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Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations

Jacek Jendrej

Abstract

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 $κ= κ(v) > 0$.

Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations

Abstract

We consider the generalized Korteweg-de Vries equation , where is an odd function of class . Under some assumptions on , this equation admits \emph{solitary waves}, that is solutions of the form , for in some range . We study pure two-solitons in the case of the same limit speed, in other words global solutions such that \begin{equation} \label{eq:abstract} \tag{} \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 with and . We describe the~dynamical behavior of any solution satisfying \eqref{eq:abstract} under the assumption that is linearly unstable (which corresponds to 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 between the solitons equals for some .

Paper Structure

This paper contains 20 sections, 36 theorems, 312 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Formal prediction of multi-soliton dynamics
  4. Statements of the results
  5. Main elements of the proof
  6. Acknowledgments
  7. Notation
  8. Analysis of the stationary equation
  9. Existence and asymptotic behavior of $Q_v$ and $\widetilde{Q}_v$
  10. Spectral properties of $L_v$
  11. Coercivity near a two-soliton
  12. Modulation near a two-soliton
  13. Definition of the position parameters
  14. Computation of the interaction force
  15. Stable and unstable directions
  16. ...and 5 more sections

Key Result

Proposition 1.1

Let $\sigma_1, \sigma_2 \in \{{-}1, 1\}$ and let $v^\infty \in (0, v_*) \setminus V_\mathrm{crit}$. If then the equation eq:constrained has a solution $(\boldsymbol{x}(t), \boldsymbol{v}(t)) = (x_1(t), x_2(t), v_1(t), v_2(t))$ such that for some $\beta > 0$, where $\kappa = \kappa(v^\infty) > 0$ is an explicit constant. In particular, If $(\boldsymbol{x}^\sharp, \boldsymbol{v}^\sharp)$ is any s

Theorems & Definitions (84)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 2.1
  • Remark 2.2
  • proof
  • ...and 74 more