In a Kerr-nonlinear photonic band-gap (PBG) material, optical solitary waves with frequencies inside a complete band gap are described, in the slowly varying envelope approximation, by an effective nonlinear Dirac equation for multi-component spinor envelopes. Aközbek and John (Phys. Rev. E 57, 2287 (1998)) constructed two- and three-dimensional gap solitary waves of these equations by variational and finite-element methods and proposed a Vakhitov–Kolokolov-type slope criterion, ∂Q/∂δ<0\partial Q/\partial\delta<0, for their linear stability, concluding that two-dimensional gap solitons are stable while three-dimensional ones are stable only in parts of the gap. The criterion was, by the authors' own statement, heuristic: a rigorous spectral-theoretic proof (or counterexamples) for general photonic-crystal data was explicitly left open. This problem asks for such a rigorous justification or refutation of the stability criterion for the multi-dimensional nonlinear Dirac-type coupled-mode equations, including the case where the self-adjoint block of the linearized operator has several negative eigenvalues and the case of general, non-separable band-edge geometry, together with the rigorous derivation of the envelope equations themselves from the vector Kerr–Maxwell system in these settings. Later literature has rigorously justified the coupled-mode approximation and soliton existence in several two- and higher-dimensional model problems, and has proved spectral or asymptotic stability only for one-dimensional or small-amplitude Dirac solitary waves; the finite-amplitude multi-dimensional stability core remains open.

Background

A photonic band-gap (PBG) or photonic crystal is a dielectric material with a periodic refractive index, whose linear electromagnetic spectrum splits into Bloch bands separated by frequency gaps (photonic band gaps) in which no linear propagation is possible. Here a Bloch wave is a solution of the linear Maxwell equations in the periodic medium, labeled by a quasi-momentum in the Brillouin zone; special quasi-momenta at which band edges occur are the symmetry points of the crystal. In a Kerr medium the polarization acquires a cubic nonlinear term (the third-order susceptibility χ(3)\chi^{(3)}, the 'instantaneous Kerr response'), which can balance the strong Bragg scattering of the periodic structure and localize light inside the band gap as a gap soliton, the multi-dimensional analogue of one-dimensional Bragg and fiber-grating gap solitons. Aközbek and John (1998) analyzed such solitary waves in PBG structures of two-dimensional square and triangular and three-dimensional face-centered-cubic symmetry. Using the slowly varying envelope approximation (SVEA) — an expansion of the field in linear Bloch modes at a band edge with slowly varying complex amplitudes — they reduced the scalar Maxwell problem to an effective nonlinear Dirac equation: a first-order system, of Dirac type, for a spinor of envelope amplitudes, in which the band-edge frequency plays the role of a mass term and the frequency detuning inside the gap plays the role of the relativistic energy parameter. At higher-order symmetry points several Bloch modes couple, giving a large spinor system. They found two classes of solitary-wave solutions (associated with higher-order and with twofold symmetry points, the latter existing only for a positive Kerr coefficient), computed their size, shape, peak intensity, and energy variationally and confirmed the results by a finite-element Ritz–Galerkin method. For stability they proposed a criterion of Vakhitov–Kolokolov type. The Vakhitov–Kolokolov criterion for the nonlinear Schrödinger equation is the classical result that a standing wave is stable when the conserved mass (charge) is an increasing function of the propagation constant; for first-order Dirac-type systems no such theorem exists, because the linearized operator about the wave is non-self-adjoint with essential spectrum on the imaginary axis, the conserved energy functionals are sign-indefinite, and the Grillakis–Shatah–Strauss orbital-stability machinery does not apply. Aközbek and John stated explicitly that an analytical stability criterion for spinor fields satisfying nonlinear Dirac-type equations 'may exist' and that their criterion merely reproduces the stability conditions of other nonlinear Dirac equations that had been studied numerically; on its basis they predicted stability of 2D gap solitons and stability of 3D gap solitons only in certain regions of the gap. Rigorous work since then has, on the one hand, justified the envelope reduction and the existence of gap solitons for coupled-mode equations in two and higher dimensions (Dohnal, Pelinovsky, Schneider, and others, including a justification inside the two-dimensional Kerr nonlinear Maxwell system) and, on the other hand, proved spectral or asymptotic stability of nonlinear Dirac solitary waves only in one dimension or in the small-amplitude nonrelativistic limit, as reviewed in the monograph of Boussaïd and Comech. The conjunction — rigorous stability theory for finite-amplitude multi-dimensional gap solitons of the Dirac-type envelope equations of PBG structures — is what remains unresolved.

Problem Statement

Establish a rigorous spectral-stability theory for the standing-wave gap solitons of the nonlinear Dirac-type coupled-mode (envelope) equations that govern two- and three-dimensional optical solitary waves in Kerr-nonlinear photonic band-gap structures, as derived by Aközbek and John. Concretely: prove or disprove, by rigorous spectral-theoretic analysis or explicit counterexamples, that the proposed Vakhitov–Kolokolov-type slope criterion ∂Q/∂δ<0\partial Q/\partial\delta<0 — where QQ is the conserved charge of the spinor solitary wave and δ\delta is the frequency detuning of the solitary wave inside the band gap — correctly identifies the linear (spectral) stability of these solitary waves in spatial dimension d≥2d\ge 2. The analysis must cover (i) the self-adjoint block of the linearized operator having multiple negative eigenvalues, where simple negative-eigenvalue-counting arguments do not apply and Krein-signature-type information is needed (the Krein signature of an eigenvalue on the imaginary axis is the sign of the quadratic form of the conserved energy functional restricted to the associated invariant eigenspace; a collision of eigenvalues of opposite Krein signature is the mechanism by which a real-axis eigenvalue pair, i.e. oscillatory instability, can be born); (ii) general photonic-crystal band-edge data, i.e. non-separable periodic structures and the multiple coupled Bloch-mode components arising at higher-order symmetry points, rather than only separable or single-mode model geometries; and (iii) the interaction of the discrete eigenvalues with the essential spectrum of the linearized Dirac-type operator (the part of the spectrum generated by the ambient periodic medium rather than by the localized wave), which lies on the imaginary axis and touches it at the threshold frequencies, so that eigenvalues may bifurcate from embedded thresholds. Where the criterion fails, identify the corrected criterion and the resulting stability classification, and decide in particular whether the source predictions — two-dimensional gap solitons stable for ideal Kerr media, three-dimensional ones stable only in parts of the gap — are rigorously true. The same program includes the rigorous justification of the underlying envelope reduction from the vector Kerr–Maxwell system in the two- and three-dimensional multi-symmetry-point settings, which later work has so far established only for restricted two-dimensional single-band-edge geometries.

References

  1. Neşet Aközbek and Sajeev John, "Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures," Phys. Rev. E 57, 2287–2319 (1998), DOI 10.1103/PhysRevE.57.2287, https://doi.org/10.1103/PhysRevE.57.2287
  2. Tomáš Dohnal, Dmitry Pelinovsky, and Guido Schneider, "Coupled-Mode Equations and Gap Solitons in a Two-Dimensional Nonlinear Elliptic Problem with a Separable Periodic Potential," Journal of Nonlinear Science 19, 95–131 (2008), DOI 10.1007/s00332-008-9027-9, arXiv:0707.3731, https://doi.org/10.1007/s00332-008-9027-9
  3. Tomáš Dohnal and Hannes Uecker, "Coupled mode equations and gap solitons for the 2D Gross–Pitaevskii equation with a non-separable periodic potential," Physica D 238, 860–879 (2009), DOI 10.1016/j.physd.2009.02.013, arXiv:0810.4499, https://doi.org/10.1016/j.physd.2009.02.013
  4. Tomáš Dohnal and Willy Dörfler, "Coupled Mode Equation Modeling for Out-of-Plane Gap Solitons in 2D Photonic Crystals," Multiscale Modeling and Simulation 11, 162–191 (2013), DOI 10.1137/120865914, arXiv:1202.3583, https://doi.org/10.1137/120865914
  5. Tomáš Dohnal and Lisa Wahlers, "Coupled mode equations and gap solitons in higher dimensions," Journal of Differential Equations 269, 2386–2418 (2020), DOI 10.1016/j.jde.2020.01.037, arXiv:1810.04944, https://doi.org/10.1016/j.jde.2020.01.037
  6. Tomáš Dohnal and Lisa Wahlers, "Bifurcation of Gap Solitons in Coupled Mode Equations in d Dimensions," Journal of Dynamics and Differential Equations 33, 891–919 (2021), DOI 10.1007/s10884-021-09971-7, arXiv:1903.02631, https://doi.org/10.1007/s10884-021-09971-7
  7. Tomáš Dohnal and Giulio Romani, "Justification of the asymptotic coupled mode approximation of out-of-plane gap solitons in Maxwell equations," Nonlinearity 34, 5261–5318 (2021), DOI 10.1088/1361-6544/ac0485, arXiv:2010.03473, https://doi.org/10.1088/1361-6544/ac0485
  8. Marina Chugunova and Dmitry Pelinovsky, "Block-diagonalization of the linearized coupled-mode system," arXiv:math/0504442 (2005), https://arxiv.org/abs/math/0504442
  9. Dmitry E. Pelinovsky and Atanas Stefanov, "Asymptotic stability of small gap solitons in nonlinear Dirac equations," Journal of Mathematical Physics 53 (2012), DOI 10.1063/1.4731477, arXiv:1008.4514, https://doi.org/10.1063/1.4731477
  10. Gregory Berkolaiko and Andrew Comech, "On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D," Mathematical Modelling of Natural Phenomena 7 (2), 13–31 (2012), DOI 10.1051/mmnp/20127202, arXiv:0910.0917, https://doi.org/10.1051/mmnp/20127202
  11. Nabile Boussaïd and Andrew Comech, "Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity," Journal of Functional Analysis 277, 108289 (2019), DOI 10.1016/j.jfa.2019.108289, arXiv:1705.05481, https://doi.org/10.1016/j.jfa.2019.108289
  12. Nabile Boussaïd and Andrew Comech, "Nonlinear Dirac Equation: Spectral Stability of Solitary Waves," Mathematical Surveys and Monographs vol. 244, American Mathematical Society, Providence, RI (2019), DOI 10.1090/surv/244, ISBN 978-1-4704-4395-5, https://doi.org/10.1090/surv/244
  13. Dmitry E. Pelinovsky, "Book Review: Nonlinear Dirac equations: Spectral stability of solitary waves, by N. Boussaïd and A. Comech," Bulletin of the American Mathematical Society 58, 289–294 (2021), DOI 10.1090/bull/1720, https://doi.org/10.1090/bull/1720
  14. Dmitry E. Pelinovsky, Andrey A. Sukhorukov, and Yuri S. Kivshar, "Bifurcations and stability of gap solitons in periodic potentials," Physical Review E 70 (2004), DOI 10.1103/PhysRevE.70.036618, arXiv:nlin/0405019, https://doi.org/10.1103/PhysRevE.70.036618
  15. Jason W. Fleischer, Mordechai Segev, Nikolaos K. Efremidis, and Demetrios N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature (2003), DOI 10.1038/nature01452, https://doi.org/10.1038/nature01452
  16. Xinyuan Qi, Simin Liu, Guoquan Zhang, Ru Guo, Zhaohong Liu, Lipeng Zhou, and Yuan Li, "Gap solitons in optically induced two-dimensional square photonic lattices in LiNbO3:Fe crystals," Applied Physics Letters (2007), DOI 10.1063/1.2793629, https://doi.org/10.1063/1.2793629