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Solitons and the Inverse Scattering Transform

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Nonlinear Partial Differential Equations for Scientists and Engineers

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Abstract

Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.

True Laws of Nature cannot be linear.

Albert Einstein

… the great primary waves of translation cross each other without change of any kind in the same manner as the small oscillations produced on the surface of a pool by a falling stone.

Scott Russell

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  1. Department of Mathematics, University of Texas, Pan American, 1201 W. University Drive, Edinburg, TX, 78539, USA

    Lokenath Debnath

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Debnath, L. (2012). Solitons and the Inverse Scattering Transform. In: Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8265-1_9

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