arXiv 0812.15792008-12nlin.SInlin.PS

半直线上Korteweg-de Vries方程的显式孤子渐近性

Explicit soliton asymptotics for the Korteweg-de Vries equation on the half-line

A. S. Fokas, J. Lenells

备注：: Corrected typos.

摘要：

线上可积偏微分方程可以用所谓的逆散射变换（IST）方法进行分析。IST的一个特别强大的方面是它能够预测解决方案的大$t$行为。也就是说，从初始数据$u(x,0)$开始，IST意味着解$u(x,t)$渐近线到孤子的集合，如$t \to \infty$、$x/t = O(1)$ ;此外，这些孤子的形状和速度可以仅使用{ \it linear}运算从$u(x,0)$计算。在这一领域的最重要的发展之一一直是推广的IST从初始到初始边界值（IBV）的问题制定的半线。可以证明，$u(x,t)$又渐近线化为一组孤子，此时这些孤子的形状和速度既取决于$u(x,0)$，也取决于$x = 0$处给定的边界条件。IBV问题的一个主要的复杂性是孤子的形状和速度的计算涉及到{ \it非线性} Volterra积分方程的解。然而，对于某种类型的边界条件，称为线性化，这种复杂性可以绕过，相关的计算与在线问题的情况一样有效。在这里，在回顾了KdV的一般理论之后，我们分析了三种不同类型的线性化边界条件。对于这些情况，初始条件是：（a）在$t = 0$上的一个和两个孤子解的限制;（b）某些指数型轮廓;（c）盒形轮廓。对于每一种情况，通过明确计算的形状和速度的渐近孤子，我们阐明了边界的影响。

英文摘要：

Integrable PDEs on the line can be analyzed by the so-called Inverse Scattering Transform (IST) method. A particularly powerful aspect of the IST is its ability to predict the large $t$ behavior of the solution. Namely, starting with initial data $u(x,0)$, IST implies that the solution $u(x,t)$ asymptotes to a collection of solitons as $t \to \infty$, $x/t = O(1)$; moreover the shapes and speeds of these solitons can be computed from $u(x,0)$ using only {\it linear} operations. One of the most important developments in this area has been the generalization of the IST from initial to initial-boundary value (IBV) problems formulated on the half-line. It can be shown that again $u(x,t)$ asymptotes into a collection of solitons, where now the shapes and the speeds of these solitons depend both on $u(x,0)$ and on the given boundary conditions at $x = 0$. A major complication of IBV problems is that the computation of the shapes and speeds of the solitons involves the solution of a {\it nonlinear} Volterra integral equation. However, for a certain class of boundary conditions, called linearizable, this complication can be bypassed and the relevant computation is as effective as in the case of the problem on the line. Here, after reviewing the general theory for KdV, we analyze three different types of linearizable boundary conditions. For these cases, the initial conditions are: (a) restrictions of one and two soliton solutions at $t = 0$; (b) profiles of certain exponential type; (c) box-shaped profiles. For each of these cases, by computing explicitly the shapes and the speeds of the asymptotic solitons, we elucidate the influence of the boundary.

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英文摘要：

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