# American Institute of Mathematical Sciences

November  2011, 30(4): 995-1035. doi: 10.3934/dcds.2011.30.995

## Dispersive estimates using scattering theory for matrix Hamiltonian equations

 1 Department of Mathematics, UNC-Chapel Hill, CB#3250, Phillips Hall, Chapel Hill, NC 27599-3250, United States

Received  February 2010 Revised  February 2011 Published  May 2011

We develop the techniques of [25] and [11] in order to derive dispersive estimates for a matrix Hamiltonian equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schrödinger equation

$\i u_t + \Delta u + \beta (|u|^2) u = 0$
$\u(0,x) = u_0 (x),$

in $\mathbb{R}^3$. These results have been seen before, though we present a new approach using scattering theory techniques. In further works, we will numerically and analytically study the existence of a minimal mass soliton, as well as the spectral assumptions made in the analysis presented here.
Citation: Jeremy L. Marzuola. Dispersive estimates using scattering theory for matrix Hamiltonian equations. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 995-1035. doi: 10.3934/dcds.2011.30.995
##### References:
 [1] S. Agmon, Spectral properties for Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 151-218.  Google Scholar [2] R. Beals, Characterization of pseudodifferential operators and applications, Duke Mathematical Journal, 44 (1977), 45-57. doi: 10.1215/S0012-7094-77-04402-7.  Google Scholar [3] H. Berestycki and P. L. Lion, Nonlinear scalar field equations, I: Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar [4] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [5] J. Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 25 (1998), 197-215.  Google Scholar [6] T. 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Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or eigenvalue at zero energy in dimension three. II, J. Anal. Math., 99 (2006), 199-248. doi: 10.1007/BF02789446.  Google Scholar [12] L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar [13] L. C. Evans and M. Zworski, Lectures on semiclassical analysis, Unpublished Lecture Notes, 2006. Available from: http://math.berkeley.edu/ zworski/semiclassical.pdf. Google Scholar [14] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar [15] P. Hislop and I. M. Sigal, "Introduction to Spectral Theory. With Applications to Schrödinger Operators," Applied Mathematical Sciences, 113, Springer-Verlag, New York, 1996.  Google Scholar [16] L. Hörmander, "The Analysis of Linear Partial Differential Operators I," Classics in Mathematics, Springer-Verlag, Berlin, 2003.  Google Scholar [17] L. Hörmander, "The Analysis of Linear Partial Differential Operators II," Classics in Mathematics, Springer-Verlag, Berlin, 2005.  Google Scholar [18] L. Hörmander, "The Analysis of Linear Partial Differential Operators III," Grundlehren der Mathematischen Wissenschaften, 274, Springer-Verlag, Berlin, 1994.  Google Scholar [19] L. Hörmander, "The Analysis of Linear Partial Differential Operators IV," Grundlehren der Mathematischen Wissenschaften, 275, Springer-Verlag, Berlin, 1994.  Google Scholar [20] J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Amer. Math. Soc., 19 (2006), 815-920. doi: 10.1090/S0894-0347-06-00524-8.  Google Scholar [21] J. L. Marzuola, A class of stable perturbations for a minimal mass soliton in three dimensional saturated nonlinear Schrödinger equations, SIAM J. of Math. Anal., 42 (2010), 1382-1403. doi: 10.1137/09075175X.  Google Scholar [22] J. L. Marzuola and G. Simpson, Spectral analysis for matrix Hamiltonian operators, Nonlinearity, 24 (2010), 389-429. doi: 10.1088/0951-7715/24/2/003.  Google Scholar [23] K. McLeod, Uniqueness of positive radial solutions of $\Delta u + f(u) = 0$ in $\mathbb R^n$, II, Transactions of the American Mathematical Society, 339 (1993), 495-505. doi: 10.2307/2154282.  Google Scholar [24] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of $N$-soliton states of NLS, preprint, (2003), arXiv:math/0309114. Google Scholar [25] W. Schlag, Stable manifolds for an orbitally unstable NLS, Annals of Math., 169 (2009), 139-227. doi: 10.4007/annals.2009.169.139.  Google Scholar [26] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Communications in Mathematical Physics, 91 (1983), 313-327. doi: 10.1007/BF01208779.  Google Scholar [27] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Transactions of the American Mathematical Society, 290 (1985), 701-710. doi: 10.1090/S0002-9947-1985-0792821-7.  Google Scholar [28] J. Shatah and W. Strauss, Instability of nonlinear bound states, Communications in Mathematical Physics, 100 (1985), 173-190. doi: 10.1007/BF01212446.  Google Scholar [29] E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.  Google Scholar [30] C. Sulem and P. Sulem, "The nonlinear Schrödinger Equation. Self-Focusing and Wave-Collapse," Applied Mathematical Sciences, 39, Springer-Verlag, New York, 1999.  Google Scholar [31] M. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations, SIAM Journal of Mathematical Analysis, 16 (1985), 472-491. doi: 10.1137/0516034.  Google Scholar [32] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Communications on Pure and Applied Mathematics, 39 (1986), 51-68. doi: 10.1002/cpa.3160390103.  Google Scholar

