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What physical mechanisms govern waves in non-conservative systems?

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Questions About Elastic Waves

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Abstract

Most mathematical models described in the previous chapters are conservative like their prime example, the classical wave equation. The celebrated KdV equation is also conservative and admits infinitely many conserved quantities [2, 59]:

$$\displaystyle\begin{array}{rcl} \int _{-\infty }^{+\infty }u\mathit{dx} = \mathit{const}.,& &{}\end{array}$$
(7.1)
$$\displaystyle\begin{array}{rcl} \int _{-\infty }^{+\infty }u^{2}\mathit{dx} = \mathit{const}.,& &{}\end{array}$$
(7.2)
$$\displaystyle\begin{array}{rcl} \int _{-\infty }^{+\infty }(u^{3} + \frac{1} {2}u_{x}^{2})\mathit{dx} = \mathit{const}.,\ldots & &{}\end{array}$$
(7.3)

These equations express the conservation of mass , momentum, and energy, respectively. The accuracy of a numerical method can be checked by calculating these conserved quantities at every time step.

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  1. Centre for Nonlinear Studies (CENS), Tallinn University of Technology Institute of Cybernetics, Tallinn, Estonia

    Jüri Engelbrecht

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Engelbrecht, J. (2015). What physical mechanisms govern waves in non-conservative systems?. In: Questions About Elastic Waves. Springer, Cham. https://doi.org/10.1007/978-3-319-14791-8_7

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