# American Institute of Mathematical Sciences

May  2008, 20(2): 177-218. doi: 10.3934/dcds.2008.20.177

## Refined asymptotics around solitons for gKdV equations

 1 Université de Versailles Saint-Quentin-en-Yvelines, Mathématiques, 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, France

Received  May 2007 Revised  October 2007 Published  November 2007

We consider the generalized Korteweg-de Vries equation

$\partial_t u + \partial_x (\partial_x^2 u + f(u))=0, (t,x)\in \mathbb{R}\times \mathbb{R}$(1)

with $C^3$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution of (1) in the energy space $H^1$ of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton.
In [11], [13], it was proved that for $f(u)=u^p$, $p=2,3,4$, the family of solitons is asymptotically stable in some local sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$, then $u(t,.+\rho(t))$ locally converges in the energy space to some $Q_{c_+}$ as $t\to +\infty$, for some $c^+$~$c$ and some function $\rho(t)$ such that $\rho'(t)$~$c^+$. Then, in [9] and [14], these results were extended with shorter proofs under general assumptions on $f$.
The first objective of this paper is to give more information about the function $\rho(t)$. In the case $f(u)=u^p$, $p=2,3,4$ and under the additional assumption $x_+ u\in L^2(\mathbb{R})$, we prove that the function $\rho(t)-c^+ t$ has a finite limit as $t\to +\infty$.
Second, we prove stability and asymptotic stability results for two solitons for a general nonlinearity $f(u)$, in the case where the ratio of the speeds of the two solitons is small.

Citation: Yvan Martel, Frank Merle. Refined asymptotics around solitons for gKdV equations. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 177-218. doi: 10.3934/dcds.2008.20.177
 [1] Rowan Killip, Soonsik Kwon, Shuanglin Shao, Monica Visan. On the mass-critical generalized KdV equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 191-221. doi: 10.3934/dcds.2012.32.191 [2] Annie Millet, Svetlana Roudenko. Generalized KdV equation subject to a stochastic perturbation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1177-1198. doi: 10.3934/dcdsb.2018147 [3] S. Raynor, G. Staffilani. Low regularity stability of solitons for the KDV equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 277-296. doi: 10.3934/cpaa.2003.2.277 [4] Esha Chatterjee, Sk. Sarif Hassan. On the asymptotic character of a generalized rational difference equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1707-1718. doi: 10.3934/dcds.2018070 [5] Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401 [6] María-Santos Bruzón, Elena Recio, Tamara-María Garrido, Rafael de la Rosa. Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020222 [7] Mamoru Okamoto. Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity. Evolution Equations & Control Theory, 2019, 8 (3) : 567-601. doi: 10.3934/eect.2019027 [8] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 [9] Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063 [10] Denis Matignon, Christophe Prieur. Asymptotic stability of Webster-Lokshin equation. Mathematical Control & Related Fields, 2014, 4 (4) : 481-500. doi: 10.3934/mcrf.2014.4.481 [11] Juan-Ming Yuan, Jiahong Wu. The complex KdV equation with or without dissipation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 489-512. doi: 10.3934/dcdsb.2005.5.489 [12] Jungho Park. Bifurcation and stability of the generalized complex Ginzburg--Landau equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1237-1253. doi: 10.3934/cpaa.2008.7.1237 [13] Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971 [14] Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583 [15] 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 [16] Shengfu Deng. Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3671-3716. doi: 10.3934/dcds.2019150 [17] Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035 [18] María Santos Bruzón, Tamara María Garrido. Symmetries and conservation laws of a KdV6 equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 631-641. doi: 10.3934/dcdss.2018038 [19] Masakazu Kato, Yu-Zhu Wang, Shuichi Kawashima. Asymptotic behavior of solutions to the generalized cubic double dispersion equation in one space dimension. Kinetic & Related Models, 2013, 6 (4) : 969-987. doi: 10.3934/krm.2013.6.969 [20] 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

2018 Impact Factor: 1.143