show all references

##### References:
 [1] S. Agmon, Spectral properties for Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 151-218.  Google Scholar [2] R. Beals, Characterization of pseudodifferential operators and applications, Duke Mathematical Journal, 44 (1977), 45-57. doi: 10.1215/S0012-7094-77-04402-7.  Google Scholar [3] H. Berestycki and P. L. Lion, Nonlinear scalar field equations, I: Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar [4] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [5] J. Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 25 (1998), 197-215.  Google Scholar [6] T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.  Google Scholar [7] M. Christ and A. Kiselev, Maximal functions associated with filtrations, Comm. Pure Appl. Math., 56 (2003), 1565-1607.  Google Scholar [8] A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., 56 (2003), 1565-1607. doi: 10.1002/cpa.10104.  Google Scholar [9] S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 409-425.  Google Scholar [10] 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 [11] B. Erdogan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or eigenvalue at zero energy in dimension three. II, J. Anal. Math., 99 (2006), 199-248. doi: 10.1007/BF02789446.  Google Scholar [12] L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar [13] L. C. Evans and M. Zworski, Lectures on semiclassical analysis, Unpublished Lecture Notes, 2006. Available from: http://math.berkeley.edu/ zworski/semiclassical.pdf. Google Scholar [14] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar [15] P. Hislop and I. M. Sigal, "Introduction to Spectral Theory. With Applications to Schrödinger Operators," Applied Mathematical Sciences, 113, Springer-Verlag, New York, 1996.  Google Scholar [16] L. Hörmander, "The Analysis of Linear Partial Differential Operators I," Classics in Mathematics, Springer-Verlag, Berlin, 2003.  Google Scholar [17] L. Hörmander, "The Analysis of Linear Partial Differential Operators II," Classics in Mathematics, Springer-Verlag, Berlin, 2005.  Google Scholar [18] L. Hörmander, "The Analysis of Linear Partial Differential Operators III," Grundlehren der Mathematischen Wissenschaften, 274, Springer-Verlag, Berlin, 1994.  Google Scholar [19] L. Hörmander, "The Analysis of Linear Partial Differential Operators IV," Grundlehren der Mathematischen Wissenschaften, 275, Springer-Verlag, Berlin, 1994.  Google Scholar [20] J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Amer. Math. Soc., 19 (2006), 815-920. doi: 10.1090/S0894-0347-06-00524-8.  Google Scholar [21] J. L. Marzuola, A class of stable perturbations for a minimal mass soliton in three dimensional saturated nonlinear Schrödinger equations, SIAM J. of Math. Anal., 42 (2010), 1382-1403. doi: 10.1137/09075175X.  Google Scholar [22] J. L. Marzuola and G. Simpson, Spectral analysis for matrix Hamiltonian operators, Nonlinearity, 24 (2010), 389-429. doi: 10.1088/0951-7715/24/2/003.  Google Scholar [23] K. McLeod, Uniqueness of positive radial solutions of $\Delta u + f(u) = 0$ in $\mathbb R^n$, II, Transactions of the American Mathematical Society, 339 (1993), 495-505. doi: 10.2307/2154282.  Google Scholar [24] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of $N$-soliton states of NLS, preprint, (2003), arXiv:math/0309114. Google Scholar [25] W. Schlag, Stable manifolds for an orbitally unstable NLS, Annals of Math., 169 (2009), 139-227. doi: 10.4007/annals.2009.169.139.  Google Scholar [26] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Communications in Mathematical Physics, 91 (1983), 313-327. doi: 10.1007/BF01208779.  Google Scholar [27] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Transactions of the American Mathematical Society, 290 (1985), 701-710. doi: 10.1090/S0002-9947-1985-0792821-7.  Google Scholar [28] J. Shatah and W. Strauss, Instability of nonlinear bound states, Communications in Mathematical Physics, 100 (1985), 173-190. doi: 10.1007/BF01212446.  Google Scholar [29] E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.  Google Scholar [30] C. Sulem and P. Sulem, "The nonlinear Schrödinger Equation. Self-Focusing and Wave-Collapse," Applied Mathematical Sciences, 39, Springer-Verlag, New York, 1999.  Google Scholar [31] M. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations, SIAM Journal of Mathematical Analysis, 16 (1985), 472-491. doi: 10.1137/0516034.  Google Scholar [32] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Communications on Pure and Applied Mathematics, 39 (1986), 51-68. doi: 10.1002/cpa.3160390103.  Google Scholar
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