American Institute of Mathematical Sciences

Advanced
January & February  2004, 10(1&2): 387-396. doi: 10.3934/dcds.2004.10.387

Scattering theory for a particle coupled to a scalar field

Valery Imaikin 1, , Alexander Komech 2, and  Herbert Spohn 3,

1. 

Institute of Mathematics, University of Vienna, Boltzmanngasse 9, 1090 Vienna, Austria
2. 

Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russian Federation
3. 

Zentrum Mathematik, TU München, 80290 München, Germany

Received  February 2002 Revised  April 2003 Published  October 2003

We establish soliton-like asymptotics for finite energy solutions to classical particle coupled to a scalar wave field. Any solution that goes to infinity as $t\to\infty$ converges to a sum of traveling wave and of outgoing free wave. The convergence holds in global energy norm. The proof uses a non-autonomous integral inequality method.
Keywords: soliton-like asymptotics, solitons, scattering states., scattering solutions, weak interaction, classical particle, particle density, Scalar wave field, smooth coupling.
Mathematics Subject Classification: 37K4.
Citation: 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
[1]

Anton Trushechkin. Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres. Kinetic & Related Models, 2014, 7 (4) : 755-778. doi: 10.3934/krm.2014.7.755
[2]

Jennifer Weissen, Simone Göttlich, Dieter Armbruster. Density dependent diffusion models for the interaction of particle ensembles with boundaries. Kinetic & Related Models, 2021, 14 (4) : 681-704. doi: 10.3934/krm.2021019
[3]

Conrad Bertrand Tabi, Alidou Mohamadou, Timoleon Crepin Kofane. Soliton-like excitation in a nonlinear model of DNA dynamics with viscosity. Mathematical Biosciences & Engineering, 2008, 5 (1) : 205-216. doi: 10.3934/mbe.2008.5.205
[4]

Nicolo' Catapano. The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit. Kinetic & Related Models, 2018, 11 (3) : 647-695. doi: 10.3934/krm.2018027
[5]

Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217
[6]

Eliot Fried. New insights into the classical mechanics of particle systems. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469
[7]

Shijin Ding, Bingyuan Huang, Xiaoyan Hou. Strong solutions to a fluid-particle interaction model with magnetic field in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021042
[8]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic & Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395
[9]

Yuri Kozitsky, Krzysztof Pilorz. Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 725-752. doi: 10.3934/dcds.2020059
[10]

Charles Bordenave, David R. McDonald, Alexandre Proutière. A particle system in interaction with a rapidly varying environment: Mean field limits and applications. Networks & Heterogeneous Media, 2010, 5 (1) : 31-62. doi: 10.3934/nhm.2010.5.31
[11]

Lingbing He. On the global smooth solution to 2-D fluid/particle system. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 237-263. doi: 10.3934/dcds.2010.27.237
[12]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[13]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681
[14]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016
[15]

Pedro M. Jordan. Finite-amplitude acoustics under the classical theory of particle-laden flows. Evolution Equations & Control Theory, 2019, 8 (1) : 101-116. doi: 10.3934/eect.2019006
[16]

Yipeng Chen, Yicheng Liu, Xiao Wang. Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021023
[17]

Justin Holmer, Maciej Zworski. Slow soliton interaction with delta impurities. Journal of Modern Dynamics, 2007, 1 (4) : 689-718. doi: 10.3934/jmd.2007.1.689
[18]

Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923
[19]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences