Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T21:41:58.538Z Has data issue: false hasContentIssue false

Bibliography

Published online by Cambridge University Press:  17 September 2021

Alexander Komech  and
Elena Kopylova
Alexander Komech
Affiliation:
Universität Wien, Austria
Elena Kopylova
Affiliation:
Universität Wien, Austria
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Attractors of Hamiltonian Nonlinear Partial Differential Equations , pp. 200 - 211
Publisher: Cambridge University Press
Print publication year: 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, V. I., Mathematical Methods of Classical Mechanics, Springer, New York, 1989.Google Scholar
Cazenave, T., Semilinear Schrödinger Equations, AMS, New York, 2003.CrossRefGoogle Scholar
Cazenave, T., Haraux, A., An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, 1998.CrossRefGoogle Scholar
Gaudry, G. I., Quasimeasures and operators commuting with convolution, Pacific J. Math. 18 (1966), 461–476.CrossRefGoogle Scholar
Hopf, E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213–231.CrossRefGoogle Scholar
Hörmander, L., The Analysis of Linear Partial Differential Operators, vol. I, 2nd ed., Springer, Berlin, 1990.Google Scholar
Jörgens, K., Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295–308.Google Scholar
Komech, A. I., Attractors of nonlinear Hamiltonian PDEs, Discrete Continuous Dyn. Syst. A 36 (2016), 6201–6256. arXiv:1409.2009Google Scholar
Komech, A. I., Komech, A. A., Principles of Partial Differential Equations, Springer, Berlin, 2009.CrossRefGoogle Scholar
Komech, A. I., Merzon, A. E., Stationary Diffraction by Wedges: Method of Automorphic Functions on Complex Characteristics, Lecture Notes in Mathematics 2249, Springer, Berlin, 2019.Google Scholar
Komech, A. I., Kopylova, E. A., Attractors of Hamiltonian nonlinear partial differential equations, Russ. Math. Surv. 75 (2020), 1–87.CrossRefGoogle Scholar
Lions, J.-L., Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier–Villars, Paris, 1969.Google Scholar
Reed, M., Simon, B., Methods of Modern Mathematical Physics, Academic Press, New York, I (1980), II (1975), III (1979), IV (1978).Google Scholar
Riesz, F., Sz.-Nagy, B., Functional Analysis, Dover, New York, 1990.Google Scholar
Rudin, W., Functional Analysis, McGraw-Hill, New York, 1977.Google Scholar
Babin, A. V., Vishik, M. I., Attractors of Evolution Equations, Studies in Mathematics and Its Applications 25, North-Holland, Amsterdam, 1992.Google Scholar
Chepyzhov, V. V., Vishik, M. I., Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications 49, American Mathematical Society, Providence, RI, 2002.Google Scholar
Foias, C., Manley, O., Rosa, R., Temam, R., Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications 83, Cambridge University Press, Cambridge, 2001.Google Scholar
Hale, J. K., Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988.Google Scholar
Haraux, A., Systémes Dynamiques Dissipatifs et Applications, R.M.A. 17, Collection dirigé par Ph. Ciarlet et J.-L. Lions, Masson, Paris, 1990.Google Scholar
Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 480, Springer, Berlin, 1981.Google Scholar
Landau, L., On the problem of turbulence, C. R. (Doklady) Acad. Sci. URSS (N.S.) 44 (1944), 311–314.Google Scholar
Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.Google Scholar
Berestycki, H., Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–345.Google Scholar
Berestycki, H., Lions, P.-L., Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), 347–375.CrossRefGoogle Scholar
Coclite, G. M., Georgiev, V., Solitary waves for Maxwell–Schrödinger equations, Electron. J. Differential Equations 94 (2004), 31 pp.Google Scholar
Esteban, M. J., Georgiev, V., Séré, E., Stationary solutions of the Maxwell-Dirac and the Klein–Gordon–Dirac equations, Calc. Var. Partial Differential Equations 4 (1996), 265–281.CrossRefGoogle Scholar
Lusternik, L., Schnirelmann, L., Méthodes Topologiques dans les Problèmes Variationels, Hermann, Paris, 1934.Google Scholar
Lusternik, L., Schnirelmann, L., Topological methods in variational problems and their applications to differential geometry of surfaces, Uspekhi Mat. Nauk 2 (1947), 166–217.Google Scholar
Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.CrossRefGoogle Scholar
Lax, P. D., Morawetz, C. S., Phillips, R. S., Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math., 16 (1963), 477–486.Google Scholar
Morawetz, C. S., Time decay for the nonlinear Klein–Gordon equations, Proc. R. Soc. London Ser. A 306 (1968), 291–296.Google Scholar
Morawetz, C. S., Strauss, W. A., Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1–31.Google Scholar
Segal, I., Quantization and dispersion for nonlinear relativistic equations, pp. 79–108 in: Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965), MIT Press, Cambridge, MA, 1966.Google Scholar
Segal, I., Dispersion for non-linear relativistic equations. II, Ann. Sci. École Norm. Sup. (4) 1 (1968), 459–497.CrossRefGoogle Scholar
Strauss, W. A., Decay and asymptotics for Du = F(u), J. Funct. Anal. 2 (1968), 409–457.Google Scholar
Vainberg, B. R., Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989.Google Scholar
Freidlin, M., Komech, A. I., On metastable regimes in stochastic Lamb system, J. Math. Phys. 47 (2006), 043301.Google Scholar
Keller, J. B., Bonilla, L. L., Irreversibility and nonrecurrence, J. Stat. Physics 42 (1986), 1115–1125.Google Scholar
Komech, A. I., On the stabilization of interaction of a string with a nonlinear oscillator, Moscow Univ. Math. Bull. 46, no. 6 (1991), 34–39.Google Scholar
Komech, A. I., On stabilization of string-nonlinear oscillator interaction, J. Math. Anal. Appl. 196 (1995), 384–409.Google Scholar
Komech, A. I., On the stabilization of string-oscillator interaction, Russ. J. Math. Phys. 3 (1995), 227–247.Google Scholar
Komech, A. I., On transitions to stationary states in one-dimensional nonlinear wave equations, Arch. Ration. Mech. Anal. 149 (1999), 213–228.CrossRefGoogle Scholar
Komech, A. I., Spohn, H., Kunze, M., Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Differential Equations 22 (1997), 307–335.Google Scholar
Komech, A. I., Spohn, H., Long-time asymptotics for the coupled Maxwell– Lorentz equations, Comm. Partial Differential Equations 25 (2000), 559–584.Google Scholar
Komech, A., Attractors of non-linear Hamiltonian one-dimensional wave equations, Russ. Math. Surv. 55, no. 1 (2000), 43–92.Google Scholar
Komech, A. I., Merzon, A., Scattering in the nonlinear Lamb system, Phys. Lett. A 373 (2009), 1005–1010.Google Scholar
Komech, A. I., Merzon, A., On asymptotic completeness for scattering in the nonlinear Lamb system, J. Math. Phys. 50 (2009), 023514.Google Scholar
Komech, A. I., Merzon, A., On asymptotic completeness of scattering in the nonlinear Lamb system, II, J. Math. Phys. 54 (2013), 012702.Google Scholar
Kopylova, E., On global attraction to stationary states for wave equation with concentrated nonlinearity, J. Dyn. Diff. Equations 30, no. 1 (2018), 107–116.Google Scholar
Lamb, H., On a peculiarity of the wave-system due to the free vibrations of a nucleus in an extended medium, Proc. London Math. Soc. 32 (1900), 208–211.Google Scholar
Spohn, H., Dynamics of Charged Particles and Their Radiation Field, Cambridge University Press, Cambridge, 2004.CrossRefGoogle Scholar
Eckhaus, W., van Harten, A., The Inverse Scattering Transformation and the Theory of Solitons, North-Holland, Amsterdam, 1981.Google Scholar
Komech, A. I., Spohn, H., Soliton asymptotics for a classical particle interacting with a scalar wave field, Nonlinear Anal. 33 (1998), 13–24.Google Scholar
Imaykin, V., Komech, A. I., Mauser, N., Soliton-type asymptotics for the coupled Maxwell-Lorentz equations, Ann. Henri Poincaré P (2004), 1117–1135.Google Scholar
Imaykin, V., Komech, A. I., Spohn, H., Soliton-type asymptotics and scattering for a charge coupled to the Maxwell field, Russ. J. Math. Phys. 9 (2002), 428–436.Google Scholar
Imaykin, V., Komech, A. I., Spohn, H., Scattering theory for a particle coupled to a scalar field, Discrete Continuous Dyn. Syst. 10 (2004), 387–396.Google Scholar
Komech, A. I., Mauser, N. J., Vinnichenko, A. P., Attraction to solitons in relativistic nonlinear wave equations, Russ. J. Math. Phys. 11 (2004), 289–307.Google Scholar
Lamb, G. L. Jr., Elements of Soliton Theory, John Wiley, New York, 1980.Google Scholar
Comech, A., Weak attractor of the Klein–Gordon field in discrete space-time interacting with a nonlinear oscillator, Discrete Continuous Dyn. Syst. 33 (2013), 2711–2755.Google Scholar
Comech, A., On global attraction to solitary waves: Klein–Gordon equation with mean field interaction at several points, J. Differential Equations 252 (2012), 5390–5413.Google Scholar
Komech, A. I., On attractor of a singular nonlinear U(1)-invariant Klein–Gordon equation, pp. 599–611 in: Progress in Analysis, World Scientific, River Edge, NJ, 2003.Google Scholar
Komech, A. I., Komech, A. A., On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator, C. R. Math. Acad. Sci. Paris 343 (2006), 111–114.CrossRefGoogle Scholar
Komech, A. I., Komech, A. A., Global attractor for a nonlinear oscillator coupled to the Klein–Gordon field, Arch. Ration. Mech. Anal. 185 (2007), 105–142.Google Scholar
Komech, A. I., Komech, A. A., On global attraction to solitary waves for the Klein–Gordon field coupled to several nonlinear oscillators, J. Math. Pure Appl. 93 (2010), 91–111.Google Scholar
Komech, A. I., Komech, A. A., Global attraction to solitary waves for Klein– Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 855–868.CrossRefGoogle Scholar
Komech, A. I., Komech, A. A., Global attraction to solitary waves for a nonlinear Dirac equation with mean field interaction, SIAM J. Math. Anal. 42 (2010), 2944–2964.Google Scholar
Komech, A. A., Komech, A. I., A variant of the Titchmarsh convolution theorem for distributions on the circle, Funktsional. Anal. Prilozhen. 47 (2013), 26–32.Google Scholar
Kopylova, E., Global attraction to solitary waves for Klein–Gordon equation with concentrated nonlinearity, Nonlinearity 30, no. 11 (2017), 4191–4207.Google Scholar
Kopylova, E., Komech, A. I., On global attractor of 3D Klein–Gordon equation with several concentrated nonlinearities, Dyn. PDE 16, no. 2 (2019), 105–124.Google Scholar
Kopylova, E., Komech, A. I., Global attractor for 1D Dirac field coupled to nonlinear oscillator, Comm. Math. Phys. 375, no. 1 (2020), 573–603.CrossRefGoogle Scholar
Ladyženskaya, O. A., On the limiting amplitude principle, Uspekhi Mat. Nauk 12 (1957), 161–164. [Russian]Google Scholar
Levin, B. Y., Lectures on Entire Functions, American Mathematical Society, Providence, RI, 1996.Google Scholar
Lewin, L., Advanced Theory of Waveguides, Iliffe, London, 1951.Google Scholar
Morawetz, C. S., The limiting amplitude principle, Comm. Pure Appl. Math. 15 (1962), 349–361.Google Scholar
Sigal, I. M., Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions, Comm. Math. Phys. 153 (1993), 297–320.Google Scholar
Strauss, W. A., Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981), 110–133.Google Scholar
Strauss, W. A., Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal. 43 (1981), 281–293.Google Scholar
Titchmarsh, E. C., The zeros of certain integral functions, Proc. London Math. Soc. S2–25, no. 1 (1926), 283–302.Google Scholar
Bach, V., Chen, T., Faupin, J., Fröhlich, J., Sigal, I. M., Effective dynamics of an electron coupled to an external potential in non-relativistic QED, Ann. Henri Poincaré 14 (2013), 1573–1597.CrossRefGoogle Scholar
Demoulini, S., Stuart, D., Adiabatic limit and the slow motion of vortices in a Chern–Simons–Schrödinger system, Comm. Math. Phys. 290 (2009), 597–632.Google Scholar
Fröhlich, J., Gustafson, S., Jonsson, B. L. G., Sigal, I. M., Solitary wave dynamics in an external potential, Comm. Math. Phys. 250 (2004), 613–642.Google Scholar
Fröhlich, J., Tsai, T.-P., Yau, H.-T., On the point-particle (Newtonian) limit of the nonlinear Hartree equation, Comm. Math. Phys. 225 (2002), 223–274.Google Scholar
Komech, A., Kunze, M., Spohn, H., Effective dynamics for a mechanical particle coupled to a wave field, Comm. Math. Phys. 203 (1999), 1–19.Google Scholar
Kunze, M., Spohn, H., Adiabatic limit for the Maxwell–Lorentz equations, Ann. Henri Poincaré 1 (2000), 625–653.Google Scholar
Long, E., Stuart, D., Effective dynamics for solitons in the nonlinear Klein– Gordon–Maxwell system and the Lorentz force law, Rev. Math. Phys. 21 (2009), 459–510.Google Scholar
Stuart, D., Existence and Newtonian limit of nonlinear bound states in the Einstein–Dirac system, J. Math. Phys. 51 (2010), 032501, 13.Google Scholar
Imaykin, V., Komech, A. I., Spohn, H., Rotating charge coupled to the Maxwell field: Scattering theory and adiabatic limit, Monatsh. Math. 142 (2004), 143–156.Google Scholar
Adami, R., Noja, D., Ortoleva, C., Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three, J. Math. Phys. 54, no. 1 (2013), 013501.Google Scholar
Albeverio, S., Figari, R., Quantum fields and point interactions, Rend. Mat. Appl. (7) 39 (2018), 161–180.Google Scholar
Berezin, F. A., Faddeev, L. D., Remark on the Schrödinger equation with singular potential, Soviet Math. Dokl. 2 (1961), 372–375.Google Scholar
Cornish, F. H., Classical radiation theory and point charges, Proc. Phys. Soc. 86, no 3 (1965), 427–442.Google Scholar
Dirac, P. A. M., Classical theory of radiating electrons, Proc. R. Soc. A 167 (1938), 148–169.Google Scholar
Gittel, H.-P., Kijowski, J., Zeidler, E., The relativistic dynamics of the combined particle-field system in renormalized classical electrodynamics, Comm. Math. Phys. 198 (1998), 711–736.CrossRefGoogle Scholar
Noja, D., Posilicano, A., Wave equations with concentrated nonlinearities. J. Phys. A 38 (2005), 5011–5022.Google Scholar
Yafaev, D. R., On a zero-range interaction of a quantum particlewith the vacuum, J. Phys. A 25 (1992), 963–978.Google Scholar
Yafaev, D. R., A point interaction for the discrete Schrödinger operator and generalized Chebyshev polynomial, J. Math. Phys. 58, no. 6 (2017), 063511.Google Scholar
Zel’dovich, Ya. B., Scattering by a singular potential in perturbation theory and in the momentum representation, JETP 11 (1960), 594–597.Google Scholar
Bambusi, D., Galgani, L., Some rigorous results on the Pauli–Fierz model of classical electrodynamics, Ann. H. Poincaré Phys. Theor. 58 (1993), 155–171.Google Scholar
Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160–197.Google Scholar
Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), 308–348.Google Scholar
Marsden, J. E., Ratiu, T. S., Introduction to Mechanics and Symmetry, Springer, Berlin, 1994.Google Scholar
Oh, Y. G., A stability criterion for Hamiltonian systems with symmetry, J. Geom. Phys. 4 (1987), 163–182.Google Scholar
Poincaré, H., Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation, Acta. Math. 7 (1885), 259–380.Google Scholar
Andersson, L., Blue, P., Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, J. Hyperbolic Diff. Equations 12 (2015), 689–743.Google Scholar
Bambusi, D., Cuccagna, S., On dispersion of small energy solutions to the nonlinear Klein–Gordon equation with a potential, Am. J. Math. 133 (2011), 1421–1468.Google Scholar
Bensoussan, A., Iliine, C., Komech, A. I., Breathers for a relativistic nonlinear wave equation, Arch. Ration. Mech. Anal. 165 (2002), 317–345.Google Scholar
Boussaid, N., Stable directions for small nonlinear Dirac standing waves, Comm. Math. Phys. 268 (2006), 757–817.Google Scholar
Boussaid, N., Cuccagna, S., On stability of standing waves of nonlinear Dirac equations, Comm. Partial Diff. Equations 37 (2012), 1001–1056.Google Scholar
Buslaev, V. S., Perelman, G. S., Scattering for the nonlinear Schrödinger equation: States that are close to a soliton, Algebra Anal. 4 (1992), 63–102.Google Scholar
Buslaev, V. S., Perelman, G. S., On the stability of solitary waves for nonlinear Schrödinger equations, pp. 75–98 in: Nonlinear Evolution Equations, Amer. Math. Soc. Transl. Ser. 2 164, AMS, Providence, RI, 1995.Google Scholar
Buslaev, V., Sulem, C., On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 419–475.Google Scholar
Buslaev, V., Komech, A. I., Kopylova, E., Stuart, D., On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Comm. Partial Diff. Equations 33 (2008), 669–705.Google Scholar
Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (2001), 1110–1145.Google Scholar
Cuccagna, S., The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Comm. Math. Phys. 305 (2011), 279–331.Google Scholar
Cuccagna, S., Mizumachi, T., On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys. 284 (2008), 51–77.Google Scholar
Dafermos, M., Rodnianski, I., A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, Invent. Math. 185 (2011), 467–559.Google Scholar
Donninger, R., Schlag, W., Soffer, A., On pointwise decay of linear waves on a Schwarzschild black hole background, Comm. Math. Phys. 309 (2012), 51–86.Google Scholar
Duyckaerts, T., Kenig, C., Merle, F., Profiles of bounded radial solutions of the focusing, energy-critical wave equation, Geom. Funct. Anal. 22 (2012), 639–698.Google Scholar
Duyckaerts, T., Kenig, C., Merle, F., Scattering for radial, bounded solutions of focusing supercritical wave equations, Int. Math. Res. Not. IMRN 2014, no. 1 (2014), 224–258.Google Scholar
Duyckaerts, T., Kenig, C., Merle, F., Concentration-compactness and universal profiles for the non-radial energy critical wave equation, Nonlinear Anal. 138 (2016), 44–82.Google Scholar
Fröhlich, J., Gang, Z., Emission of Cherenkov radiation as a mechanism for Hamiltonian friction, Adv. Math. 264 (2014), 183–235.Google Scholar
Harada, T., Maeda, H., Stability criterion for self-similar solutions with a scalar field and those with a stiff fluid in general relativity, Classical Quantum Gravity 21 (2004), 371–389.Google Scholar
Imaikin, V., Soliton asymptotics for systems of field-particle type, Russ. Math. Surv. 68 (2013), 227–281.Google Scholar
Imaykin, V., Komech, A. I., Spohn, H., Scattering asymptotics for a charged particle coupled to the Maxwell field, J. Math. Phys. 52 (2011), 042701.CrossRefGoogle Scholar
Imaykin, V., Komech, A. I., Vainberg, B., On scattering of solitons for the Klein–Gordon equation coupled to a particle, Comm. Math. Phys. 268 (2006), 321–367.Google Scholar
Imaykin, V., Komech, A. I., Vainberg, B., Scattering of solitons for coupled wave-particle equations, J. Math. Anal. Appl. 389 (2012), 713–740.Google Scholar
Kenig, C., Lawrie, A., Liu, B., Schlag, W., Stable soliton resolution for exterior wave maps in all equivariance classes, Adv. Math. 285 (2015), 235–300.Google Scholar
Kenig, C. E., Lawrie, A., Schlag, W., Relaxation of wave maps exterior to a ball to harmonic maps for all data, Geom. Funct. Anal. 24 (2014), 610–647.Google Scholar
Kenig, C. E., Merle, F., Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), 645–675.Google Scholar
Kenig, C. E., Merle, F., Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), 147–212.Google Scholar
Kenig, C. E., Merle, F., Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, Am. J. Math. 133 (2011), 1029–1065.CrossRefGoogle Scholar
Komech, A. I., Komech, A. A., Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator, Russ. J. Math. Phys. 14 (2007), 164–173.Google Scholar
Komech, A. I., Kopylova, E., Scattering of solitons for the Schrödinger equation coupled to a particle, Russ. J. Math. Phys. 13 (2006), 158–187.Google Scholar
Kopylova, E., Komech, A. I., On asymptotic stability of moving kink for relativistic Ginzburg–Landau equation, Comm. Math. Phys. 302 (2011), 225–252.Google Scholar
Kopylova, E., Komech, A. I., On asymptotic stability of kink for relativistic Ginzburg–Landau equations, Arch. Ration. Mech. Anal. 202 (2011), 213–245.Google Scholar
Komech, A. I., Kopylova, E., On eigenfunction expansion of solutions to the Hamiltonian equations, J. Stat. Phys. 154 (2014), 503–521.Google Scholar
Komech, A. I., Kopylova, E., On the eigenfunction expansion for the Hamiltonian operators, J. Spectr. Theory 5 (2015), 331–361.Google Scholar
Komech, A. I., Kopylova, E., Kopylov, S., On nonlinear wave equations with parabolic potentials, J. Spectr. Theory 3 (2013), 485–503.Google Scholar
Komech, A. I., Kopylova, E., Spohn, H., Scattering of solitons for Dirac equation coupled to a particle, J. Math. Anal. Appl. 383 (2011), 265–290.Google Scholar
Komech, A. I., Kopylova, E., Stuart, D., On asymptotic stability of solitons in a nonlinear Schrödinger equation, Comm. Pure Appl. Anal. 11 (2012), 1063–1079.Google Scholar
Kopylova, E., On asymptotic stability of solitary waves in discrete Schrödinger equation coupled to nonlinear oscillator, Nonlinear Anal. Ser. A Theory Methods Appl. 71 (2009), 3031–3046.Google Scholar
Kopylova, E., On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to nonlinear oscillator, Applicable Anal. 89 (2010), 1467–1493.Google Scholar
Kopylova, E., Asymptotic stability of solitons for non-linear hyperbolic equations, Russ. Math. Surv. 68 (2013), 283–334.Google Scholar
Krein, M. G., Langer, H. K., The spectral function of a self-adjoint operator in a space with indefinite metric, Sov. Math. Dokl. 4 (1963), 1236–1239.Google Scholar
Krieger, J., Nakanishi, K., Schlag, W., Center-stable manifold of the ground state in the energy space for the critical wave equation, Math. Ann. 361 (2015), 1–50.Google Scholar
Krieger, J., Schlag, W., Concentration Compactness for Critical Wave Maps, EMS Monographs in Mathematics, European Mathematical Society, Zürich, 2012.Google Scholar
Langer, H., Spectral functions of definitizable operators in Krein spaces, pp. 1–46 in: Butkovic, D., Kraljevic, H., Kurepa, S. (eds.), Functional Analysis: Proceedings of a Conference Held in Dubrovnik, November 2–14, 1981, Lecture Notes in Mathematics 948, Springer, Berlin, 1982.Google Scholar
Martel, Y., Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg–de Vries equations, Am. J. Math. 127 (2005), 1103–1140.Google Scholar
Martel, Y., Merle, F., Asymptotic stability of solitons of the subcritical gKdV equations revisited, Nonlinearity 18 (2005), 55–80.Google Scholar
Martel, Y., Merle, F., Tsai, T. P., Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations, Comm. Math. Phys. 231 (2002), 347–373.Google Scholar
Merkli, M., Sigal, I. M., A time-dependent theory of quantum resonances, Comm. Math. Phys. 201 (1999), 549–576.Google Scholar
Miller, J. R., Weinstein, M. I., Asymptotic stability of solitary waves for the regularized long-wave equation, Comm. Pure Appl. Math. 49 (1996), 399–441.Google Scholar
Nakanishi, K., Schlag, W., Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society, Zürich, 2011.Google Scholar
Pillet, C. A., Wayne, C. E., Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Diff. Equations 141 (1997), 310–326.Google Scholar
Pego, R. L., Weinstein, M. I., Asymptotic stability of solitary waves, Comm. Math. Phys. 164 (1994), 305–349.Google Scholar
Perelman, G., Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Diff. Equations 29 (2004), 1051–1095.Google Scholar
Reed, M., Simon, B., Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.Google Scholar
Rodnianski, I., Schlag, W., Soffer, A., Asymptotic stability of N-soliton states of NLS, ArXiv: math/ 0309114.Google Scholar
Rodnianski, I., Schlag, W., Soffer, A., Dispersive analysis of charge transfer models, Comm. Pure Appl. Math. 58 (2005), 149–216.Google Scholar
Soffer, A., Soliton dynamics and scattering, pp. 459–471 in: International Congress of Mathematicians, vol. III, European Mathematical Society, Zürich, 2006.Google Scholar
Soffer, A., Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), 119–146.Google Scholar
Soffer, A., Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data, J. Diff. Equations 98 (1992), 376–390.Google Scholar
Soffer, A., Weinstein, M. I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), 9–74.Google Scholar
Soffer, A., Weinstein, M. I., Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys. 16 (2004), 977–1071.Google Scholar
Stuart, D. M. A., Modulational approach to stability of non-topological solitons, J. Math. Pure. Appl. 80, no. 1 (2001), 51–83.Google Scholar
Tsai, T. P., Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Diff. Equations 192 (2003), 225–282.Google Scholar
Tsai, T. P., Yau, H. T., Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. Theor. Math. Phys. 6 (2002), 107–139.Google Scholar
Tsai, T. P., Yau, H. T., Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions, Comm. Pure Appl. Math. 55 (2002), 153–216.Google Scholar
Weinstein, M. I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472–491.Google Scholar
Agmon, S., Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 151–218.Google Scholar
D’Ancona, P., Fanelli, L., Vega, L., Visciglia, N., Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal. 258 (2010), 3227–3240.Google Scholar
D’Ancona, P., Kato smoothing and Strichartz estimates for wave equations with magnetic potentials, Comm. Math. Phys. 335 (2015), 1–16.Google Scholar
Beals, M., Strauss, W., Lp estimates for the wave equation with a potential, Comm. Partial Diff. Equations 18 (1993), 1365–1397.Google Scholar
Beceanu, M., Goldberg, M., Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys. 314 (2012), 471–481.Google Scholar
Beceanu, M., Goldberg, M., Strichartz estimates and maximal operators for the wave equation in R3, J. Funct. Anal. 266 (2014), 1476–1510.Google Scholar
Egorova, I., Kopylova, E., Marchenko, V. A., Teschl, G., Dispersion estimates for one-dimensional Schrödinger and Klein–Gordon equations. Revisited. Russ. Math. Surv. 71 (2016), 391–415.Google Scholar
Egorova, I., Kopylova, E., Teschl, G., Dispersion estimates for one-dimensional discrete Schrödinger and wave equations, J. Spectr. Theory 5 (2015), 663–696.Google Scholar
Erdoğan, M. B., Goldberg, M., Green, W. R., Dispersive estimates for four-dimensional Schrödinger and wave equations with obstructions at zero energy, Comm. Partial Diff. Equations 39 (2014), 1936–1964.Google Scholar
Goldberg, M., Schlag, W., Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys. 251 (2004), 157–178.Google Scholar
Goldberg, M., Green, W. R., Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues I: The odd dimensional case, J. Funct. Anal. 269 (2015), 633–682.Google Scholar
Goldberg, M., Green, W. R., Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues II: The even dimensional case, J. Spectr. Theory 7, no. 1 (2017), 33–86.Google Scholar
Jensen, A., Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46 (1979), 583–611.Google Scholar
Journé, J.-L., Soffer, A., Sogge, C. D., Decay estimates for Schrödinger operators, Comm. Pure Appl. Math. 44 (1991), 573–604.Google Scholar
Komech, A. I., Kopylova, E., Long time decay for 2D Klein–Gordon equation, J. Funct. Anal. 259 (2010), 477–502.Google Scholar
Komech, A. I., Kopylova, E., Weighted energy decay for 3D Klein–Gordon equation, J. Diff. Equations 248 (2010), 501–520.Google Scholar
Komech, A. I., Kopylova, E., Weighted energy decay for 1D Klein–Gordon equation, Comm. PDE 35 (2010), 353–374.Google Scholar
Komech, A. I., Kopylova, E., Dispersion Decay and Scattering Theory, John Wiley, Hoboken, NJ, 2012.Google Scholar
Komech, A. I., Kopylova, E., Dispersion decay for the magnetic Schrödinger equation, J. Funct. Anal. 264 (2013), 735–751.Google Scholar
Komech, A. I., Kopylova, E., Weighted energy decay for magnetic Klein–Gordon equation, Appl. Anal. 94 (2015), 219–233.Google Scholar
Zygmund, A., Trigonometric Series I, Cambridge University Press, Cambridge, 1968.Google Scholar
Komech, A. I., Kopylova, E. A., Kunze, M., Dispersive estimates for 1D discrete Schrödinger and Klein–Gordon equations, Appl. Anal. 85 (2006), 1487–1508.Google Scholar
Komech, A. I., Kopylova, E. A., Vainberg, B. R., On dispersive properties of discrete 2D Schrödinger and Klein–Gordon equations, J. Funct. Anal. 254 (2008), 2227–2254.Google Scholar
Kopylova, E. A., Dispersive estimates for discrete Schrödinger and Klein– Gordon equations, Algebra Anal. 21 (2009), 87–113.Google Scholar
Kopylova, E., On dispersion decay for 3D Klein–Gordon equation, Discrete Continuous Dyn. Syst. A 38 (2018), 5765–5780.Google Scholar
Kopylova, E. A., Dispersion estimates for the Schrödinger and Klein–Gordon equations, Uspekhi Mat. Nauk 65 (2010), 97–144.Google Scholar
Kopylova, E., Teschl, G., Dispersion estimates for one-dimensional discrete Dirac equations, J. Math. Anal. Appl. 434 (2016), 191–208.Google Scholar
Marshall, B., Strauss, W., Wainger, S., Lp − Lq estimates for the Klein–Gordon equation, J. Math. Pure. Appl. (9) 59 (1980), 417–440.Google Scholar
Rodnianski, I., Schlag, W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), 451–513.Google Scholar
Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Series 43, Princeton University Press, Princeton, NJ, 1993.Google Scholar
Tataru, D., Local decay of waves on asymptotically flat stationary space-times, Am. J. Math. 135 (2013), 361–401.Google Scholar
Yajima, K., Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue, Comm. Math. Phys. 259 (2005), 475–509.Google Scholar
Abraham, M., Prinzipien der Dynamik des Elektrons, Phys. Z. 4 (1902), 57–63.Google Scholar
Abraham, M., Theorie der Elektrizität, Bd.2: Elektromagnetische Theorie der Strahlung, Teubner, Leipzig, 1905.Google Scholar
Bohr, N., Discussion with Einstein on epistemological problems in atomic physics, pp. 201–241 in: Schilpp, P. A., ed., Albert Einstein: Philosopher-Scientist, Library of Living Philosophers, Evanston, Illinois, 1949.Google Scholar
Einstein, A., Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?, Ann. Phys. 18 (1905), 639–643.Google Scholar
Feynman, R. P., Leighton, R. B., Sands, M., The Feynman Lectures on Physics, vol. 2, Mainly Electromagnetism and Matter, Addison-Wesley, Reading, MA, 1964.Google Scholar
Heisenberg, W., Der derzeitige Stand der nichtlinearen Spinortheorie der Elementarteilchen, Acta Phys. Austriaca 14 (1961), 328–339.Google Scholar
Heisenberg, W., Introduction to the Unified Field Theory of Elementary Particles, Interscience, London, 1966.Google Scholar
Houllevigue, L., L’É volution des Sciences, A. Collin, Paris, 1908.Google Scholar
Itzykson, C., Zuber, J. B., Quantum Field Theory, McGraw-Hill, New York, 1980.Google Scholar
Jackson, J. D., Classical Electrodynamics, 3rd ed., John Wiley, New York, 1999.Google Scholar
Komech, A. I., Quantum Mechanics: Genesis and Achievements, Springer, Dordrecht, 2013.Google Scholar
Komech, A. I., Quantum jumps and attractors of Maxwell–Schrödinger equations, arXiv 1907.04297 math-ph. https://arxiv.org/abs/1907.04297Google Scholar
von Neumann, J., Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955.Google Scholar
Newton, R., Quantum Physics, Springer, New York, 2002.Google Scholar
Sakurai, J. J., Advanced Quantum Mechanics, Addison-Wesley, Reading, MA, 1967.Google Scholar
Schiff, L. I., Quantum Mechanics, McGraw-Hill, New York, 1955.Google Scholar
Schrödinger, E., Quantisierung als Eigenwertproblem, Ann. Phys. I, II 79 (1926) 361, 489; III 80 (1926) 437; IV 81 (1926) 109. (English translation in E. Schrödinger, Collected Papers on Wave Mechanics, Blackie & Sohn, London, 1928.)Google Scholar
Woodgate, G. K., Elementary Atomic Structure, Clarendon Press, Oxford, 2002.Google Scholar
Barnes, V. E. et al., Observation of a hyperon with strangeness minus three, Phys. Rev. Lett. 12 (1964), 204–206.CrossRefGoogle Scholar
Gell-Mann, M., Symmetries of baryons and mesons, Phys. Rev. (2) 125 (1962), 1067–1084.Google Scholar
Halzen, F., Martin, A., Quarks and Leptons: An Introductory Course in Modern Particle Physics, John Wiley, New York, 1984.Google Scholar
Néeman, Y., Unified interactions in the unitary gauge theory, Nuclear Phys. 30 (1962), 347–349.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Bibliography
  • Alexander Komech, Universität Wien, Austria, Elena Kopylova, Universität Wien, Austria
  • Book: Attractors of Hamiltonian Nonlinear Partial Differential Equations
  • Online publication: 17 September 2021
  • Chapter DOI: https://doi.org/10.1017/9781009025454.011
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Bibliography
  • Alexander Komech, Universität Wien, Austria, Elena Kopylova, Universität Wien, Austria
  • Book: Attractors of Hamiltonian Nonlinear Partial Differential Equations
  • Online publication: 17 September 2021
  • Chapter DOI: https://doi.org/10.1017/9781009025454.011
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Bibliography
  • Alexander Komech, Universität Wien, Austria, Elena Kopylova, Universität Wien, Austria
  • Book: Attractors of Hamiltonian Nonlinear Partial Differential Equations
  • Online publication: 17 September 2021
  • Chapter DOI: https://doi.org/10.1017/9781009025454.011
Available formats
×