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Prof. Dr. Alexander Komech

Profil

Derzeitige StellungPost Doc
FachgebietTheoretische Physik,Analysis, Differentialgleichungen,Stochastik, Wahrscheinlichkeitstheorie
Keywordslong time asymptotics, nonlinear Hamiltonian partial differential equns, quantum electrodynamics, quantum fields, quantum mechanics, differential crossection, soliton, scattering, radiation

Aktuelle Kontaktadresse

LandRussische Föderation
OrtMoscow
Universität/InstitutionMoscow M.V. Lomonosov State University
Institut/AbteilungDepartment of Mechanics and Mathematics

Gastgeber*innen während der Förderung

Prof. Dr. Eberhard ZeidlerMax-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig
Prof. Dr. Dr. h.c. mult. Herbert SpohnFakultät für Mathematik / Zentrum Mathematik, Technische Universität München, München
Prof. Dr. Dr. h.c. mult. Herbert SpohnFakultät für Mathematik / Zentrum Mathematik, Technische Universität München, Garching
Beginn der ersten Förderung01.07.2007

Programm(e)

2006Forschungspreis-Programm auf Gegenseitigkeit für Wissenschaftler*innen aus dem Ausland

Projektbeschreibung der*des Nominierenden

Wave phenomena are ubiquitous in nature and in science, including engineering, applications. Linear wave equations have a long mathematical tradition and there is a very well developed theory. Nonlinear wave equations are more difficult and one has to rely on a case by case study. The more recent mathematical work of Alexander Komech is centered around the understanding of the long time asymptotics of solutions to particular nonlinear wave equations, as for example the nonlinear Schrödinger equation. In one space-dimension such equations admit travelling wave solutions (soliton solutions). Numerical work indicates that, starting from well localized initial conditions, for long times the solution converges to one of the one-parameter family of solitons. However there is always some "background radiation", i.e. some part of the solution which escapes to infinity and carries away energy and momentum. This is the central mathematical difficulty: there is no simple algorithm to predict the asymptotic velocity of the soliton on the basis of the initial data. Nevertheless, no matter how one starts, the solution seems to settle for long times to some definite soliton solution. One outstanding achievement of Alexander Komech is to prove such a behavior for one-dimensional relativistic nonlinear wave equations with a suitable choice of the nonlinear force term. A physically important variant are (classical) charges coupled to the Maxwell field. The nonlinearity now resides in the particle-field interaction. A soliton consists of a charge travelling at constant velocity and dressed by the nearby Maxwell field. An outstanding theorem of Alexander Komech (and coworkers) proves the long time asymptotics of the motion of a single charge and the scattered part of the field. One future project is to extend such results to the case of several charges. Of course, now the dynamical behavior is much richer and depends on the relative sign of the charges. In particular, charges can form bound states and the soliton is then such a "molecule" travelling at constant velocity and dressed by the Maxwell field. The goal of the project is to establish the existence and to characterize the stability of such asymptotic soliton solutions.

Publikationen (Auswahl)

