Asymmetric localized states in periodic potentials with a domain wall-like Kerr nonlinearity
JJ. Phys. Commun. ( ))
JJ. Phys. Commun. ( )) // doi.org / / / ab07d1 PAPER
Asymmetric localized states in periodic potentials with a domain-wall-like Kerr nonlinearity
Jincheng Shi and Jianhua Zeng State Key Laboratory of Transient Optics and Photonics, Xi ’ an Institute of Optics and Precision Mechanics of CAS, Xi ’ an 710119,People ʼ s Republic of China University of Chinese Academy of Sciences, Beijing 100084, People ʼ s Republic of China Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of Information Photonic Technique,Xi ’ an Jiaotong University, Xi ’ an 710049, People ʼ s Republic of China E-mail: [email protected]:
Bose – Einstein condensates in periodic potentials, solitons, photonic crystals, nonlinear optics at surfaces, fi ber Bragg gratings,Kerr effect Abstract
We study the existence of one-dimensional localized states supported by linear periodic potentials anda domain-wall-like Kerr nonlinearity. The model gives rise to several new types of asymmetriclocalized states, including single- and double-hump soliton pro fi les, and multihump structures.Exploiting the linear stability analysis and direct simulations, we prove that these localized states areexceptional stable in the respective fi nite band gaps. The model applies to Bose – Einstein condensatesloaded onto optical lattices, and in optics with period potentials, e.g., the photonic crystals and opticalwaveguide arrays, thereby the predicted solutions can be implemented in the state-of-the-artexperiments.
1. Introduction
Periodic potentials ( structures ) are well-known and commonly used in physical science, among whichundoubtedly the most familiar case is the crystal lattice in solids [
1, 2 ] . In fact, we can say, without exaggeration,that the most intriguing feature of such periodic system is the emergence of forbidden band gaps where thewaves pertaining to the corresponding wave spectra are not permitted to transmit. Because of the existence ofresonant Bragg scattering from the periodic structures, such spectral gaps, in particular, give rise to a strongdestructive interference of multiple re fl ections of waves, and accordingly, can greatly constrain the wavedispersion and diffraction in nature [ ] . The control of wave dynamics in periodic potentials, during the lastdecades, has aroused growing interest of numerous researchers from different disciplines and fi elds, andsigni fi cant progress has been made in exploring new and more complex periodic structures, such as the photoniccrystal fi bers / waveguides [ – ] , optically induced photonic lattices [ – ] , optical lattices ( optical periodicpotentials formed by counter-propagating laser beams ) [
7, 8 ] , etc.When the self-focusing or defocusing nonlinearity comes into play, the physical systems with periodicpotentials show a lot of new phenomena [ – ] ; most notably, besides the ordinary ( fundamental ) solitonssupported by self-focusing effect, they can generate a novel class of solitons — the so-called bright gap solitons – the localized states residing inside the fi nite band gaps of the underlying linear spectrum under the action ofdefocusing nonlinearity and the strong Bragg scattering ( which leads to anomalous dispersion because of thenegative effective mass ) described above, contrary to the well established concept that defocusing effect can onlysupport dark solitons in uniform media [ ] . It have demonstrated, over the past years, that the gap solitons canbe created in diverse types of nonlinear periodic structures such as the optical fi ber Bragg gratings [ ] , photoniccrystals [ ] and lattices [ – ] , atomic Bose – Einstein condensates ( BECs ) trapped in optical lattices [
14, 18, 19 ] ,and quite recently, the exciton-polariton BECs in semiconductor microcavities with recon fi gurable latticestructures [ – ] . Further, other types of spatially localized states can be found in such nonlinear periodicsettings as well, for example, vortex solitons carrying topological charge [ – ] , multipole solitons [ ] and OPEN ACCESS
RECEIVED
24 January 2019
ACCEPTED FOR PUBLICATION
18 February 2019
