American Institute of Mathematical Sciences

Advanced
June  2016, 36(6): 2991-3009. doi: 10.3934/dcds.2016.36.2991

A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation

Matt Coles 1, and  Stephen Gustafson 1,

1. 

Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2, Canada, Canada

Received  July 2015 Revised  September 2015 Published  December 2015

This work deals with the focusing Nonlinear Schrödinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.
Keywords: Nonlinear Schrödinger equation, linearized operator, edge bifurcation, resolvent expansion, Lyapunov-Schmidt reduction, Birman-Schwinger formulation.
Mathematics Subject Classification: 35P15, 35Q5.
Citation: Matt Coles, Stephen Gustafson. A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2991-3009. doi: 10.3934/dcds.2016.36.2991
References:
[1]

D. Bambusi, Asymptotic stability of ground states in some hamiltonian pde with symmetry,, Comm. Math. Phys., 320 (2013), 499. doi: 10.1007/s00220-013-1684-3. Google Scholar

[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. doi: 10.1016/S0294-1449(02)00018-5. Google Scholar

[3]

T. Cazenave, Semilinear Schrödginer Equations,, American Mathematical Soc., (2003). Google Scholar

[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. doi: 10.1137/050648389. Google Scholar

[5]

S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 54 (2001), 1110. doi: 10.1002/cpa.1018. Google Scholar

[6]

S. Cuccagna, On asymptotic stability of ground states of NLS,, Rev. Math. Phys., 15 (2003), 877. doi: 10.1142/S0129055X03001849. Google Scholar

[7]

S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation,, Trans. Amer. Math. Soc., 366 (2014), 2827. doi: 10.1090/S0002-9947-2014-05770-X. Google Scholar

[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). doi: 10.1063/1.1901345. Google Scholar

[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. doi: 10.1002/cpa.20050. Google Scholar

[10]

S. Cuccagna and D. Pelinovsky, The asymptotic stability of solitons in the cubic NLS equation on the line,, Applicable Analysis, 93 (2014), 791. doi: 10.1080/00036811.2013.866227. Google Scholar

[11]

Z. Gang and I. M. Sigal, Asymptotic stability of nonlinear Schrödinger equations with potential,, Rev. Math. Phys., 17 (2005), 1143. doi: 10.1142/S0129055X05002522. Google Scholar

[12]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations,, Comm. Pure Appl. Anal., 41 (1988), 747. doi: 10.1002/cpa.3160410602. Google Scholar

[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. doi: 10.1016/0022-1236(87)90044-9. Google Scholar

[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. Google Scholar

[15]

G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse,, Springer, (2015). doi: 10.1007/978-3-319-12748-4. Google Scholar

[16]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions,, Duke Math. J., 46 (1979), 583. Google Scholar

[17]

A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds,, Rev. Math. Phys., 13 (2001), 717. doi: 10.1142/S0129055X01000843. Google Scholar

[18]

T. Kapitula, Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation,, Physica D, 116 (1998), 95. doi: 10.1016/S0167-2789(97)00245-5. Google Scholar

[19]

T. Kapitula and B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,, Physica D, 124 (1998), 58. doi: 10.1016/S0167-2789(98)00172-9. Google Scholar

[20]

T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans functions,, SIAM J. Math. Anal., 33 (2002), 1117. doi: 10.1137/S0036141000372301. Google Scholar

[21]

T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans functions,, Discrete Contin. Dyn. Syst., 10 (2004), 857. doi: 10.3934/dcds.2004.10.857. Google Scholar

[22]

D. Pelinovsky, Y. Kivshar and V. Afanasjev, Internal modes of envelope solitons,, Physica D, 116 (1998), 121. doi: 10.1016/S0167-2789(98)80010-9. Google Scholar

[23]

G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations,, Comm. Partial Differential Equations, 29 (2004), 1051. doi: 10.1081/PDE-200033754. Google Scholar

[24]

W. Schlag, Stabile manifolds for an orbitally unstable nonlinear Schrödinger equation,, Ann. of Math., 169 (2009), 139. doi: 10.4007/annals.2009.169.139. Google Scholar

[25]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation,, Springer, (1999). Google Scholar

[26]

V. Vougalter, On threshold eigenvalues and resonances for the linearized NLS equation,, Math. Model. Nat. Phenom., 5 (2010), 448. doi: 10.1051/mmnp/20105417. Google Scholar

[27]

V. Vougalter, On the negative index theorem for the linearized NLS problem,, Canad. Math. Bull., 53 (2010), 737. doi: 10.4153/CMB-2010-062-4. Google Scholar

[28]

V. Vougalter and D. Pelinovsky, Eigenvalues of zero energy in the linearized NLS problem,, Journal of Mathematical Physics, 47 (2006). doi: 10.1063/1.2203233. Google Scholar

[29]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,, SIAM J. Math Anal., 16 (1985), 472. doi: 10.1137/0516034. Google Scholar

[30]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolutions equations,, Comm. Pure Appl. Math., 39 (1986), 51. doi: 10.1002/cpa.3160390103. Google Scholar

show all references

References:
[1]

