Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations
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arXiv
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| Natura: | Preprint |
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2018
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| _version_ | 1866910378044162048 |
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| author | Jendrej, Jacek |
| author_facet | Jendrej, Jacek |
| contents | We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is solutions of the form $u(t, x) = Q_v(x - vt - x_0)$, for $v$ in some range $(0, v_*)$. We study pure two-solitons in the case of the same limit speed, in other words global solutions $u(t)$ such that \begin{equation} \label{eq:abstract} \tag{$\ast$} \lim_{t\to\infty}\|u(t) - (Q_v(\cdot - x_1(t)) \pm Q_v(\cdot - x_2(t)))\|_{H^1} = 0, \qquad \text{with}\quad\lim_{t \to \infty}x_2(t) - x_1(t) = \infty. \end{equation} Existence of such solutions is known for $f(u) = |u|^{p-1}u$ with $p \in \mathbb{Z} \setminus \{5\}$ and $p > 2$. We describe the~dynamical behavior of any solution satisfying \eqref{eq:abstract} under the assumption that $Q_v$ is linearly unstable (which corresponds to $p > 5$ for power nonlinearities). We prove that in this case the sign in \eqref{eq:abstract} is necessarily "$+$", which corresponds to an attractive interaction. We also prove that the~distance $x_2(t) - x_1(t)$ between the solitons equals $\frac{2}{\sqrt v}\log(κt) + o(1)$ for some $κ= κ(v) > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_06294 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations Jendrej, Jacek Analysis of PDEs We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is solutions of the form $u(t, x) = Q_v(x - vt - x_0)$, for $v$ in some range $(0, v_*)$. We study pure two-solitons in the case of the same limit speed, in other words global solutions $u(t)$ such that \begin{equation} \label{eq:abstract} \tag{$\ast$} \lim_{t\to\infty}\|u(t) - (Q_v(\cdot - x_1(t)) \pm Q_v(\cdot - x_2(t)))\|_{H^1} = 0, \qquad \text{with}\quad\lim_{t \to \infty}x_2(t) - x_1(t) = \infty. \end{equation} Existence of such solutions is known for $f(u) = |u|^{p-1}u$ with $p \in \mathbb{Z} \setminus \{5\}$ and $p > 2$. We describe the~dynamical behavior of any solution satisfying \eqref{eq:abstract} under the assumption that $Q_v$ is linearly unstable (which corresponds to $p > 5$ for power nonlinearities). We prove that in this case the sign in \eqref{eq:abstract} is necessarily "$+$", which corresponds to an attractive interaction. We also prove that the~distance $x_2(t) - x_1(t)$ between the solitons equals $\frac{2}{\sqrt v}\log(κt) + o(1)$ for some $κ= κ(v) > 0$. |
| title | Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1802.06294 |