MODEL EQUATIONS FOR LONG WAVES IN NONLINEAR DISPERSIVE SYSTEMS
MODEL EQUATIONS FOR LONG WAVES IN NONLINEAR DISPERSIVE SYSTEMS
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DOI：
10.1098/rsta.1972.0032
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发表时间：
1972-01-01
影响因子：
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通讯作者：
MAHONY, JJ
中科院分区：
文献类型：
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作者：
BENJAMIN, TB;BONA, JL;MAHONY, JJ
研究了有关系统中长波单向传播的数学模型的几个主题，这些系统表现出特定但常见的非线性和色散效应。大多数新材料都涉及方程的初值问题，其解 u(x,t) 被认为是一类为 ࢤ∞ <x< ∞,t≥0 定义的实数非周期函数。作为特定物理系统中小但有限振幅的中等长波的近似推导，该方程与 Korteweg-de Vries 方程具有相同的形式证明，(a) 可以通过多种方式与该方程进行比较。有人认为，(a) 在重要方面是更可取的模型，消除了 (b) 的某些有问题的方面，并且通常具有更有利的数学属性。本文分为两个部分，分别强调描述性和严格数学。第 2 部分一般性地讨论了方程 (a) 和 (b) 的起源和直接性质，并回顾了 (b) 的比较缺点。在本文的其余部分（§§ 3,4）中 - 可以独立阅读前面的讨论 - 发展了（a）的精确理论。在第 3 节中，证明了经典解的存在性：并根据我们的主要结果、定理 1，提出了一些扩展和补充。在第 4 节中，解被证明是唯一的，持续依赖于它们的初始值，并且持续依赖于添加到 (a) 右侧的强制函数。因此，初值问题被证实在哈达玛意义上是经典的。在附录 1 中，考虑了 (a) 的推广，其中宽类内的色散效应由抽象伪微分算子表示。这种方程的物理起源以§2的风格进行了解释，给出了两个源自确定物理问题的例子，并概述了存在理论。附录 2 中确立了第 3 节中使用的技术事实。
Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation, whose solutionu(x,t) is considered in a class of real nonperiodic functions defined for ࢤ∞ <x< ∞,t≥0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equationwith which (a) is to be compared in various ways. It is contended that (a) is in important respects the preferable model, obviating certain problematical aspects of (b) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics In §2 the origins and immediate properties of equations (a) and (b) are discussed in general terms, and the comparative shortcomings of (b) are reviewed. In the remainder of the paper (§§ 3,4) - which can be read independently Preceding discussion _ an exact theory of (a) is developed. In § 3 the existence of classical solutions is proved: and following our main result, theorem 1, several extensions and sidelights are presented. In § 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of (a). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of (a) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of § 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in § 3 is established.