Abstract
Many physical systems involving nonlinear wave propagation include the effects of dispersion, dissipation, and/or the inhomogeneous property of the medium. The governing equations are usually derived from conservation laws. In simple cases, these equations are hyperbolic. However, in general, the physical processes involved are so complex that the governing equations are very complicated, and hence, are not integrable by analytic methods. So, special attention is given to seeking mathematical methods which lead to a less complicated problem, yet retain all of the important physical features. In recent years, several asymptotic methods have been developed for the derivation of the evolution equations which describe how some dynamical variables evolve in time and space.
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The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena.
David Hilbert
It seems to be one of the fundamental features of nature that fundamental physics laws are described in terms of great beauty and power.
As time goes on, it becomes increasingly evident that the rules that the mathematician finds interesting are the same as those that nature has chosen.
Paul Dirac
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Debnath, L. (2012). Asymptotic Methods and Nonlinear Evolution Equations. In: Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8265-1_12
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DOI: https://doi.org/10.1007/978-0-8176-8265-1_12
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