-
Previous Article
Mathematical theory of viscous fluids: Retrospective and future perspectives
- DCDS Home
- This Issue
-
Next Article
Mathematical strategies for filtering turbulent dynamical systems
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 |
$ (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).
| [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] |
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 |
| [9] |
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 |
| [10] |
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 |
| [11] |
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 |
| [12] |
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 |
| [13] |
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 |
| [14] |
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 |
| [15] |
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 |
| [16] |
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 |
| [17] |
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 |
| [18] |
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 |
| [19] |
Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 |
| [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
Tools
Metrics
Other articles
by authors
[Back to Top]