Stability of Two Soliton Collision for Nonintegrable gKdV Equations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Stability of Two Soliton Collision for Nonintegrable gKdV Equations

Authors:

Institution:

(1) Université de Versailles Saint-Quentin-en-Yvelines, Mathématiques, UMR 8100, 45, av. des Etats-Unis, 78035 Versailles cedex, France;(2) Université de Cergy-Pontoise, IHES and CNRS, Mathématiques, 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

Abstract:

We continue our study of the collision of two solitons for the subcritical generalized KdV equations

$\label{kdvabs}\partial_{t}u\,{+}\,\partial_{x}(\partial_{x}^{2}u\,{+}\,f(u)) = 0.\quad \quad (0.1)$

Solitons are solutions of the type ${u(t, x) = Q_{c_{0}}(x - x_{0} - c_{0}t)}$ where c ₀ > 0. In 21], mainly devoted to the case f (u) = u ⁴, we have introduced a new framework to understand the collision of two solitons ${Q_{c_{1}}}$ , ${Q_{c_{2}}}$ for (0.1) in the case ${c_{2} \ll c_{1}}$ (or equivalently, ${\|Q_{c_{2}}\|_{H^{1}}\ll \|Q_{c_{1}}\|_{H^{1}}}$ ). In this paper, we consider the case of a general nonlinearity f (u) for which ${Q_{c_{1}}}$ , ${Q_{c_{2}}}$ are nonlinearly stable. In particular, since f is general and c ₁ can be large, the results are not perturbations of the ones for the power case in 21]. First, we prove that the two solitons survive the collision up to a shift in their trajectory and up to a small perturbation term whose size is explicitly controlled from above: after the collision, ${u(t) \sim Q_{c_{1}^{+}}+Q_{c_{2}^{+}}}$ , where ${c_{j}^{+}}$ is close to c _j (j = 1, 2). Then, we exhibit new exceptional solutions similar to multi-soliton solutions: for all ${c_{1}, c_{2} > 0, c_{2} \ll c_{1}}$ , there exists a solution ${\varphi(t)}$ such that

$\begin{aligned} \varphi(t,x)\,&=\,Q_{c_{1}}(x-\rho_{1}(t)) + Q_{c_{2}}(x-\rho_{2}(t)) + \eta(t,x), {\rm for}\, t\ll -1,\\ \varphi(t,x)\,&=\,Q_{c_{1}}(x-\rho_{1}(t)) + Q_{c_{2}}(x-\rho_{2}(t)) + \eta(t,x), {\rm for}\, t\gg 1, \end{aligned}$

where ${\rho_{j}(t) \to c_{j}}$ (j = 1, 2) and ${\eta(t)}$ converges to 0 in a neighborhood of the solitons as ${t \to \pm \infty}$ . The analysis is split in two distinct parts. For the interaction region, we extend the algebraic tools developed in 21] for the power case, by expanding f (u) as a sum of powers plus a perturbation term. To study the solutions in large time, we rely on previous tools on asymptotic stability in 17,22] and 18], refined in 19,20]. This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN).

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