American Institute of Mathematical Sciences

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

 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.

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

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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
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.
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 $A=-3$.
An island $D$ with $v$-curve $\beta$, $v$-strip $V$, $h$-curve $\alpha$ and $h$-strip $H$.
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 $D_2$.
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.
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 $H_{01(-1)0}$.
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 $h_{0\times M}^+$.
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).
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).
(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 $m=10$.
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 $\ldots$ $D_0$ $\ldots$ $D_0$ $D_{i_1}$ $\ldots$ $D_{i_m}$ $D_0$ $\ldots$ $\widehat\psi(x)$ $\ldots$ $\ldots$ $\widehat{\psi}(-M\pi)$ $\ldots$ $\widehat{\psi}(0)$ $\widehat{\psi}(\pi)$ $\widehat{\psi}(m\pi)$ $\widehat{\psi}((m+1)\pi)$ $\ldots$
 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 $\ldots$ $D_0$ $\ldots$ $D_0$ $D_{i_1}$ $\ldots$ $D_{i_m}$ $D_0$ $\ldots$ $\widehat\psi(x)$ $\ldots$ $\ldots$ $\widehat{\psi}(-M\pi)$ $\ldots$ $\widehat{\psi}(0)$ $\widehat{\psi}(\pi)$ $\widehat{\psi}(m\pi)$ $\widehat{\psi}((m+1)\pi)$ $\ldots$
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