Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity. (English) Zbl 1527.35043
Summary: 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 \(\pm 2\omega \mathrm{i}\).
MSC:
| 35B35 | Stability in context of PDEs |
| 35C08 | Soliton solutions |
| 35P05 | General topics in linear spectral theory for PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
Keywords:
nonlinear Dirac equation; Soler model; solitary waves; spectral stability; concentrated nonlinearityReferences:
| [1] | R. G. R. Adami Dell’Antonio Figari, The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20, 477-500 (2003) · Zbl 1028.35137 · 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. · Zbl 1078.81003 |
| [3] | D. J. E. Aldunate Ricaud Stockmeyer, Results on the spectral stability of standing wave solutions of the Soler model in 1-D, Commun. Math. Phys., 401, 227-273 (2023) · Zbl 1522.35428 · doi:10.1007/s00220-023-04646-4 |
| [4] | R. A. Adami Teta, A class of nonlinear Schrödinger equations with concentrated nonlinearity, J. Funct. Anal., 180, 148-175 (2001) · Zbl 0979.35130 · doi:10.1006/jfan.2000.3697 |
| [5] | S. L. Benvegnù Dabrowski, Relativistic point interaction, Lett. Math. Phys., 30, 159-167 (1994) · Zbl 0788.47054 · doi:10.1007/BF00939703 |
| [6] | G. A. Berkolaiko Comech, On spectral stability of solitary waves of nonlinear Dirac equation in 1D, Math. Model. Nat. Phenom., 7, 13-31 (2012) · Zbl 1247.35117 · doi:10.1051/mmnp/20127202 |
| [7] | W. R. L. Borrelli Carlone Tentarelli, Nonlinear Dirac equation on graphs with localized nonlinearities: bound states and nonrelativistic limit, SIAM J. Math. Anal., 51, 1046-1081 (2019) · Zbl 1411.35263 · doi:10.1137/18M1211714 |
| [8] | W. R. L. Borrelli Carlone Tentarelli, On the nonlinear Dirac equation on noncompact metric graphs, J. Differ. Equ., 278, 326-357 (2021) · Zbl 1457.35045 · doi:10.1016/j.jde.2021.01.005 |
| [9] | N. S. Boussaïd Cuccagna, On stability of standing waves of nonlinear Dirac equations, Commun. Partial Differ. Equ., 37, 1001-1056 (2012) · Zbl 1251.35098 · doi:10.1080/03605302.2012.665973 |
| [10] | N. A. Boussaïd Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal., 271, 1462-1524 (2016) · Zbl 1457.35046 · doi:10.1016/j.jfa.2016.04.013 |
| [11] | N. A. Boussaïd Comech, Nonrelativistic asymptotics of solitary waves in the Dirac equation with Soler-type nonlinearity, SIAM J. Math. Anal., 49, 2527-2572 (2017) · Zbl 1375.35427 · doi:10.1137/16M1081385 |
| [12] | N. A. Boussaïd Comech, Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models, Commun. Pure Appl. Anal., 17, 1331-1347 (2018) · Zbl 1393.35196 · 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. · Zbl 1426.35026 |
| [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. · Zbl 1426.35026 |
| [15] | N. Boussaïd and A. Comech, Virtual levels and virtual states of linear operators in Banach spaces. arXiv: 2101.11979. · Zbl 1485.35300 |
| [16] | N. A. Boussaïd Comech, Limiting absorption principle and virtual levels of operators in Banach spaces, Ann. Math. Quebec, 46, 161-180 (2022) · Zbl 1485.35300 · doi:10.1007/s40316-021-00181-7 |
| [17] | N. Boussaïd, Stable directions for small nonlinear Dirac standing waves, Commun. Math. Phys., 268, 757-817 (2006) · Zbl 1127.35060 · 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, 1621-1670 (2008) · Zbl 1167.35438 · doi:10.1137/070684641 |
| [19] | F. E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann., 142, 22-130 (1961) · Zbl 0104.07502 · doi:10.1007/BF01343363 |
| [20] | V. A. E. Buslaev Komech Kopylova, On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Commun. Partial Differ. Equ., 33, 669-705 (2008) · Zbl 1185.35247 · doi:10.1080/03605300801970937 |
| [21] | C. R. D. Cacciapuoti Carlone Noja, The one-dimensional Dirac equation with concentrated nonlinearity, SIAM J. Math. Anal., 49, 2246-2268 (2017) · Zbl 1371.35236 · doi:10.1137/16M1084420 |
| [22] | C. D. D. Cacciapuoti Finco Noja, The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit, Lett. Math. Phys., 104, 1557-1570 (2014) · Zbl 1303.81072 · doi:10.1007/s11005-014-0725-y |
| [23] | R. M. L. Carlone Correggi Tentarelli, Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36, 257-294 (2019) · Zbl 1410.35196 · doi:10.1016/j.anihpc.2018.05.003 |
| [24] | A. E. Comech Kopylova, Orbital stability and spectral properties of solitary waves of Klein-Gordon equation with concentrated nonlinearity, Comm. Pure Appl. Anal., 20, 2187-2209 (2021) · Zbl 1478.35080 · doi:10.3934/cpaa.2021063 |
| [25] | A. T. V. A. Comech Phan Stefanov, Asymptotic stability of solitary waves in generalized Gross-Neveu model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34, 157-196 (2017) · Zbl 1357.35083 · 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. · Zbl 1447.47006 |
| [27] | M. B. W. R. E. Erdoğan Green Toprak, Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies, Amer. J. Math., 141, 1217-1258 (2019) · Zbl 1428.35432 · doi:10.1353/ajm.2019.0031 |
| [28] | A. Galindo, A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento, 20, 210-212 (1977) · doi:10.1007/BF02785129 |
| [29] | F. P. Gesztesy Šeba, New analytically solvable models of relativistic point interactions, Lett. Math. Phys., 13, 345-358 (1987) · Zbl 0646.60107 · doi:10.1007/BF00401163 |
| [30] | A. T. Jensen Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46, 583-611 (1979) · Zbl 0448.35080 · doi:10.1215/S0012-7094-79-04631-3 |
| [31] | A. G. Jensen Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., 13, 717-754 (2001) · Zbl 1029.81067 · doi:10.1142/S0129055X01000843 |
| [32] | A. I. A. A. Komech Komech, Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator, Russ. J. Math. Phys., 14, 164-173 (2007) · Zbl 1125.35092 · doi:10.1134/S1061920807020057 |
| [33] | A. A. Komech Komech, On global attraction to solitary waves for the Klein-Gordon field coupled to several nonlinear oscillators, J. Math. Pures Appl., 93, 91-111 (2010) · Zbl 1180.35124 · doi:10.1016/j.matpur.2009.08.011 |
| [34] | A. E. D. Komech Kopylova Stuart, On asymptotic stability of solitons in a nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 11, 1063-1079 (2012) · Zbl 1282.35352 · 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, 426-428 (1973) · doi:10.1007/BF00850963 |
| [36] | D. A. Noja Posilicano, Wave equations with concentrated nonlinearities, J. Phys. A, 38, 5011-5022 (2005) · Zbl 1085.81039 · 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. · Zbl 1279.35083 |
| [38] | M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1, 2766-2769 (1970) · 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. · Zbl 1197.35006 |
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