Stable NLS solitons in a cubic-quintic medium with a delta-function potential. (English) Zbl 1398.35213
Summary: We study the one-dimensional nonlinear Schrödinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a positive soliton profile through explicit formulas and, using bifurcation theory, we describe their behavior with respect to the propagation constant. This information is used to prove their stability by means of the rigorous theory of orbital stability of Hamiltonian systems. The presence of the trapping potential gives rise to a regime where two stable bound states coexist, with different powers and same propagation constant.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B32 | Bifurcations in context of PDEs |
| 35C08 | Soliton solutions |
| 35J61 | Semilinear elliptic equations |
| 37C75 | Stability theory for smooth dynamical systems |
| 74J30 | Nonlinear waves in solid mechanics |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 35J60 | Nonlinear elliptic equations |
Keywords:
nonlinear Schrödinger equation; cubic-quintic nonlinearity; trapping delta potential; bifurcation; stabilityReferences:
| [1] | Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Holden, H., Solvable Models in Quantum Mechanics (2005), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI · Zbl 1078.81003 |
| [2] | Bao, W.; Du, Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25, 1674-1697 (2004) · Zbl 1061.82025 |
| [3] | Berezin, F. A.; Shubin, M. A., The Schrödinger Equation (1991), Kluwer Academic Publishers · Zbl 0749.35001 |
| [4] | Birnbaum, Z.; Malomed, B. A., Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity, Physica D, 237, 3252-3262 (2008) · Zbl 1153.78323 |
| [5] | Boudebs, G.; Cherukulappurath, S.; Leblond, H.; Troles, J.; Smektala, F.; Sanchez, F., Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun., 219, 427-433 (2003) |
| [6] | Cazenave, T., (Semilinear Schrödinger Equations. Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics (2003), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1055.35003 |
| [7] | Chiofalo, M. L.; Succi, S.; Tosi, M. P., Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62, 7438-7444 (2000) |
| [8] | Cowan, S.; Enns, R. H.; Rangnekar, S. S.; Sanghera, S. S., Quasi-soliton and other behavior of the nonlinear cubic-quintic Schrö dinger equation, Can. J. Phys., 64, 311-315 (1986) |
| [9] | Crandall, M. G.; Rabinowitz, P. H., Bifurcation from simple eigenvalues, J. Funct. Anal., 8, 321-340 (1971) · Zbl 0219.46015 |
| [10] | Crandall, M. G.; Rabinowitz, P. H., Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal., 52, 161-180 (1973) · Zbl 0275.47044 |
| [11] | de Bièvre, S.; Genoud, F.; Rota-Nodari, S., Orbital stability: analysis meets geometry, (Besse, C.; Garreau, J. C., Nonlinear Optical and Atomic Systems. Nonlinear Optical and Atomic Systems, Lecture Notes in Mathematics, vol. 2146 (2015), Springer), 147-273 · Zbl 1347.37122 |
| [12] | Falcão-Filho, E. L.; de Araújo, C. B.; Boudebs, G.; Leblond, H.; Skarka, V., Robust two-dimensional spatial solitons in liquid carbon disulfide, Phys. Rev. Lett., 110, Article 013901 pp. (2013) |
| [13] | Falcão-Filho, E. L.; de Araújo, C. B.; Rodrigues, J. J., High-order nonlinearities of aqueous colloids containing silver nanoparticles, J. Opt. Soc. Amer. B, 24, 2948-2956 (2007) |
| [14] | Fukuizumi, R.; Jeanjean, L., Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential, Discrete Contin. Dyn. Syst., 21, 121-136 (2008) · Zbl 1144.35465 |
| [15] | Fukuizumi, R.; Ohta, M.; Ozawa, T., Nonlinear Schrödinger equation with a point defect, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 837-845 (2008) · Zbl 1145.35457 |
| [16] | Gisin, B. V.; Driben, R.; Malomed, B. A., Bistable guided solitons in the cubic-quintic medium, J. Opt. B: Quantum Semiclassical Opt., 6, S259-S264 (2004) |
| [17] | Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74, 160-197 (1987) · Zbl 0656.35122 |
| [18] | Kato, T., (Perturbation Theory for Linear Operators. Perturbation Theory for Linear Operators, Classics in Mathematics (1995), Springer-Verlag: Springer-Verlag Berlin), Reprint of the 1980 edition · Zbl 0836.47009 |
| [19] | Le Coz, S.; Fukuizumi, R.; Fibich, G.; Ksherim, B.; Sivan, Y., Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Physica D, 237, 1103-1128 (2008) · Zbl 1147.35356 |
| [20] | Maeda, M., Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity, Kodai Math. J., 31, 263-271 (2008) · Zbl 1180.35483 |
| [21] | Malomed, B. A.; Azbel, M. Ya., Modulational instability of a wave scattered by a nonlinear center, Phys. Rev. B, 47, 10402-10406 (1993) |
| [22] | Ohta, M., Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity, Kodai Math. J., 18, 68-74 (1995) · Zbl 0868.35111 |
| [23] | Papagiannis, P.; Kominis, Y.; Hizanidis, K., Power- and momentum-dependent soliton dynamics in lattices with longitudinal modulation, Phys. Rev. A, 84, Article 013820 pp. (2011) |
| [24] | Pelinovsky, D. E.; Kivshar, Y. S.; Afanasjev, V. V., Internal modes of envelope solitons, Physica D, 116, 121-142 (1998) · Zbl 0934.35175 |
| [25] | Pushkarov, K. I.; Pushkarov, D. I.; Tomov, I. V., Self-action of light beams in nonlinear media: soliton solutions, Opt. Quantum Electron., 11, 471-478 (1979) |
| [26] | Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504 |
| [27] | Reyna, A. S.; Malomed, B. A.; de Araújo, C. B., Stability conditions for one-dimensional optical solitons in cubic-quintic-septimal media, Phys. Rev. A, 92, Article 033810 pp. (2015) |
| [28] | Stuart, C. A., An introduction to elliptic equations on \(R^N\), (Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997 (1998), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 237-285 · Zbl 0956.35028 |
| [29] | Stuart, C. A., Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76, 329-399 (2008) · Zbl 1179.37101 |
| [30] | Vakhitov, N. G.; Kolokolov, A. A., Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. and Quantum Electronics, 16, 783-789 (1973) |
| [31] | Yang, J., Conditions and stability analysis for saddle-node bifurcations of solitary waves in generalized nonlinear Schrödinger equations, (Malomed, B. A., Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations. Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, Progress Optical Sci., Photonics, vol. 1 (2013)), 639-655 · Zbl 1291.35371 |
| [32] | Yang, J., Stability switching at transcritical bifurcations of solitary waves in generalized nonlinear Schrödinger equations, Phys. Lett. A, 377, 866-870 (2013) · Zbl 1298.35030 |
| [33] | Zegadlo, K. B.; Wasak, T.; Malomed, B. A.; Karpierz, M. A.; Trippenbach, M., Stabilization of solitons under competing nonlinearities by external potentials, Chaos, 24, Article 043136 pp. (2014) · Zbl 1361.35170 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
