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Travelling Waves for the Brio System
Journal of Nonlinear Science ( IF 3 ) Pub Date : 2021-06-08 , DOI: 10.1007/s00332-021-09727-z
C.O.R. Sarrico

In the setting of a product of distributions, we define a concept of a solution for the Brio system \(u_{t}+\frac{1}{2}(u^{2}+v^{2})_{x}=0\), \(v_{t} +(uv-v)_{x}=0\), which extends the classical solution concept. New results about that product allow us to establish necessary and sufficient conditions for the propagation of distributional travelling waves. Within this framework, we prove that continuous travelling waves are necessarily constant functions. Thus, if we want to seek for travelling waves in the Brio system, we must seek them among distributions that are not continuous functions. Examples that include discontinuous functions, measures and distributions which are not measures are given explicitly. For the reader’s convenience and completeness, a survey of the main ideas and formulas needed for multiplying distributions is also provided.



中文翻译:

Brio 系统的行波

在分布乘积的设置中,我们定义了 Brio 系统的解的概念\(u_{t}+\frac{1}{2}(u^{2}+v^{2})_{ x}=0\) , \(v_{t} +(uv-v)_{x}=0\),它扩展了经典解决方案的概念。关于该产品的新结果使我们能够为分布式行波的传播建立充分必要条件。在这个框架内,我们证明了连续行波必然是常数函数。因此,如果我们想在 Brio 系统中寻找行波,我们必须在非连续函数的分布中寻找它们。明确给出了包括不连续函数、度量和不是度量的分布的示例。为了读者的方便和完整性,还提供了乘法分布所需的主要思想和公式的概览。

更新日期:2021-06-08
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