PUBLISHED ( s ) and the title of the work, journal citation and DOI. © 2019 The Author ( s ) . Published by IOP Publishing Ltd runcated nonlinear Bloch waves ( or gap waves ) [ – ] . In particular, the latter case is self-trapped localizedstates with steep edges and arbitrary large atom numbers localized in a great number of deep optical lattice sites,which is distinct from the experimental realization of BEC gap solitons [ ] under constraints of low atomnumbers and densities.In the meanwhile, periodic potentials have recently been extended to the nonlinearity coef fi cient, which arethe ‘ nonlinear lattices ’ [ ] what we now call. The inhomogeneous modulation of nonlinearity has widely usedin nonlinear optics too, which is termed nonlinearity management [ ] . Contemporary and cutting-edgefabrication technologies have made it possible to create structures with the nonlinearity landscape of uniqueproperties featuring almost arbitrary transverse modulation [ ] . As such, the study of soliton phenomena intruly periodic nonlinear lattice ( the nonlinearity embedded into a linear uniform medium ) is particularlyintriguing, since it departs from the case in conventional linear lattices mentioned above with modulatedrefractive index. The stabilization of solitons and vortical ones in nonlinear lattices [ – ] , combined linearand nonlinear lattices [ – ] is increasingly being studied in past years, and recently the interest is also on thescenarios with inhomogeneous modulations of nonlinearity [ – ] . In particular, spatially inhomogeneousnonlinear media with a defocusing nonlinearity, whose local strength grows fast enough from the pivot to theperiphery, can uphold a vast variety of localized states, both the fundamental and higher-order solitons, whichare in the forms of solitary vortices ( with arbitrarily vortex charges ) [ ] , vortex rings [ ] , soliton gyroscopes [ ] and skyrmions [ ] , hop fi ons, complex hybrid modes, localized dark solitons and vortices [ ] .In this work, we study in detail, theoretically and numerically, the existence and dynamics of one-dimensional matter wave localized states in an optical lattice with a domain-wall-like Kerr nonlinearity,characterized by different local ( interatomic interaction ) constant strengths at the two semi-in fi nite regions ( andthus can be expressed by a step function ) . The model is able to support several new types of asymmetric localizedstates, including single- and double-hump soliton pro fi les, and multihump structures, which are stable in widelinear spectrum ( fi nite band gaps ) regions, veri fi ed by linear stability analysis and systematic simulations. Wediscuss the physical mechanism of the stabilization of these localized structures by such a combined linear latticeperiodic potentials and a simple step-function nonlinearity model.We stress that there is a very clear contrast between our model and the scenarios of semi-in fi nite periodiclattices ( the cases with a interface between an optical lattice and uniform media, while keeping the nonlinearityconstant ) , which were widely investigated for the surface gap solitons possessing a combination of the uniqueproperties exhibited by gap solitons and common features typical for nonlinear surface waves [ – ] .Physically, such type of nonlinear surface solitons are spatially localized at the interface, since the chemicalpotential ( or propagation constant, in optics ) , whose value is imaginary for a gap soliton residing at the opticallattice half interface, decides the penetration depth into uniform media, and therefore forming evanescent wavein this half [ ] . Since the presence of exponentially decaying tail ( evanescent wave ) in the uniform half, thesurface gap solitons existed merely at the linear interface [ ] , while in our model the nonlinear interface justtunes the amplitude and shape of solitons, we thus do not term them surface solitons. Our model is differentfrom the purely nonlinear interfaces supported by two different nonlinear coef fi cients ( while the linear potentialis constant ) [ – ] , either [ in particular, the theory of light-beam propagation at nonlinear interfaces had beenwell developed in the literatures [
63, 64 ] three decades ago ] . Because of lacking of linear lattices with spatiallymodulated refractive index and thus the fascinating tunable band gap in the system ’ s linear spectrum, suchnonlinear interfaces model, however, cannot support the localized states ( both solitons and vortical ones ) of gaptypes. The model proposed here therefore introduces a full linear periodic potential and shares the properties ofpurely nonlinear interfaces.The rest of this work is organized as follows. In section 2, we present our model and analyze the band-gapdiagram of Bloch modes in the linear periodic systems ( the non-interacting BECs in optical lattices ) , the linearstability analysis for the spatially localized nonlinear modes based on eigenvalue problem is also introduced inthis section. Several kinds of asymmetric localized states, including single- and double-hump solitons, andmultihump structures, are predicted and studied systematically by numerical computation in section 3.Section 4 discusses the experimental conditions for realizing the predicted solutions, concludes with a summaryand further extension.
2. Model
We consider dynamics of atomic BECs loaded into an optical lattice with a domain-wall-type Kerr cubicnonlinearity described by the mean- fi eld Gross-Pitaevskii equation ( or nonlinear Schrödinger equation ) fordimensionless scale of the wave function ψ ( x , t ) :2 J. Phys. Commun. ( ))
We consider dynamics of atomic BECs loaded into an optical lattice with a domain-wall-type Kerr cubicnonlinearity described by the mean- fi eld Gross-Pitaevskii equation ( or nonlinear Schrödinger equation ) fordimensionless scale of the wave function ψ ( x , t ) :2 J. Phys. Commun. ( )) V x g x
12 sin , 1 t x y y y y y = - ¶ + + ( ) ( )∣ ∣ ( ) where V is the strength of the optical lattice, and the nonlinear coef fi cient g ( x ) yields: g x g xg x , 0,, 0. 2 lr = > ⎧⎨⎩ ( ) ( ) Here the real constant coef fi cients g l and g r are in the same sign, and thus the nonlinearity suffers a suddenchange only in magnitude. We de fi ne the parameter γ = g r / g l , for the sake of discussion, and set g l º ∣ ∣ . Forcomparison, we also discuss the uniform nonlinearity at two constant nonlinear coef fi cients g and g whosevalues will be speci fi cally given below. The nonlinearity inherent to BECs because of the inevitable atom-atomcollisions could be tuned with the popular technique called Feshbach resonance. We stress that the equation ( ) also describes optical wave propagation in nonlinear optics with a replacement of time t by propagationdistance z .The stationary solution of wave function ψ ( x , t ) is usually sought as x t x i t , exp y f m = - ( ) ( ) ( ) withchemical potential μ , in doing so the equation ( ) satis fi es V x g x
12 sin . 3 x mf f f f = - ¶ + + ( ) ( ) ( ) To understand the matter-wave localized modes and their properties in the model with linear periodic lattice ( optical lattice ) it is necessary to give fi rst the relevant band-gap structure. The linearization of equation ( ) results into the following eigenvalue equation: V x
12 sin . 4 x mf f f = - ¶ + ( ) ( ) The solution of eigenvalue problem ( equation ( )) can produce the underlying band-gap spectrum μ ( K ) ,characterized by the momentum K inside such periodic lattice. Speci fi cally, according to the well-knownBloch ’ s theorem borrowing from solid-state physics, eigenfunctions of equation ( ) f are periodic solutionsknown as Floquet-Bloch modes, and represented by their momentum K inside the lattice, provided thateigenvalues μ inside the energy ( Bloch ) bands. Typical Floquet-Bloch mode at momentum K is written as x x iKx exp K K f = F ( ) ( ) ( ) , here periodic function Φ K ( x ) with a period equaling to that of the lattice. In doing so,we can get the band-gap spectrum for an optical lattice ( we refer the readers to consult the very relevant papers in [
65, 66 ] and books in [
2, 3 ] for more details ) . As shown in fi gure 1 for the example at V =
6, besides the usualsemi-in fi nite gap, the fi rst two fi nite band gaps are also existed. Inside these band gaps there are not any localizedwaves in linear case, as mentioned above, while such an acknowledgement would be overturned in the nonlinearscenario where a combination of periodicity and nonlinearity would lead to the appearance of families ofdifferent localized states in such linearly forbidden band gaps. To settle the stability problem of localized modes, we take the perturbed solutions as t i t i t exp exp exp , 5 y f u l w l m = + + - [ ( ) ( )] ( ) ( )
Figure 1.
Band-gap structure of linear spectrum μ ( K ) , induced by the optical lattice V x sin ( ) for V =
6. Regions SIG, 1st BG and2nd BG correspond, respectively, to the semi-in fi nite gap, the fi rst and second band gaps ( similarly hereinafter in fi gures 3 and 8 ) . J. Phys. Commun. ( ))
6. Regions SIG, 1st BG and2nd BG correspond, respectively, to the semi-in fi nite gap, the fi rst and second band gaps ( similarly hereinafter in fi gures 3 and 8 ) . J. Phys. Commun. ( )) ere υ and ω are the real and imaginary parts of in fi nitesimal perturbation eigenfunctions, λ is the homologousperturbation eigenvalue ( or growth rate ) . The linearization of equation ( ) around stationary solution f foundfrom equation ( ) results in the eigenvalue problem for λ i V x g x
12 sin , 6 xx lu w m w f w = - + + + [ ( )] ( ) ( ) i V x g x
12 sin 3 . 7 xx lw u m u f u = - + + + [ ( )] ( ) ( ) The linear stability analysis of perturbed localized solutions and associated growth rate λ based on theperturbation equations ( ) and ( ) can be solved numerically, and evidently, the perturbed localized modes arestable as long as Re ( λ ) = ( λ ) .
3. Asymmetric spatially localized modes We fi rst investigate the formation of fundamental localized modes, e.g., the aforementioned single-hump wavestructures in the forms of ordinary solitons and gap ones existing within semi-in fi nite gap and fi nite band gaps ofthe relevant linear spectrum, separatively, and under the self-attractive and self-repulsive nonlinearities. As amatter of fact, it should be noticed, in the condition of uniform nonlinearity, that such single-hump structuresare known as symmetrical modes. For comparison, we have plotted these ordinary solitons in fi gure 2 ( a ) , and thegap solitons as populated in the fi rst two band gaps in fi gures 2 ( b ) and ( c ) , for the two constant nonlinearities g and g ( the former is in dotted red lines, and dashed blue lines for the latter ) , under the same chemical potential μ . One can clearly observe that both kinds of solitons shrink at larger nonlinearity g , conforming to thenonlinear saturation mechanism of the physical system. And due to this fact, the corresponding single-humplocalized wave structures, supported by the current domain-wall-like nonlinearity given in equation ( ) , stand inthe middle of the cases with constant nonlinearities g and g .Interestingly, the unique property of our nonlinearity renders the localized waves stay away from the centre x =
0, resulting in asymmetrical shapes of the wave structures ( besides the single-hump structures here in fi gures 2 ( a ) – ( c ) , asymmetric spatially localized modes are also for the double-hump structures and multihumpones below in fi gures 5 and 6 ) . To elucidate the principle of this deviation, we de fi ne a parameter Δ x to measurethe off-centre value between peak position of the solitons and geometric center x =
0. A scrutinized analysisfound that the deviation Δ x > fi gure 1 ( a ) , while Δ x < fi gures 1 ( b ) , ( c ) . The former case may be explained by the fact that, to maintain a balance in thecurrent step nonlinearity, the asymmetric localized waves prefer to stay at the ( right ) side of larger self-focusingnonlinearity ( g r = − ) in which the threshold value of norm N for generating a stable ordinary soliton issmaller compared to the other ( left ) side with lower nonlinearity ( g l = − ) . For the latter in band gaps, by Figure 2.
Pro fi les of the stable ( single-hump ) fundamental solitons supported by the optical lattice and different types of nonlinearity:uniform nonlinearities ( the dotted red and dashed blue lines ) and the domain-wall-like shape given by equation ( ) ( black solid lines ) ,in the semi-in fi nite gap ( a ) , and the fi rst ( b ) and second ( c ) band gaps. Here and in fi gures 4 and 5, the parameter Δ x represents theshifting value of peak positions ( of the solitons ) between the cases in uniform and domain-wall-like nonlinearities. ( d ) The changes of Δ x with the variation of γ at μ = fi rst band gap, dashed blue line is the fi tting one. J. Phys. Commun. ( ))
