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 - A, 2010, 27 (2) : 487-532. doi: 10.3934/dcds.2010.27.487
[1]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[2]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033

[3]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[4]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[5]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388

[6]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312

[7]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[8]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[9]

Ludovick Gagnon, José M. Urquiza. Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation. Evolution Equations & Control Theory, 2021, 10 (1) : 129-153. doi: 10.3934/eect.2020054

[10]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[11]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[12]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[13]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[14]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025

[15]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[16]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[17]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[18]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[19]

Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020370

[20]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]