# American Institute of Mathematical Sciences

May  2010, 27(2): 487-532. doi: 10.3934/dcds.2010.27.487

## Inelastic interaction of nearly equal solitons for the BBM equation

 1 Université de Versailles Saint-Quentin-en-Yvelines and IUF, Laboratoire de mathématiques de Versailles, UMR CNRS 8100, 45, av. des Etats-Unis, 78035 Versailles cedex, France 2 Université de Cergy-Pontoise and IHES, Laboratoire de mathématiques, UMR CNRS 8088, 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex

Received  October 2009 Revised  February 2010 Published  February 2010

This paper is concerned with the interaction of two solitons of nearly equal speeds for the (BBM) equation. This work is an extension of [31] addressing the same question for the quartic (gKdV) equation. We consider the (BBM) equation, for $\lambda \in [0,1)$,

$(1-\lambda \partial_x^2) \partial_t u + \partial_x (\partial_x^2 u - u + u^2) =0.$     (BBM)

Solitons are solutions of the form $R_{\mu,x_0}(t,x)=Q_{\mu}(x-\mu t -x_0),$ for $\mu > -1$, $x_0\in \RR$.
For $\mu_0>0$ small, let $U(t,x)$ be the unique solution of (BBM) such that

$\lim_{t\to -\infty} \||U(t) - Q_{-\mu_0}(.+\mu_0 t) - Q_{\mu_0}(.-\mu_0 t )\||_{H^1} = 0.$

First, we prove that $U(t)$ remains close to the sum of two solitons, for all time $t\in \RR$,

$U(t,x) = Q_{\mu_1(t)}(x-y_1(t)) + Q_{\mu_2(t)}(x-y_2(t)) +$ε$\ (t) \quad where \quad |\|\varepsilon(t)\| | \leq \mu_0^{2^-},$

with $y_1(t)-y_2(t)> 2 |\ln \mu_0| + O(1),$ which means that at the main order the situation is similar to the integrable KdV case. However, we show that the collision is perfectly elastic if and only if $\lambda=0$ (i.e. only in the integrable case).

Citation: Yvan Martel, Frank Merle. Inelastic interaction of nearly equal solitons for the BBM equation. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 487-532. doi: 10.3934/dcds.2010.27.487
 [1] Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905 [2] Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824 [3] Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583 [4] C. H. Arthur Cheng, John M. Hong, Ying-Chieh Lin, Jiahong Wu, Juan-Ming Yuan. Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 763-779. doi: 10.3934/dcdsb.2016.21.763 [5] Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171 [6] Wenxia Chen, Ping Yang, Weiwei Gao, Lixin Tian. The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium. Communications on Pure & Applied Analysis, 2018, 17 (3) : 823-848. doi: 10.3934/cpaa.2018042 [7] Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851 [8] Qiangheng Zhang. Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021293 [9] Hui Yin, Huijiang Zhao. Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space. Kinetic & Related Models, 2009, 2 (3) : 521-550. doi: 10.3934/krm.2009.2.521 [10] Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262 [11] Yangrong Li, Renhai Wang, Jinyan Yin. Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2569-2586. doi: 10.3934/dcdsb.2017092 [12] Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146 [13] Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695 [14] G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 [15] H. A. Erbay, S. Erbay, A. Erkip. The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6101-6116. doi: 10.3934/dcds.2016066 [16] Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941 [17] Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27 [18] Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055 [19] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [20] Tran Ngoc Thach, Nguyen Huy Tuan, Donal O'Regan. Regularized solution for a biharmonic equation with discrete data. Evolution Equations & Control Theory, 2020, 9 (2) : 341-358. doi: 10.3934/eect.2020008

2020 Impact Factor: 1.392