Pro fi les of the stable ( single-hump ) fundamental solitons supported by the optical lattice and different types of nonlinearity:uniform nonlinearities ( the dotted red and dashed blue lines ) and the domain-wall-like shape given by equation ( ) ( black solid lines ) ,in the semi-in fi nite gap ( a ) , and the fi rst ( b ) and second ( c ) band gaps. Here and in fi gures 4 and 5, the parameter Δ x represents theshifting value of peak positions ( of the solitons ) between the cases in uniform and domain-wall-like nonlinearities. ( d ) The changes of Δ x with the variation of γ at μ = fi rst band gap, dashed blue line is the fi tting one. J. Phys. Commun. ( )) ontrast, a bigger defocusing nonlinearity ( g r = ) , which repels the wave on the right half side, squeezes thewave to the left side ( g l = ) , leading to a negative Δ x. Such deviation Δ x ( absolute value ) grows in much higherband gaps owning to their much stronger Bragg scattering, as seen from a comparison of fi gures 1 ( b ) and ( c ) . Wefurther found, from numerous calculations, that the Δ x decreases quickly with an increase of γ , and then reachesto a certain value at large γ , as shown for the case at μ = − ( in fi rst band gap ) in fi gure 1 ( d ) .A series of numerical computations, relied on the linear stability analysis ( perturbation equations ( ) and ( )) and dynamical equation ( ) , demonstrate that such single-hump matter-wave structures are very stable,exceptional cases are those near the band edges where exist weak oscillatory instability ( λ r ∼ − − − ) which, particularly, grows when going deeper inside band gaps, see the relevant linear stability results in the toppanel of fi gure 4 and its representative dynamical evolution for both stable and unstable perturbation modes inthe bottom left and right panels.Besides the single-hump wave structures, the present model also supports different double-hump localizedmodes, which, depending on their shapes, can be categorized as two types — with one zero ( node ) and lackthereof, examples of them are plotted in fi gure 5. A common feature of both types localized modes is that theyare composed by a single-hump localized state and a triple-hump one at constant nonlinearities g and g ,respectively. Their stability regions within the fi rst two band gaps are collected in fi gure 3. Figure 3.
Number of atoms N versus chemical potential μ for different types of localized states, induced by the optical lattice at V = γ =
4. The structure of the single-hump modes corresponding to the marked black points I ,I and I is shown in fi gures 2 ( a ) , ( b ) , ( c ) . The double-hump modes for the marked points II a1 and II a2 are respectively shown in fi gures 5 ( a ) , ( b ) , while II b1 and II b2 are in fi gures 5 ( c ) , ( d ) . Shaded regions depict the linear Bloch bands. Figure 4.
Top: real part of the perturbation growth rate ( λ r ) versus chemical potential ( μ ) for the single-hump fundamental solitons.Bottom: Temporal evolution of stable ( left ) and unstable ( right ) localized modes corresponding to the marked points I and I on thetop and in fi gure 3. J. Phys. Commun. ( ))
Top: real part of the perturbation growth rate ( λ r ) versus chemical potential ( μ ) for the single-hump fundamental solitons.Bottom: Temporal evolution of stable ( left ) and unstable ( right ) localized modes corresponding to the marked points I and I on thetop and in fi gure 3. J. Phys. Commun. ( )) espite surface localized states, including surface solitons and gap ones, have been widely considered at twoclasses of interfaces: the purely nonlinear interface ( or with an additional aperiodic potential ) [
61, 62 ] , and at theinterface between a semi-in fi nite lattice and uniform media ( with a constant nonlinearity ) [ – ] , these modelshave restrictions: the former does not exhibit unique feature of linear periodic potential, the latter loses thenonlinearity-mediated ( e.g., step nonlinearity ) stabilization, and the lattice potential is only limited to half plate ( not the whole space ) . Therefore, the investigation of localized states in the present model — full periodpotentials with a domain-wall-like nonlinearity — is relatively new. Our physical setting ( and thereby thepredicted asymmetric localized states ) integrates speci fi c properties typical for periodic potentials and nonlinearinterfaces. It is also worthwhile mentioning here that the localized states, supported by the above two kinds ofinterfaces, are spatially localized at and near the interface, while a notable characteristic of the predictedasymmetric localized states ( in our model ) can be loosely localized and occupy many lattice sites ( thus across theinterface in both directions ) , see the following multiple-peak waves structures in fi gure 6. The above predicted localized states, including both single-hump fundamental solitons and double-hump ones,are all populated at and near the nonlinear interface, a natural question one may ask lies in whether localizedstates can be existed across such interface? If possible, subsequent questions arise: what are the formation
Figure 5.
Two families of double-hump localized states: with one zero nodes ( a ) , ( b ) and without nodes ( c ) , ( d ) . Figure 6.
Two families of triple-peak localized states: with one zero nodes ( dashed line ) and the impending patterns ( solid line ) originating from the fi rst ( a ) and second ( b ) band gaps. ( c ) , ( d ) : The similar cases for quadruple-peak localized structures. J. Phys. Commun. ( ))
Two families of triple-peak localized states: with one zero nodes ( dashed line ) and the impending patterns ( solid line ) originating from the fi rst ( a ) and second ( b ) band gaps. ( c ) , ( d ) : The similar cases for quadruple-peak localized structures. J. Phys. Commun. ( )) onditions and how to generate them? Are they stable in fi nite band gaps? To this end, We now test the possibilityfor creating multiple-peak ( or multihump ) wave structures, which are of particular interest in experiments too,since the experimental observed BEC gap solitons were at very low atom numbers ( it is thus a challenging task torealize them in conventional labs ) [ ] .Numerical simulations suggest that the model also supports triple-peak and quadruple-peak localized states,as seen from the typical examples of the former lying in the fi rst and second band gaps portrayed in fi gures 6 ( a ) and ( b ) . Representative modes for the latter are shown in the fi gures 6 ( c ) and ( d ) . We emphasize that bothlocalized states occupy several lattice sites ( hence they go beyond the nonlinear interface ) , and can as well beclassi fi ed as two kinds depending on the number of zero ( node ) — with one and null, like their double-peakcounterparts in fi gure 5. Stable evolutions of the triple-peak and quadruple-peak localized states aredemonstrated in fi gure 7 through direct numerical simulations of them in real time. Their stability regions arecollected in fi gure 8. Based on these results, we speculate and fi rmly believe that different matter-wave structureswith more peaks — broad asymmetric localized states — can be constructed theoretically and realized inexperiments. Figure 8.
Norm N versus chemical potential μ for families of different localized states: triple-peak structures ( the two lines III a and III b at the bottom ) and the quadruple-peak modes ( the fi rst two lines IV a and IV b at the top ) . Marked points III a1 ( dark ) and III b1 ( gray ) are plotted in fi gure 6 ( a ) , the points III a2 and III b2 are in fi gure 6 ( b ) ; similarly, IV a1 ( blue ) and IV b1 ( red ) are in ( c ) , IV a2 and IV b2 are in ( d ) of the same fi gure. Figure 7.
Temporal evolutions of triple-peak ( a ) , ( c ) and quadruple-peak ( b ) , ( d ) impending localized states in fi gures 6 ( a ) , ( b ) and ( c ) , ( d ) . The stable localized states are shown in panels ( a, b ) and the unstable patterns for panels ( c ) , ( d ) . J. Phys. Commun. ( ))
Temporal evolutions of triple-peak ( a ) , ( c ) and quadruple-peak ( b ) , ( d ) impending localized states in fi gures 6 ( a ) , ( b ) and ( c ) , ( d ) . The stable localized states are shown in panels ( a, b ) and the unstable patterns for panels ( c ) , ( d ) . J. Phys. Commun. ( )) . Conclusion We studied dynamics of one-dimensional atomic Bose – Einstein condensates ( BECs ) in an optical lattice with adomain-wall-like Kerr cubic nonlinearity, whose strength is made up of two different constants in bothdirections ( the left and right half-planes ) , in the framework of the mean- fi eld Gross-Pitaevskii equation. Severalnew types of localized states with asymmetric shape — asymmetric localized states, including single-, double- andmultiple-hump modes, inside relevant gaps ( both semi-in fi nite and the fi rst two fi nite band gaps ) of theunderlying linear spectrum, are found. We have analyzed the physical mechanism of such localized wavestructures, and addressed their stability properties by means of linear stability analysis and direct numericalsimulations. A notable feature is the existence of stable multiple-peak wave structures in fi nite band gaps,making the observation of localized waves of gap type more accessible in conventional ultracold atomslaboratories, since a facing challenge to observe localized gap solitons is the low atom densities [ the reportedresult is only about 250 atoms ] [ ] .Since Gross-Pitaevskii equation is the fundamental governing system equivalent to the nonlinearSchrödinger equation in nonlinear optics, thereby besides the BECs loaded into an optical lattice, the predictedsolutions can also be realized in other periodic potentials, including fi ber Bragg gratings, photonic crystals andlattice, as well as the ordinary waveguide arrays. As such, a natural extension is to study the asymmetric localizedstates and their propagation dynamics in such nonlinear optical systems, particularly in experiments. An issue ofgreat interest is to consider more complicated situations with increasing dimensions and ingredients like thetwo-component system. Acknowledgments
This work was supported, in part, by the NSFC, China ( project Nos. 61690224, 61690222 ) , and by the YouthInnovation Promotion Association of the Chinese Academy of Sciences ( project No. 2016357 ) . ORCID iDs
Jincheng Shi https: // orcid.org / // orcid.org / References [ ] Pelinovsky D E 2011
Localization in Periodic Potential: From Schr ö dinger Operators to the Gross-Pitaevskii equation ( Cambridge:Cambridge University Press ) [ ] Kivshar Y S and Agrawal G P 2003
Optical Solitons: From Fibers to Photonic Crystals ( San Diego, CA: Academic ) [ ] Joannopoulos J D, Johnson S G, Winn J N and Meade R D 2008
Photonic Crystals: Molding the Flow of Light ( Princeton: PrincetonUniversity Press ) [ ] Christodoulides D N, Lederer F and Silberberg Y 2003
Nature (London) [ ] Lederer F, Stegeman G I, Christodoulides D N, Assanto G, Segev M and Silberberg Y 2008
Phys. Rep. [ ] Garanovich I L, Longhi S, Sukhorukov A A and Kivshar Y S 2012
Phys. Rep. [ ] Brazhnyi V A and Konotop V V 2004
Mod. Phys. Lett. B [ ] Morsch O and Oberthaler M 2006
Rev. Mod. Phys. [ ] Malomed B A, Mihalache D, Wise F and Torner L 2005
J. Optics B: Quant. Semicl. Opt. R53 [ ] Kartashov Y V, Vysloukh V A and Torner L 2009
Progress in Optics ed E Wolf 52 ( North Holland: Amsterdam ) – [ ] Malomed B A, Mihalache D, Wise F and Torner L 2005
J. Optics B R53 [ ] Baizakov B B, Malomed B A and Salerno M 2003
Europhys. Lett. Phys. Rev. A [ ] Yang J and Musslimani Z H 2003
Opt. Lett. [ ] Kevrekidis P G, Frantzeskakis D J and Carretero-González R ( ed ) Emergent Nonlinear Phenomena in Bose – Einstein Condensates ( Berlin: Springer ) [ ] Pethick C J and Smith H 2008
Bose – Einstein Condensate in Dilute Gas ( Cambridge: Cambridge University Press ) [ ] Carretero- González R et al ( ed ) Localized Excitations in Nonlinear Complex Systems ( Heidelberg: Springer ) [ ] Eggleton B J, Slusher R E, de Sterke C M, Krug P A and Sipe J E 1996
Phys. Rev. Lett. [ ] Ostrovskaya E A and Kivshar Y S 2004
Phys. Rev. Lett. [ ] Eiermann B, Anker T, Albiez M, Taglieber M, Treutlein P, Marzlin K-P and Oberthaler M K 2004
Phys. Rev. Lett. [ ] Ostrovskaya E A, Abdullaev J, Fraser M D, Desyatnikov A S and Kivshar Y S 2013
Phys. Rev. Lett. [ ] Cerda-Méndez E A, Sarkar D, Krizhanovskii D N, Gavrilov S S, Biermann K, Skolnick M S and Santos P V 2013
Phys. Rev. Lett. [ ] Tanese D et al
Nat. Commun. [ ] Ostrovskaya E A and Kivshar Y S 2004
Phys. Rev. Lett. [ ] Sakaguchi H and Malomed B A 2009
Phys. Rev. A [ ] Zeng J and Malomed B A 2017
Vortex Structures in Fluid Dynamic Problems ( Rijeka: InTech ) ch 10 ( https: // doi.org / / ) [ ] Lobanov V E, Kartashov Y V and Konotop V V 2014
Phys. Rev. Lett. J. Phys. Commun. ( ))
Phys. Rev. Lett. J. Phys. Commun. ( )) ] Anker T, Albiez M, Gati R, Hunsmann S, Eiermann B, Trombettoni A and Oberthaler M K 2005
Phys. Rev. Lett. [ ] Alexander T J, Ostrovskaya E A and Kivshar Y S 2006
Phys. Rev. Lett. [ ] Zhang Y and Wu B 2009
Phys. Rev. Lett. [ ] Bennet F H, Alexander T J, Haslinger F, Mitchell A, Neshev D N and Kivshar Y S 2011
Phys. Rev. Lett. [ ] Bersch C, Onishchukov G and Peschel U 2012
Phys. Rev. Lett. [ ] Kartashov Y V, Malomed B A and Torner L 2011
Rev. Mod. Phys. [ ] Malomed B A 2006
Soliton Management in Periodic Systems ( Heidelberg: Springer ) [ ] Sakaguchi H and Sakaguchi B A 2005
Phys. Rev. E [ ] Theocharis G, Schmelcher P, Kevrekidis P G and Frantzeskakis D J 2005
Phys. Rev. A [ ] Abdullaev F K and Garnier J 2005
Phys. Rev. A ( R ) [ ] Sivan Y, Fibich G and Weinstein M I 2006
Phys. Rev. Lett. [ ] Belmonte-Beitia J, Pérez-García V M, Vekslerchik V and Torres P J 2007
Phys. Rev. Lett. [ ] Abdullaev F K, Gammal A, Salerno M and Tomio L 2008
Phys. Rev. A [ ] Kartashov Y V, Malomed B A, Vysloukh V A and Torner L 2009
Opt. Lett. Opt. Lett. [ ] Zeng J and Malomed B A 2012
Phys. Rev. A [ ] Shi J, Zeng J and Malomed B A 2018
Chaos [ ] Gao X and Zeng J 2018
Front. Phys. [ ] Salasnich L, Cetoli A, Malomed B A, Toigo F and Reatto L 2007
Phys. Rev. A [ ] Kartashov Y V, Vysloukh V A and Torner L 2008
Opt. Lett. Opt. Lett. [ ] Sakaguchi H and Malomed B A 2010
Phys. Rev. A [ ] Zeng J and Malomed B A 2012
Phys. Scr. T [ ] Borovkova O V, Kartashov Y V, Torner L and Malomed B A 2011
Phys. Rev. E ( R ) [ ] Driben R, Kartashov Y V, Malomed B A, Meier T and Torner L 2014
Phys. Rev. Lett. [ ] Kartashov Y V, Malomed B A, Shnir Y and Torner L 2014
Phys. Rev. Lett. [ ] Zeng J and Malomed B A 2012
Phys. Rev. E [ ] Tian Q, Wu L, Zhang Y and Zhang J-F 2012
Phys. Rev. E [ ] Cardoso W B, Zeng J, Avelar A T, Bazeia D and Malomed B A 2013
Phys. Rev. E [ ] Wu Y, Xie Q, Zhong H, Wen L and Hai W 2013
Phys. Rev. A [ ] Driben R, Kartashov Y V, Malomed B A, Meier T and Torner L 2014
New J. Phys. [ ] Zeng J and Malomed B A 2017
Phys. Rev. E [ ] Kartashov Y V, Vysloukh V A and Torner L 2006
Phys. Rev. Lett. [ ] Suntsov S, Makris K G, Christodoulides D N, Stegeman G I, Haché A, Morandotti R, Yang H, Salamo G and Sorel M 2006
Phys. Rev.Lett. [ ] Rosberg C R, Neshev D N, Krolikowski W, Mitchell A, Vicencio R A, Molina M I and Kivshar Y S 2006
Phys. Rev. Lett. [ ] Szameit A, Kartashov Y V, Dreisow F, Pertsch T, Nolte S, Tünnermann A and Torner L 2007
Phys. Rev. Lett. [ ] Dong L and Li H 2010
J. Opt. Soc. Am. B [ ] Ye F, Kartashov Y V and Torner L 2006
Phys. Rev. A [ ] Aceves A B, Moloney J V and Newell A C 1989
Phys. Rev. A [ ] Aceves A B, Moloney J V and Newell A C 1989
Phys. Rev. A [ ] Louis P J Y, Ostrovskaya E A, Savage C M and Kivshar Y S 2003
Phys. Rev. A [ ] Efremidis N K and Christodoulides D N 2003
Phys. Rev. A J. Phys. Commun. ( ))