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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 and 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.

[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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