Description of two soliton collision for the quartic gKdV equation

• 成果类型: Article
• 署名作者: Martel, Yvan; Merle, Frank
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.2.2
• 发表日期: 2011
• 页码: 757-857
• 关键词: korteweg-devries equation; generalized kdv equations; de-vries equation; asymptotic stability; multisoliton solutions; waves; CONSTRUCTION; SCATTERING; EVOLUTION; BEHAVIOR

摘要

In this paper, we give the first description of the collision of two solitons for a nonintegrable equation in a special regime. We consider solutions of the quartic gKdV equation partial derivative(t)u + partial derivative(x)(partial derivative(2)(x)u + u(4)) = 0, which behave as t -> -infinity like u(t, x) = Q(c1) (x - c(1)t) + Q(c2) (x - c(2)t) + eta(t, x), where Q(c)(x - ct) is a soliton and parallel to eta(t)parallel to(H1) << parallel to Q(c2)parallel to(H1) << parallel to Q(c1)parallel to(H1). The global behavior of u(t) is given by the following stability result: for all t is an element of R, u(t, x) = Q(c1(t)) (x - y(1)(t)) + Q(c2(t)) (x - y(2)(t)) + eta(t, x), where parallel to eta(t)parallel to(H1) << parallel to Q(c2)parallel to(H1) and lim(t ->+infinity) c(1)(t) = c(1)(,)(+) lim(t ->+infinity) c(2)(t) = c(2)(+). In the case where u(t) is a pure 2-soliton solution as t -> -infinity (i.e. lim(t)->+infinity parallel to eta(t)parallel to(H1) > 0), we obtain c(1)(+) > c(1), c(2)(+) < c(2) and for the residual part, lim(t ->+infinity) parallel to eta(t)parallel to(H1) > 0. Therefore, in contrast with the integrable KdV equation (or mKdV equation), no global pure 2-soliton solution exists and the collision is inelastic. A different notion of global 2-soliton is then proposed.