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Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity

  • *Corresponding author: Andrew Comech

    *Corresponding author: Andrew Comech 

D. Noja acknowledges for funding the EC grant IPaDEGAN (MSCA-RISE-778010). This work was supported by a grant from the Simons Foundation (851052, A.C.).

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  • We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the $ \mathbf{SU}(1,1) $ symmetry: the first preserving and the second breaking the parity symmetry. We show that a particular perturbation which breaks the $ \mathbf{SU}(1,1) $ symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a particular perturbation which breaks both the $ \mathbf{SU}(1,1) $ symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. This instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues $ \pm 2\omega \mathrm{i} $.

    Mathematics Subject Classification: Primary: 34L40, 35B35, 35P05; Secondary: 35Q41.

    Citation:

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  • Figure 1.  Location of eigenvalues of the linearized operator $ \mathbf{A}(\omega,\kappa) $ for different values of parameters $ \omega\in(-m,m) $ and $ \kappa\in \mathbb{R} $. In the while area, there are no eigenvalues besides $ \lambda = 0 $ and $ \lambda = \pm 2\omega \mathrm{i} $. The virtual level curves $ \omega = \mathcal{T}_\kappa $ (dotted curves for $ \kappa<2^{-1/2}-1 $ and $ \kappa>2^{-1/2} $) correspond to virtual levels at thresholds $ \pm \mathrm{i}(m-\vert {\omega} \vert) $. The lightly shaded regions between the virtual level curves and the Kolokolov curves $ \omega = \Omega_\kappa $ (solid curves for $ \kappa<-1/3 $ and $ \kappa>1 $) correspond to two purely imaginary eigenvalues in the spectral gap. The Kolokolov curves correspond to the collision of two eigenvalues at $ \lambda = 0 $ (as indicated by the Kolokolov condition). On the other side of the Kolokolov curves, in the crosshatched regions on the plot, there is a pair of real eigenvalues $ \pm\lambda\in \mathbb{R} $ (linear instability). In these regions, one has $ \pm\lambda\to\pm\infty $ as $ \omega $ and $ \kappa $ approach the curve $ \omega = 2\Omega_\kappa $ (dash-dot curve for $ \kappa<-1/2 $). At $ \kappa = -1 $, $ \omega = 0 $ (point of intersection of the three curves), the spectrum of the linearized operator $ \mathbf{A} $ consists of the whole complex plane

