldots$ $p_0$ $p_1$ $\ldots$ $p_m$ $p_{m+1}$ $\ldots$
Island $\ldots$ $D_0$ $\ldots$ $D_0$ $D_{i_1}$ $\ldots$ $D_{i_m}$ $D_0$ $\ldots$
$\widehat\psi(x)$ $\ldots$ $\ldots$ $\widehat{\psi}(-M\pi)$ $\ldots$ $\widehat{\psi}(0)$ $\widehat{\psi}(\pi)$ $\widehat{\psi}(m\pi)$ $\widehat{\psi}((m+1)\pi)$ $\ldots$
[1]

Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225
[2]

Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214
[3]

Patrick Henning, Johan Wärnegård. Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation. Kinetic & Related Models, 2019, 12 (6) : 1247-1271. doi: 10.3934/krm.2019048
[4]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505
[5]

Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481
[6]

E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120
[7]

Philipp Bader, Sergio Blanes, Fernando Casas, Mechthild Thalhammer. Efficient time integration methods for Gross-Pitaevskii equations with rotation term. Journal of Computational Dynamics, 2019, 6 (2) : 147-169. doi: 10.3934/jcd.2019008
[8]

Thomas Chen, Nataša Pavlović. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 715-739. doi: 10.3934/dcds.2010.27.715
[9]

Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905
[10]

Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306
[11]

Yujin Guo, Xiaoyu Zeng, Huan-Song Zhou. Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3749-3786. doi: 10.3934/dcds.2017159
[12]

Shuai Li, Jingjing Yan, Xincai Zhu. Constraint minimizers of perturbed gross-pitaevskii energy functionals in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 65-81. doi: 10.3934/cpaa.2019005
[13]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[14]

A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419
[15]

Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[16]

Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637
[17]

Dan Endres, Martin Kummer. Nonlinear normal modes for the isosceles DST. Conference Publications, 1998, 1998 (Special) : 231-241. doi: 10.3934/proc.1998.1998.231
[18]

André de Laire, Pierre Mennuni. Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 635-682. doi: 10.3934/dcds.2020026
[19]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174
[20]

Vianney Combet. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1961-1993. doi: 10.3934/dcds.2014.34.1961

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (108)
  • HTML views (99)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences