# American Institute of Mathematical Sciences

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 - A, 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. [2] R. Beals, Characterization of pseudodifferential operators and applications,, Duke Mathematical Journal, 44 (1977), 45. doi: 10.1215/S0012-7094-77-04402-7. [3] H. Berestycki and P. L. Lion, Nonlinear scalar field equations, I: Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. [4] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Grundlehren der Mathematischen Wissenschaften, 223 (1976). [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., 25 (1998), 197. [6] T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003). [7] M. Christ and A. Kiselev, Maximal functions associated with filtrations,, Comm. Pure Appl. Math., 56 (2003), 1565. [8] A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy,, Comm. Pure Appl. Math., 56 (2003), 1565. doi: 10.1002/cpa.10104. [9] S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 54 (2001), 409. [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. doi: 10.1002/cpa.20050. [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. doi: 10.1007/BF02789446. [12] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). [13] L. C. Evans and M. Zworski, Lectures on semiclassical analysis,, Unpublished Lecture Notes, (2006). [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. doi: 10.1016/0022-1236(90)90016-E. [15] P. Hislop and I. M. Sigal, "Introduction to Spectral Theory. With Applications to Schrödinger Operators,", Applied Mathematical Sciences, 113 (1996). [16] L. Hörmander, "The Analysis of Linear Partial Differential Operators I,", Classics in Mathematics, (2003). [17] L. Hörmander, "The Analysis of Linear Partial Differential Operators II,", Classics in Mathematics, (2005). [18] L. Hörmander, "The Analysis of Linear Partial Differential Operators III,", Grundlehren der Mathematischen Wissenschaften, 274 (1994). [19] L. Hörmander, "The Analysis of Linear Partial Differential Operators IV,", Grundlehren der Mathematischen Wissenschaften, 275 (1994). [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. doi: 10.1090/S0894-0347-06-00524-8. [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. doi: 10.1137/09075175X. [22] J. L. Marzuola and G. Simpson, Spectral analysis for matrix Hamiltonian operators,, Nonlinearity, 24 (2010), 389. doi: 10.1088/0951-7715/24/2/003. [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. doi: 10.2307/2154282. [24] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of $N$-soliton states of NLS,, preprint, (2003). [25] W. Schlag, Stable manifolds for an orbitally unstable NLS,, Annals of Math., 169 (2009), 139. doi: 10.4007/annals.2009.169.139. [26] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations,, Communications in Mathematical Physics, 91 (1983), 313. doi: 10.1007/BF01208779. [27] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations,, Transactions of the American Mathematical Society, 290 (1985), 701. doi: 10.1090/S0002-9947-1985-0792821-7. [28] J. Shatah and W. Strauss, Instability of nonlinear bound states,, Communications in Mathematical Physics, 100 (1985), 173. doi: 10.1007/BF01212446. [29] E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,", Princeton Mathematical Series, 43 (1993). [30] C. Sulem and P. Sulem, "The nonlinear Schrödinger Equation. Self-Focusing and Wave-Collapse,", Applied Mathematical Sciences, 39 (1999). [31] M. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations,, SIAM Journal of Mathematical Analysis, 16 (1985), 472. doi: 10.1137/0516034. [32] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Communications on Pure and Applied Mathematics, 39 (1986), 51. doi: 10.1002/cpa.3160390103.

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##### References:
 [1] S. Agmon, Spectral properties for Schrödinger operators and scattering theory,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 151. [2] R. Beals, Characterization of pseudodifferential operators and applications,, Duke Mathematical Journal, 44 (1977), 45. doi: 10.1215/S0012-7094-77-04402-7. [3] H. Berestycki and P. L. Lion, Nonlinear scalar field equations, I: Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. [4] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Grundlehren der Mathematischen Wissenschaften, 223 (1976). [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., 25 (1998), 197. [6] T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003). [7] M. Christ and A. Kiselev, Maximal functions associated with filtrations,, Comm. Pure Appl. Math., 56 (2003), 1565. [8] A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy,, Comm. Pure Appl. Math., 56 (2003), 1565. doi: 10.1002/cpa.10104. [9] S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 54 (2001), 409. [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. doi: 10.1002/cpa.20050. [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. doi: 10.1007/BF02789446. [12] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). [13] L. C. Evans and M. Zworski, Lectures on semiclassical analysis,, Unpublished Lecture Notes, (2006). [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. doi: 10.1016/0022-1236(90)90016-E. [15] P. Hislop and I. M. Sigal, "Introduction to Spectral Theory. With Applications to Schrödinger Operators,", Applied Mathematical Sciences, 113 (1996). [16] L. Hörmander, "The Analysis of Linear Partial Differential Operators I,", Classics in Mathematics, (2003). [17] L. Hörmander, "The Analysis of Linear Partial Differential Operators II,", Classics in Mathematics, (2005). [18] L. Hörmander, "The Analysis of Linear Partial Differential Operators III,", Grundlehren der Mathematischen Wissenschaften, 274 (1994). [19] L. Hörmander, "The Analysis of Linear Partial Differential Operators IV,", Grundlehren der Mathematischen Wissenschaften, 275 (1994). [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. doi: 10.1090/S0894-0347-06-00524-8. [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. doi: 10.1137/09075175X. [22] J. L. Marzuola and G. Simpson, Spectral analysis for matrix Hamiltonian operators,, Nonlinearity, 24 (2010), 389. doi: 10.1088/0951-7715/24/2/003. [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. doi: 10.2307/2154282. [24] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of $N$-soliton states of NLS,, preprint, (2003). [25] W. Schlag, Stable manifolds for an orbitally unstable NLS,, Annals of Math., 169 (2009), 139. doi: 10.4007/annals.2009.169.139. [26] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations,, Communications in Mathematical Physics, 91 (1983), 313. doi: 10.1007/BF01208779. [27] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations,, Transactions of the American Mathematical Society, 290 (1985), 701. doi: 10.1090/S0002-9947-1985-0792821-7. [28] J. Shatah and W. Strauss, Instability of nonlinear bound states,, Communications in Mathematical Physics, 100 (1985), 173. doi: 10.1007/BF01212446. [29] E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,", Princeton Mathematical Series, 43 (1993). [30] C. Sulem and P. Sulem, "The nonlinear Schrödinger Equation. Self-Focusing and Wave-Collapse,", Applied Mathematical Sciences, 39 (1999). [31] M. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations,, SIAM Journal of Mathematical Analysis, 16 (1985), 472. doi: 10.1137/0516034. [32] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Communications on Pure and Applied Mathematics, 39 (1986), 51. doi: 10.1002/cpa.3160390103.
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