Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-21T08:32:46.807Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Negative Index Theorem for the Linearized Non-Linear Schrödinger Problem

Published online by Cambridge University Press:  20 November 2018

Vitali Vougalter
Vitali Vougalter*
Affiliation:
University of Toronto, Department of Mathematics, Toronto, ON e-mail: vitali@math.toronto.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new and elementary proof is given of the recent result of Cuccagna, Pelinovsky, and Vougalter based on the variational principle for the quadratic form of a self-adjoint operator. It is the negative index theorem for a linearized $\text{NLS}$ operator in three dimensions.

Keywords

35Q5581Q10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 4 , 01 December 2010 , pp. 737 - 745
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Adachi, S., A positive solution of a nonhomogeneous elliptic equation in ℝ N with G-invariant nonlinearity. Comm. Partial Differential Equations 27(2002), no. 1–2, 1–22. doi:10.1081/PDE-120002781Google Scholar
[2] Berestycki, H. and Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82(1983), no. 4, 313–345.Google Scholar
[3] Berestycki, H. and Lions, P.-L., Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82(1983), no. 4, 347–375.Google Scholar
[4] Berestycki, H., Lions, P.-L., and Peletier, L., An ODE approach to the existence of positive solutions for semilinear problems in ℝ N. Indiana Univ. Math. J. 30(1981), no. 1, 141–157. doi:10.1512/iumj.1981.30.30012Google Scholar
[5] Buslaev, V. S. and Perel’man, G. S., Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. St. Petersburg Math. J. 4(1993), no. 6, 1111–1142.Google Scholar
[6] Buslaev, V. S. and Sulem, C., On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(2003), no. 3, 419–475. doi:10.1016/S0294-1449(02)00018-5Google Scholar
[7] Chugunova, M. and Pelinovsky, D., Count of eigenvalues in the generalized eigenvalue problem. http://arxiv.org/abs/math/0602386.Google Scholar
[8] Cuccagna, S., On asymptotic stability of ground states of NLS. Rev. Math. Phys. 15(2003), no. 8, 877–903. doi:10.1142/S0129055X03001849Google Scholar
[9] Comech, A. and Pelinovsky, D., Purely nonlinear instability of standing waves with minimal energy. Comm. Pure Appl. Math. 56(2003), no. 11, 1565–1607. doi:10.1002/cpa.10104Google Scholar
[10] Comech, A. and Pelinovsky, D., Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem. J. Math. Phys. 46(2005), no. 5. doi:10.1063/1.1901345Google Scholar
[11] Cuccagna, S., Pelinovsky, D., and Vougalter, V., Spectra of positive and negative energies in the linearized NLS problem. Comm. Pure Appl. Math. 58(2005), no. 1, 1–29. doi:10.1002/cpa.20050Google Scholar
[12] Erdogan, B. and Schlag, W., Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II. J. Anal. Math. 99(2006), 199–248. doi:10.1007/BF02789446Google Scholar
[13] Frölich, J., Gustafson, S., Jonsson, B. L. G., and Sigal, I. M., Solitary wave dynamics in an external potential. Comm. Math. Phys. 250(2004), no. 3, 613–642. doi:10.1007/s00220-004-1128-1Google Scholar
[14] Grillakis, M., Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system. Comm. Pure Appl. Math. 43(1990), no. 3, 299–333. doi:10.1002/cpa.3160430302Google Scholar
[15] Gang, Z. and Sigal, I. M., Asymptotic stability of nonlinear Schrödinger equations with potential. Rev. Math. Phys. 17(2005), no. 10, 1143–1207. doi:10.1142/S0129055X05002522Google Scholar
[16] McLeod, K., Uniqueness of positive radial solutions of Δu + f (u) = 0 in ℝ n. II. Trans. Amer. Math. Soc. 339(1993), no. 2, 495–505. doi:10.2307/2154282Google Scholar
[17] Pelinovsky, D., Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2005), no. 2055, 783–812. doi:10.1098/rspa.2004.1345Google Scholar
[18] Perelman, G., Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations. Comm. Partial Differential Equations 29(2004), no. 7–8, 1051–1095. doi:10.1081/PDE-200033754Google Scholar
[19] Reed, M. and Simon, B., Methods of modern mathematical physics IV. Analysis of operators. Academic Press, New York–London, 1978.Google Scholar
[20] Schlag, W., Stable manifolds for an orbitally unstable nonlinear Schrödinger equation. Ann. of Math. 169(2009), no. 1, 139–227. doi:10.4007/annals.2009.169.139Google Scholar
[21] Strauss, W., Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55(1977), no. 2, 149–162. doi:10.1007/BF01626517Google Scholar
[22] Vougalter, V. and Pelinovsky, D., Eigenvalues of zero energy in the linearized NLS problem. J. Math. Phys. 47(2006), no. 6. doi:10.1063/1.2203233Google Scholar
[23] Weinstein, M. I., Modulation stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(1985), no. 3, 472–491. doi:10.1137/0516034Google Scholar