A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation
A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation
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发表时间：
2016-07
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通讯作者：
S. Sato;K. Oguma;T. Matsuo;B. Feng
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作者：
S. Sato;K. Oguma;T. Matsuo;B. Feng
Schaefer-Wayne(2004)引入了短脉冲方程来模拟超短光脉冲的传输。虽然它可以描述很大范围的解，但它的几个周期的超短脉冲解是传统的非线性薛定谔方程所不具备的，引起了人们的极大关注。在这样的区域，现有的数值方法需要非常精细的数值网格，因此计算成本很高。本文将速度图变换的思想与保结构数值方法相结合，建立了一种新的高效数值方法。所得到的格式是一种自适应移动网格格式，它不仅可以成功地捕获超短脉冲，而且可以成功地捕获奇异解，如回路孤子解。
The short pulse equation was introduced by Schaefer--Wayne (2004) for modeling the propagation of ultrashort optical pulses. While it can describe a wide range of solutions, its ultrashort pulse solutions with a few cycles, which the conventional nonlinear Schroedinger equation does not possess, have drawn much attention. In such a region, existing numerical methods turn out to require very fine numerical mesh, and accordingly are computationally expensive. In this paper, we establish a new efficient numerical method by combining the idea of the hodograph transformation and the structure-preserving numerical methods. The resulting scheme is a self-adaptive moving mesh scheme that can successfully capture not only the ultrashort pulses but also exotic solutions such as loop soliton solutions.