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**Refined asymptotics around solitons for gKdV equations.**
*(English)*
Zbl 1137.35062

Summary: We consider the generalized Korteweg-de Vries equation

\[ \partial_tu+ \partial_x(\partial_x^2u+ f(u))=0, \quad (t,x)\in\mathbb R\times\mathbb R,\tag{1} \]

with \(C^3\) nonlinearity \(f\). Under some 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 [the authors, J. Am. Math. Soc. 15, No. 3, 617–664 (2002; Zbl 0996.35064), Nonlinearity 18, No. 1, 55–80 (2005; Zbl 1064.35171)], 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 \(c^+\sim c\) and some function \(\rho(t)\) such that \(\rho'(t)\sim c^+\). Then, in [the the author, SIAM J. Math. Anal. 38, No. 3, 759–781 (2006; Zbl 1126.35055) and the authors, Math. Ann. 341, No. 2, 391–427 (2008; Zbl 1153.35068)], 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.

\[ \partial_tu+ \partial_x(\partial_x^2u+ f(u))=0, \quad (t,x)\in\mathbb R\times\mathbb R,\tag{1} \]

with \(C^3\) nonlinearity \(f\). Under some 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 [the authors, J. Am. Math. Soc. 15, No. 3, 617–664 (2002; Zbl 0996.35064), Nonlinearity 18, No. 1, 55–80 (2005; Zbl 1064.35171)], 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 \(c^+\sim c\) and some function \(\rho(t)\) such that \(\rho'(t)\sim c^+\). Then, in [the the author, SIAM J. Math. Anal. 38, No. 3, 759–781 (2006; Zbl 1126.35055) and the authors, Math. Ann. 341, No. 2, 391–427 (2008; Zbl 1153.35068)], 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.