  • [1] R. AdamiG. Dell'Antonio and R. Figari, et al., The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 477-500.  doi: 10.1016/S0294-1449(02)00022-7.
    [2] S. Albeverio, F. Gesztesy and R. Høegh-Krohn et al., Solvable Models in Quantum Mechanics, American Mathematical Society, Providence, RI, 2005, second edition. doi: 10.1090/chel/350.
    [3] D. AldunateJ. Ricaud and E. Stockmeyer, et al., Results on the spectral stability of standing wave solutions of the Soler model in 1-D, Commun. Math. Phys., 401 (2023), 227-273.  doi: 10.1007/s00220-023-04646-4.
    [4] R. Adami and A. Teta, A class of nonlinear Schrödinger equations with concentrated nonlinearity, J. Funct. Anal., 180 (2001), 148-175.  doi: 10.1006/jfan.2000.3697.
    [5] S. Benvegnù and L. Dabrowski, Relativistic point interaction, Lett. Math. Phys., 30 (1994), 159-167.  doi: 10.1007/BF00939703.
    [6] G. Berkolaiko and A. Comech, On spectral stability of solitary waves of nonlinear Dirac equation in 1D, Math. Model. Nat. Phenom., 7 (2012), 13-31.  doi: 10.1051/mmnp/20127202.
    [7] W. BorrelliR. Carlone and L. Tentarelli, Nonlinear Dirac equation on graphs with localized nonlinearities: bound states and nonrelativistic limit, SIAM J. Math. Anal., 51 (2019), 1046-1081.  doi: 10.1137/18M1211714.
    [8] W. BorrelliR. Carlone and L. Tentarelli, On the nonlinear Dirac equation on noncompact metric graphs, J. Differ. Equ., 278 (2021), 326-357.  doi: 10.1016/j.jde.2021.01.005.
    [9] N. Boussaïd and S. Cuccagna, On stability of standing waves of nonlinear Dirac equations, Commun. Partial Differ. Equ., 37 (2012), 1001-1056.  doi: 10.1080/03605302.2012.665973.
    [10] N. Boussaïd and A. Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal., 271 (2016), 1462-1524.  doi: 10.1016/j.jfa.2016.04.013.
    [11] N. Boussaïd and A. Comech, Nonrelativistic asymptotics of solitary waves in the Dirac equation with Soler-type nonlinearity, SIAM J. Math. Anal., 49 (2017), 2527-2572.  doi: 10.1137/16M1081385.
    [12] N. Boussaïd and A. Comech, Spectral stability of bi-frequency solitary waves in Soler and Dirac–Klein–Gordon models, Commun. Pure Appl. Anal., 17 (2018), 1331-1347.  doi: 10.3934/cpaa.2018065.
    [13] N. Boussaïd and A. Comech, Nonlinear Dirac equation. Spectral stability of solitary waves, vol. 244 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2019. doi: 10.1090/surv/244.
    [14] N. Boussaïd and A. Comech, Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity, J. Funct. Anal., 277 (2019), 68 pp. doi: 10.1016/j.jfa.2019.108289.
    [15] N. Boussaïd and A. Comech, Virtual levels and virtual states of linear operators in Banach spaces. arXiv: 2101.11979.
    [16] N. Boussaïd and A. Comech, Limiting absorption principle and virtual levels of operators in Banach spaces, Ann. Math. Quebec, 46 (2022), 161-180.  doi: 10.1007/s40316-021-00181-7.
    [17] N. Boussaïd, Stable directions for small nonlinear Dirac standing waves, Commun. Math. Phys., 268 (2006), 757-817.  doi: 10.1007/s00220-006-0112-3.
    [18] N. Boussaïd, On the asymptotic stability of small nonlinear Dirac standing waves in a resonant case, SIAM J. Math. Anal., 40 (2008), 1621-1670.  doi: 10.1137/070684641.
    [19] F. E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann., 142 (1961), 22-130.  doi: 10.1007/BF01343363.
    [20] V. BuslaevA. Komech and E. Kopylova, et al., On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Commun. Partial Differ. Equ., 33 (2008), 669-705.  doi: 10.1080/03605300801970937.
    [21] C. CacciapuotiR. Carlone and D. Noja, et al., The one-dimensional Dirac equation with concentrated nonlinearity, SIAM J. Math. Anal., 49 (2017), 2246-2268.  doi: 10.1137/16M1084420.
    [22] C. CacciapuotiD. Finco and D. Noja, et al., The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit, Lett. Math. Phys., 104 (2014), 1557-1570.  doi: 10.1007/s11005-014-0725-y.
    [23] R. CarloneM. Correggi and L. Tentarelli, Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 257-294.  doi: 10.1016/j.anihpc.2018.05.003.
    [24] A. Comech and E. Kopylova, Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity, Comm. Pure Appl. Anal., 20 (2021), 2187-2209.  doi: 10.3934/cpaa.2021063.
    [25] A. ComechT. V. Phan and A. Stefanov, Asymptotic stability of solitary waves in generalized Gross–Neveu model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 157-196.  doi: 10.1016/j.anihpc.2015.11.001.
    [26] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2018, 2 edn. doi: 10.1093/oso/9780198812050.001.0001.
    [27] M. B. ErdoğanW. R. Green and E. Toprak, Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies, Amer. J. Math., 141 (2019), 1217-1258.  doi: 10.1353/ajm.2019.0031.
    [28] A. Galindo, A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento, 20 (1977), 210-212.  doi: 10.1007/BF02785129.
    [29] F. Gesztesy and P. Šeba, New analytically solvable models of relativistic point interactions, Lett. Math. Phys., 13 (1987), 345-358.  doi: 10.1007/BF00401163.
    [30] A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.  doi: 10.1215/S0012-7094-79-04631-3.
    [31] A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., 13 (2001), 717-754.  doi: 10.1142/S0129055X01000843.
    [32] A. I. Komech and A. A. Komech, Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator, Russ. J. Math. Phys., 14 (2007), 164-173.  doi: 10.1134/S1061920807020057.
    [33] A. Komech and A. Komech, On global attraction to solitary waves for the Klein–Gordon field coupled to several nonlinear oscillators, J. Math. Pures Appl., 93 (2010), 91-111.  doi: 10.1016/j.matpur.2009.08.011.
    [34] A. KomechE. Kopylova and D. Stuart, On asymptotic stability of solitons in a nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 11 (2012), 1063-1079.  doi: 10.3934/cpaa.2012.11.1063.
    [35] A. A. Kolokolov, Stability of the dominant mode of the nonlinear wave equation in a cubic medium, J. Appl. Mech. Tech. Phys., 14 (1973), 426-428.  doi: 10.1007/BF00850963.
    [36] D. Noja and A. Posilicano, Wave equations with concentrated nonlinearities, J. Phys. A, 38 (2005), 5011-5022.  doi: 10.1088/0305-4470/38/22/022.
    [37] D. E. Pelinovsky and A. Stefanov, Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys., 53 (2012), 27 pp. doi: 10.1063/1.4731477.
    [38] M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), 2766-2769.  doi: 10.1103/PhysRevD.1.2766.
    [39] D. R. Yafaev, Mathematical Scattering Theory, vol. 158 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/158.
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