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A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation
1. | Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2, Canada, Canada |
References:
[1] |
D. Bambusi, Asymptotic stability of ground states in some hamiltonian pde with symmetry, Comm. Math. Phys., 320 (2013), 499-542.
doi: 10.1007/s00220-013-1684-3. |
[2] |
V. S. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 20 (2003), 419-475.
doi: 10.1016/S0294-1449(02)00018-5. |
[3] |
T. Cazenave, Semilinear Schrödginer Equations, American Mathematical Soc., Providence, RI, 2003. |
[4] |
S. Chang, S. Gustafson, K. Nakanishi and T. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math Anal., 39 (2007), 1070-1111.
doi: 10.1137/050648389. |
[5] |
S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145.
doi: 10.1002/cpa.1018. |
[6] |
S. Cuccagna, On asymptotic stability of ground states of NLS, Rev. Math. Phys., 15 (2003), 877-903.
doi: 10.1142/S0129055X03001849. |
[7] |
S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 366 (2014), 2827-2888.
doi: 10.1090/S0002-9947-2014-05770-X. |
[8] |
S. Cuccagna and D. Pelinovsky, Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem, J. Math. Phys., 46 (2005), 053520, 15pp.
doi: 10.1063/1.1901345. |
[9] |
S. Cuccagna, D. Pelinovsky and V. Vougalter, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math., 58 (2005), 1-29.
doi: 10.1002/cpa.20050. |
[10] |
S. Cuccagna and D. Pelinovsky, The asymptotic stability of solitons in the cubic NLS equation on the line, Applicable Analysis, 93 (2014), 791-822.
doi: 10.1080/00036811.2013.866227. |
[11] |
Z. Gang and I. M. Sigal, Asymptotic stability of nonlinear Schrödinger equations with potential, Rev. Math. Phys., 17 (2005), 1143-1207.
doi: 10.1142/S0129055X05002522. |
[12] |
M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Anal., 41 (1988), 747-774.
doi: 10.1002/cpa.3160410602. |
[13] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[14] |
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics (2nd ed.), Springer-Verlag Berlin Heidelberg, 2011.
doi: 10.1007/978-3-642-21866-8. |
[15] |
G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse, Springer, 2015.
doi: 10.1007/978-3-319-12748-4. |
[16] |
A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611. |
[17] |
A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., 13 (2001), 717-754.
doi: 10.1142/S0129055X01000843. |
[18] |
T. Kapitula, Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation, Physica D, 116 (1998), 95-120.
doi: 10.1016/S0167-2789(97)00245-5. |
[19] |
T. Kapitula and B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Physica D, 124 (1998), 58-103.
doi: 10.1016/S0167-2789(98)00172-9. |
[20] |
T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans functions, SIAM J. Math. Anal., 33 (2002), 1117-1143.
doi: 10.1137/S0036141000372301. |
[21] |
T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans functions, Discrete Contin. Dyn. Syst., 10 (2004), 857-869.
doi: 10.3934/dcds.2004.10.857. |
[22] |
D. Pelinovsky, Y. Kivshar and V. Afanasjev, Internal modes of envelope solitons, Physica D, 116 (1998), 121-142.
doi: 10.1016/S0167-2789(98)80010-9. |
[23] |
G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 1051-1095.
doi: 10.1081/PDE-200033754. |
[24] |
W. Schlag, Stabile manifolds for an orbitally unstable nonlinear Schrödinger equation, Ann. of Math., 169 (2009), 139-227.
doi: 10.4007/annals.2009.169.139. |
[25] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer, 1999. |
[26] |
V. Vougalter, On threshold eigenvalues and resonances for the linearized NLS equation, Math. Model. Nat. Phenom., 5 (2010), 448-469.
doi: 10.1051/mmnp/20105417. |
[27] |
V. Vougalter, On the negative index theorem for the linearized NLS problem, Canad. Math. Bull., 53 (2010), 737-745.
doi: 10.4153/CMB-2010-062-4. |
[28] |
V. Vougalter and D. Pelinovsky, Eigenvalues of zero energy in the linearized NLS problem, Journal of Mathematical Physics, 47 (2006), 062701, 13pp.
doi: 10.1063/1.2203233. |
[29] |
M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math Anal., 16 (1985), 472-491.
doi: 10.1137/0516034. |
[30] |
M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolutions equations, Comm. Pure Appl. Math., 39 (1986), 51-68.
doi: 10.1002/cpa.3160390103. |
show all references
References:
[1] |
D. Bambusi, Asymptotic stability of ground states in some hamiltonian pde with symmetry, Comm. Math. Phys., 320 (2013), 499-542.
doi: 10.1007/s00220-013-1684-3. |
[2] |
V. S. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 20 (2003), 419-475.
doi: 10.1016/S0294-1449(02)00018-5. |
[3] |
T. Cazenave, Semilinear Schrödginer Equations, American Mathematical Soc., Providence, RI, 2003. |
[4] |
S. Chang, S. Gustafson, K. Nakanishi and T. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math Anal., 39 (2007), 1070-1111.
doi: 10.1137/050648389. |
[5] |
S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145.
doi: 10.1002/cpa.1018. |
[6] |
S. Cuccagna, On asymptotic stability of ground states of NLS, Rev. Math. Phys., 15 (2003), 877-903.
doi: 10.1142/S0129055X03001849. |
[7] |
S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 366 (2014), 2827-2888.
doi: 10.1090/S0002-9947-2014-05770-X. |
[8] |
S. Cuccagna and D. Pelinovsky, Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem, J. Math. Phys., 46 (2005), 053520, 15pp.
doi: 10.1063/1.1901345. |
[9] |
S. Cuccagna, D. Pelinovsky and V. Vougalter, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math., 58 (2005), 1-29.
doi: 10.1002/cpa.20050. |
[10] |
S. Cuccagna and D. Pelinovsky, The asymptotic stability of solitons in the cubic NLS equation on the line, Applicable Analysis, 93 (2014), 791-822.
doi: 10.1080/00036811.2013.866227. |
[11] |
Z. Gang and I. M. Sigal, Asymptotic stability of nonlinear Schrödinger equations with potential, Rev. Math. Phys., 17 (2005), 1143-1207.
doi: 10.1142/S0129055X05002522. |
[12] |
M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Anal., 41 (1988), 747-774.
doi: 10.1002/cpa.3160410602. |
[13] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[14] |
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics (2nd ed.), Springer-Verlag Berlin Heidelberg, 2011.
doi: 10.1007/978-3-642-21866-8. |
[15] |
G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse, Springer, 2015.
doi: 10.1007/978-3-319-12748-4. |
[16] |
A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611. |
[17] |
A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., 13 (2001), 717-754.
doi: 10.1142/S0129055X01000843. |
[18] |
T. Kapitula, Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation, Physica D, 116 (1998), 95-120.
doi: 10.1016/S0167-2789(97)00245-5. |
[19] |
T. Kapitula and B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Physica D, 124 (1998), 58-103.
doi: 10.1016/S0167-2789(98)00172-9. |
[20] |
T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans functions, SIAM J. Math. Anal., 33 (2002), 1117-1143.
doi: 10.1137/S0036141000372301. |
[21] |
T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans functions, Discrete Contin. Dyn. Syst., 10 (2004), 857-869.
doi: 10.3934/dcds.2004.10.857. |
[22] |
D. Pelinovsky, Y. Kivshar and V. Afanasjev, Internal modes of envelope solitons, Physica D, 116 (1998), 121-142.
doi: 10.1016/S0167-2789(98)80010-9. |
[23] |
G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 1051-1095.
doi: 10.1081/PDE-200033754. |
[24] |
W. Schlag, Stabile manifolds for an orbitally unstable nonlinear Schrödinger equation, Ann. of Math., 169 (2009), 139-227.
doi: 10.4007/annals.2009.169.139. |
[25] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer, 1999. |
[26] |
V. Vougalter, On threshold eigenvalues and resonances for the linearized NLS equation, Math. Model. Nat. Phenom., 5 (2010), 448-469.
doi: 10.1051/mmnp/20105417. |
[27] |
V. Vougalter, On the negative index theorem for the linearized NLS problem, Canad. Math. Bull., 53 (2010), 737-745.
doi: 10.4153/CMB-2010-062-4. |
[28] |
V. Vougalter and D. Pelinovsky, Eigenvalues of zero energy in the linearized NLS problem, Journal of Mathematical Physics, 47 (2006), 062701, 13pp.
doi: 10.1063/1.2203233. |
[29] |
M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math Anal., 16 (1985), 472-491.
doi: 10.1137/0516034. |
[30] |
M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolutions equations, Comm. Pure Appl. Math., 39 (1986), 51-68.
doi: 10.1002/cpa.3160390103. |
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