# American Institute of Mathematical Sciences

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

## Scattering theory for a particle coupled to a scalar field

 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.
Citation: Valery Imaikin, Alexander Komech, Herbert Spohn. Scattering theory for a particle coupled to a scalar field. Discrete and 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 and 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 and 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] Yuan Li, Shou-Fu Tian. Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation. Communications on Pure and Applied Analysis, 2022, 21 (1) : 293-313. doi: 10.3934/cpaa.2021178 [5] Nicolo' Catapano. The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit. Kinetic and Related Models, 2018, 11 (3) : 647-695. doi: 10.3934/krm.2018027 [6] Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217 [7] Eliot Fried. New insights into the classical mechanics of particle systems. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469 [8] Shijin Ding, Bingyuan Huang, Xiaoyan Hou. Strong solutions to a fluid-particle interaction model with magnetic field in $\mathbb{R}^2$. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 277-300. doi: 10.3934/dcdsb.2021042 [9] Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic and Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395 [10] Yuri Kozitsky, Krzysztof Pilorz. Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 725-752. doi: 10.3934/dcds.2020059 [11] Charles Bordenave, David R. McDonald, Alexandre Proutière. A particle system in interaction with a rapidly varying environment: Mean field limits and applications. Networks and Heterogeneous Media, 2010, 5 (1) : 31-62. doi: 10.3934/nhm.2010.5.31 [12] Lingbing He. On the global smooth solution to 2-D fluid/particle system. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 237-263. doi: 10.3934/dcds.2010.27.237 [13] Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061 [14] Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems and Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681 [15] Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016 [16] Pedro M. Jordan. Finite-amplitude acoustics under the classical theory of particle-laden flows. Evolution Equations and Control Theory, 2019, 8 (1) : 101-116. doi: 10.3934/eect.2019006 [17] Yipeng Chen, Yicheng Liu, Xiao Wang. Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance. Evolution Equations and Control Theory, 2022, 11 (3) : 729-748. doi: 10.3934/eect.2021023 [18] 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 [19] Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923 [20] Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems and Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004

2020 Impact Factor: 1.392