D. Bambusi, Asymptotic stability of ground states in some hamiltonian pde with symmetry,, Comm. Math. Phys., 320 (2013), 499. doi: 10.1007/s00220-013-1684-3. Google Scholar

[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. doi: 10.1016/S0294-1449(02)00018-5. Google Scholar

[3]

T. Cazenave, Semilinear Schrödginer Equations,, American Mathematical Soc., (2003). Google Scholar

[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. doi: 10.1137/050648389. Google Scholar

[5]

S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 54 (2001), 1110. doi: 10.1002/cpa.1018. Google Scholar

[6]

S. Cuccagna, On asymptotic stability of ground states of NLS,, Rev. Math. Phys., 15 (2003), 877. doi: 10.1142/S0129055X03001849. Google Scholar

[7]

S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation,, Trans. Amer. Math. Soc., 366 (2014), 2827. doi: 10.1090/S0002-9947-2014-05770-X. Google Scholar

[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). doi: 10.1063/1.1901345. Google Scholar

[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. doi: 10.1002/cpa.20050. Google Scholar

[10]

S. Cuccagna and D. Pelinovsky, The asymptotic stability of solitons in the cubic NLS equation on the line,, Applicable Analysis, 93 (2014), 791. doi: 10.1080/00036811.2013.866227. Google Scholar

[11]

Z. Gang and I. M. Sigal, Asymptotic stability of nonlinear Schrödinger equations with potential,, Rev. Math. Phys., 17 (2005), 1143. doi: 10.1142/S0129055X05002522. Google Scholar

[12]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations,, Comm. Pure Appl. Anal., 41 (1988), 747. doi: 10.1002/cpa.3160410602. Google Scholar

[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. doi: 10.1016/0022-1236(87)90044-9. Google Scholar

[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. Google Scholar

[15]

G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse,, Springer, (2015). doi: 10.1007/978-3-319-12748-4. Google Scholar

[16]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions,, Duke Math. J., 46 (1979), 583. Google Scholar

[17]

A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds,, Rev. Math. Phys., 13 (2001), 717. doi: 10.1142/S0129055X01000843. Google Scholar

[18]

T. Kapitula, Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation,, Physica D, 116 (1998), 95. doi: 10.1016/S0167-2789(97)00245-5. Google Scholar

[19]

T. Kapitula and B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,, Physica D, 124 (1998), 58. doi: 10.1016/S0167-2789(98)00172-9. Google Scholar

[20]

T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans functions,, SIAM J. Math. Anal., 33 (2002), 1117. doi: 10.1137/S0036141000372301. Google Scholar

[21]

T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans functions,, Discrete Contin. Dyn. Syst., 10 (2004), 857. doi: 10.3934/dcds.2004.10.857. Google Scholar

[22]

D. Pelinovsky, Y. Kivshar and V. Afanasjev, Internal modes of envelope solitons,, Physica D, 116 (1998), 121. doi: 10.1016/S0167-2789(98)80010-9. Google Scholar

[23]

G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations,, Comm. Partial Differential Equations, 29 (2004), 1051. doi: 10.1081/PDE-200033754. Google Scholar

[24]

W. Schlag, Stabile manifolds for an orbitally unstable nonlinear Schrödinger equation,, Ann. of Math., 169 (2009), 139. doi: 10.4007/annals.2009.169.139. Google Scholar

[25]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation,, Springer, (1999). Google Scholar

[26]

V. Vougalter, On threshold eigenvalues and resonances for the linearized NLS equation,, Math. Model. Nat. Phenom., 5 (2010), 448. doi: 10.1051/mmnp/20105417. Google Scholar

[27]

V. Vougalter, On the negative index theorem for the linearized NLS problem,, Canad. Math. Bull., 53 (2010), 737. doi: 10.4153/CMB-2010-062-4. Google Scholar

[28]

V. Vougalter and D. Pelinovsky, Eigenvalues of zero energy in the linearized NLS problem,, Journal of Mathematical Physics, 47 (2006). doi: 10.1063/1.2203233. Google Scholar

[29]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,, SIAM J. Math Anal., 16 (1985), 472. doi: 10.1137/0516034. Google Scholar

[30]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolutions equations,, Comm. Pure Appl. Math., 39 (1986), 51. doi: 10.1002/cpa.3160390103. Google Scholar

[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201
[2]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
[3]

Shuya Kanagawa, Ben T. Nohara. The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion. Conference Publications, 2013, 2013 (special) : 415-426. doi: 10.3934/proc.2013.2013.415
[4]

Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237
[5]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[6]

J. Cruz-Sampedro. Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1061-1076. doi: 10.3934/dcds.2013.33.1061
[7]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[8]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[9]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[10]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[11]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377
[12]

Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541
[13]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505
[14]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[15]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[16]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[17]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[18]

Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
[19]

Benjamin Dodson. Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 127-140. doi: 10.3934/cpaa.2011.10.127
[20]

Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences