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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, 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), 499-542. doi: 10.1007/s00220-013-1684-3.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.  Google Scholar

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A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., 13 (2001), 717-754. doi: 10.1142/S0129055X01000843.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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), 448-469. 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), 737-745. 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), 062701, 13pp. 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), 472-491. 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), 51-68. 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), 499-542. 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), 419-475. doi: 10.1016/S0294-1449(02)00018-5.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödginer Equations, American Mathematical Soc., Providence, RI, 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), 1070-1111. doi: 10.1137/050648389.  Google Scholar

[5]

S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145. doi: 10.1002/cpa.1018.  Google Scholar

[6]

S. Cuccagna, On asymptotic stability of ground states of NLS, Rev. Math. Phys., 15 (2003), 877-903. 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), 2827-2888. 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), 053520, 15pp. 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), 1-29. 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), 791-822. 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), 1143-1207. doi: 10.1142/S0129055X05002522.  Google Scholar

[12]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Anal., 41 (1988), 747-774. 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), 160-197. 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), 583-611.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 448-469. 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), 737-745. 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), 062701, 13pp. 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), 472-491. 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), 51-68. doi: 10.1002/cpa.3160390103.  Google Scholar

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