2017A. Komech, A. E. Merzon: Asymptotic completeness of scattering in the nonlinear Lamb system for nonzero mass. In: Russ. J. Math. Phys., 2017, 336-346
2017A.Komech, E. Kopylova, H. Spohn,: On global attractors and radiation damping for nonrelativistic particle coupled to scalar field. In: Algebra and Analysis, 2017, 34-58
2017V. Imaykin, A. Komech, H. Spohn. On invariants for the Poincar\'e equations and applications. In: J. Math. Phys., 2017, 012901-1-012901-13
2017 A. Komech, E. Kopylova: On stability of ground states for finite crystals in the Schr\"odinger--Poisson model. In: J. Math. Phys., 58, 2017, 031902-1 -- 031902-18
2016Alexander Komech, Elena Kopylova: Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle . In: Russian Journal of Mathematical Physics, 2016, 93-100
2016Alexander Komech: Attractors of nonlinear Hamilton PDEs. In: Discrete and Continuous Dynamical Systems A, 2016, 6201-6256
2016Alexander Komech: On crystal ground state in the Schr\'odinger-Poisson model with point ions. In: Mathematical Notes, 2016, 886-894
2016Alexander Komech, Elena Kopylova: On the linear stability of crystals for the Schroedinger-Poisson model. In: Journal of Statistical Physics, 2016, 246-273
2015Prof. Dr. Alexander Komech, Dr. V. Imaykin, Prof. Dr. H. Spohn: On Lagrangian theory for rotating charge coupled to the Maxwell field. In: Physics Letters A, 2015, 5-10
2015Prof. Dr. Alexander Komech: On dynamical justification of quantum scattering cross section. In: J. Math. Anal. Appl., 2015, 583-602
2015Prof. Dr. Alexander Komech: On the Hartree-Fock dynamics in wave-matrix picture. In: Dynamics of PDE, 2015, 157-176
2015Prof. Dr. Alexander Komech, Prof. Dr. A.E. Merzon, Dr. J.E. De la Paz Mendez, Dr. T. J. Villalba Vega: On the Keller-Blank solution to the scattering problem of pulses by wedges. In: Mathematical Methods in Applied Sciences, 2015, 2035-2040
2015Prof. Dr. Alexander Komech: On the crystal ground state in the Schroedinger-Poisson model. In: SIAM J. Math. Anal. , 2015, 1001-1021
2015Prof. Dr. Alexander Komech, Dr. E.A. Kopylova: On the eigenfunction expansion for Hamilton operators. In: J. Spectral Theory, 2015, 331-361
2015Prof. Dr. Alexander Komech, Dr. A.E. Merzon, Dr. J.E. De la Paz Mendez: On uniqueness and stability of Sobolev's solution in scattering by wedges. In: Zeitschrift fuer angewandte Mathematik und Physik , 2015, 2485-2498
2015Prof. Dr. Alexander Komech, Dr. E.A. Kopylova: Weighted energy decay for magnetic Klein-Gordon equation. In: J. Applicable Analysis, 2015, 219-233
2014Alexander Komech, E. Kopylova: On eigenfunction expansion of solutions to the Hamilton equations. In: J. Stat. Phys., 2014, 503-521
2013Alexander Komech, Andrew Komech: A variant of the Titchmarsh convolution theorem for distributions on the circle. In: Funct. Anal. Appl., 2013, 21-26
2013Alexander Komech, Anatoly Merzon: On asymptotic completeness of scattering in the nonlinear Lamb system II. In: J. Math. Physics, 2013, 012702-012710
2013Alexander Komech, Elena Kopylova, Sergey Kopylov: On nonlinear wave equations with parabolic potentials. In: Journal of Spectral Theory, 2013, 1-19
2013Alexander Komech, Elena Kopylova, Yurii Karlovich, Anatoly Merzon: On the spreading rate of the soliton perturbation for relativistic nonlinear wave equation. In: Comm. Math. Analysis, 2013, 95-102
2013Alexander Komech: Quantum Mechanics: Genesis and Achievements. Springer, 2013
2013Alexander Komech, Elena Kopylova: Weighted decay for magnetic Schroedinger equation. In: J. Funct. Analysis , 2013, 735-751
2012Alexander Komech, Elena Kopylova: Dispersion Decay and Scattering Theory. John Wiley & Sons, 2012
2012Valery Imaykin, Alexander Komech, Boris Vainberg: Scattering of solitons for coupled wave-particle equations. In: J. Math. Anal. Appl. , 2012, 713-740
2011Alexander Komech, Elena Kopylova: On asymptotic stability of kink for relativistic Ginzburg-Landau equations. In: Arch. Ration. Mech. Anal., 2011, 213-245
2011Alexander Komech, Elena Kopylova: On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation. In: Comm. Math. Physics, 2011, 225-252
2011Alexander Komech, Elena Kopylova: On convergence to equilibrium distribution for Dirac equation. In: Markov Processes and Related Fields , 2011, 523-540
2011Alexander Komech, Andrew Komech: On global attraction to quantum stationary states. Dirac equation with mean field interaction. In: Commun. Math. Anal., 2011, 131-136
2011Alexander Komech, Andrew Komech: On global attraction to quantum stationary states. Dirac equation with mean field interaction. In: Commun. Math. Anal. , 2011, 131-136
2011Alexander Komech, Elena Kopylova, Herbert Spohn: Scattering of solitons for Dirac equation coupled to a particle. In: J. Math. Anal. Appl., 2011, 265-290
2011Andrew Comech, Alexander Komech, Well-posedness and the energy and charge conservation for nonlinear wave equations in discrete space-time. In: Russ. J. Math. Phys., 2011, 410-419
2010Alexander Komech, Elena Kopylova: Dispersive long-time decay for Klein-Gordon equation. In: Modern Problems of Analysis and Mathematical Education. Proceedings of International Conference dedicated to 105-anniversary of S.M.Nikolskii, 18-20 May 2010, 2010, 89-89
2010Alexander Komech, Andrew Komech: Global attraction to solitary waves for nonlinear Dirac equation with mean field interaction. In: SIAM J. Math. Analysis, 2010, 2944-2964
2010Alexander Komech, Elena Kopylova: Long time decay for 2D Klein-Gordon equation. In: J. Functional Analysis, 2010, 477-502
2010Alexander Komech, Andrew Komech: On global attraction to solitary waves for the Klein-Gordon field coupled to several nonlinear oscillators. In: J. des Mathematiques Pures et App., 2010, 91-111
2010Alexander Komech: On global attractors of nonlinear hyperbolic PDEs. In: Proceedings of International Conference on Differential Equations and Dynamical Systems, July 02-07, 2010, Suzdal, Vladimir region, Russia, 2010, 216-216
2010Alexander Komech, Elena Kopylova: Weighted energy decay for 1D Klein-Gordon equation. In: Comm. PDE, 2010, 353-374
2010Alexander Komech, Elena Kopylova: Weighted energy decay for 3D Klein-Gordon equation. In: J. Differential Equations, 2010, 501-520
2009Alexander Komech, Andrew Komech: Global attraction to quantum stationary states. In: XVI International Congress on Mathematical Physics, Prague, August 3-8, 2009, 2009, 30-30
2009Alexander Komech, Andrew Komech: Global attraction to solitary waves for Klein-Gordon equation with mean field interaction. In: Annales de l'IHP-ANL, 2009, 855-868
2009Alexander Komech, Anatoli Merzon: On asymptotic completeness of scattering in the nonlinear Lamb system. In: J. Math. Physics, 2009, 023514-1-023514-10
2009Alexander Komech, Elena Kopylova: On asymptotic stability of kink for relativistic Ginzburg-Landau equation. In: XVI International Congress on Mathematical Physics, Prague, August 3-8, 2009, 2009, 30-30
2009Alexander Komech: On global attraction to quantum stationary states. In: The International Conference 'Modern Problems of Mathematics, Mechanics and their Applications' 30.03-2.04.2009, Moscow State University, Moscow, 2009, 2009, 250-250
2009Alexander Komech, Andrew Komech: Principles of Partial Differential Equations. Springer, 2009
2009Alexander Komech, Anatoli Merzon: Scattering in the nonlinear Lamb system. In: Physics Letters A, 2009, 1005-1010
2009Valery Imaykin, Alexander Komech, Herbert Spohn, Boris Vainberg: Soliton-type asymptotics for wave-particle systems. In: XVI International Congress on Mathematical Physics, Prague, August 3-8, 2009, 2009, 58-58
2008Alexander Komech, Andrew Komech: Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation. In: Symmetry Integrability Geom. Methods Appl., 2008, 1-23
2008Alexander Komech, Andrew Komech: Global attraction to solitary waves in models based on the Klein-Gordon equation. In: Abstracts of 5th European Congress of Mathematics, Amsterdam 14-18 July, Amsterdam, 2008, 2008, 58-58
2008Vladimir Buslaev, Alexander Komech, Elena Kopylova, David Stuart: On asymptotic stability of solitary waves in nonlinear Schroedinger equation. In: Comm. Partial Diff. Eqns, 2008, 669-705
2008Alexander Komech, Elena Kopylova, David Stuart: On asymptotic stability of solitary waves for Schroedinger equation coupled to nonlinear oscillator. In: Oberwolfach Rep., 2008, 386-389
2008Alexander Komech, Elena Kopylova, Boris Vainberg: On dispersive properties of discrete 2D Schr\'odinger and Klein-Gordon equations. In: J. Funct. Analysis, 2008, 2227-2254
2008Alexander Komech: On global attraction to solitary waves for the Klein-Gordon equation coupled to nonlinear oscillator. In: International Conference on Differential Equations and Dynamical Systems, Suzdal, 25.06-2.07 2008, Vladimir 2008, 2008, 308-309
2008Alexander Komech, Anatoli Merzon, Taneco Hernandes: On scattering states in nonlinear Lamb systems. In: Abstracts of 5th European Congress of Mathematics, Amsterdam 14-18 July, Amsterdam, 2008, 2008, 58-58
2007Alexander Komech, Andrew Komech: Book of Practical PDEs. Max-Planck Institute for Mathematics in the Sciences, 2007
2007Alexander Komech, Andrew Komech: Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field. In: Arch. Rat. Mech. Anal., 2007, 105-142
2007Alexander Komech, Andrew Komech: Global well-posedness for the Schrodinger equation coupled to a nonlinear oscillator,. In: Russ. J. Math. Phys, 2007, 164-173
2007Alexander Komech: Lectures on elliptic partial differential equations (Pseudodifferential operator approach). Max-Planck Institute for Mathematics in the Sciences, 2007
2007Alexander Komech, Anatoly Merzon: On relation between the Cauchy data in scattering by wedge. In: Russ. J. Math. Phys., 2007, 279-303
2007Alexander Komech, Anatoli Merzon: On relation between the Cauchy data in the scattering problems on a wedge. In: 22-nd Conference of I.G. Petrovskii ``Differential equations and related topics~, Moscow 21.05--26.05.2007. Book of abstracts, 2007, 145-146
2007Valery Imaykin, Alexander Komech, Boris Vainberg: On scattering of solitons for wave equation coupled to a particle. In: Proceedings of the international conference 'Probability and Mathematical Physics' (Montreal, Canada, 2006), Centre de recherches mathematiques, Universit\'e de Montreal, CMR Proceedings and Lecture Notes , 2007, 249-273