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. 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

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

[1]

M. Burak Erdoǧan, William R. Green. Dispersive estimates for matrix Schrödinger operators in dimension two. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4473-4495. doi: 10.3934/dcds.2013.33.4473

[2]

Yonggeun Cho, Tohru Ozawa, Suxia Xia. Remarks on some dispersive estimates. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1121-1128. doi: 10.3934/cpaa.2011.10.1121

[3]

Fabio Nicola. Remarks on dispersive estimates and curvature. Communications on Pure & Applied Analysis, 2007, 6 (1) : 203-212. doi: 10.3934/cpaa.2007.6.203

[4]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57

[5]

Michael Ruzhansky, Jens Wirth. Dispersive type estimates for fourier integrals and applications to hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 1263-1270. doi: 10.3934/proc.2011.2011.1263

[6]

Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75

[7]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551

[8]

Fioralba Cakoni, Shixu Meng, Jingni Xiao. A note on transmission eigenvalues in electromagnetic scattering theory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021025

[9]

Valery Imaikin, Alexander Komech, Herbert Spohn. Scattering theory for a particle coupled to a scalar field. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 387-396. doi: 10.3934/dcds.2004.10.387

[10]

Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems & Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003

[11]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010

[12]

Oana Ivanovici. Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021093

[13]

Kirill D. Cherednichenko, Alexander V. Kiselev, Luis O. Silva. Functional model for extensions of symmetric operators and applications to scattering theory. Networks & Heterogeneous Media, 2018, 13 (2) : 191-215. doi: 10.3934/nhm.2018009

[14]

Deyue Zhang, Yukun Guo. Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory. Electronic Research Archive, 2021, 29 (2) : 2149-2165. doi: 10.3934/era.2020110

[15]

Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085

[16]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139

[17]

Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261

[18]

Vito Mandorino. Connecting orbits for families of Tonelli Hamiltonians. Journal of Modern Dynamics, 2012, 6 (4) : 499-538. doi: 10.3934/jmd.2012.6.499

[19]

Elena Cordero, Fabio Nicola, Luigi Rodino. Schrödinger equations with rough Hamiltonians. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4805-4821. doi: 10.3934/dcds.2015.35.4805

[20]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]